Difference Equations, Special Functions and Orthogonal Polynomials: Proceedings of the International Conference: Munich, Germany 25-30, July 2005

Difference Equations, Special Functions and Orthogonal Polynomials

This page intentionally left blank

Proceedings of the International Conference

Difference Equations,

Special Functions and

rthogonal OrthogonalPolynomials Polynomials Munich, Germany

25 - 30 July 2005

Editors

S Elaydi Trinity University, USA

J Cushing University of Arizona, USA

R Lasser Technical University of Munich, Germany

V Papageorgiou University of Patras, Greece

A Ruffing Technical University of Munich, Germany

W Van Assche Katholieke Universiteit Leuven, Belgium

N E W JERSEY

- LONDON

K8World Scientific *

SINGAPORE

BElJlNG

- SHANGHAI - HONG KONG

*

TAIPEI

CHENNAI

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA ofice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK oficet 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library

DIFFERENCE EQUATIONS, SPECIAL FUNCTIONS AND ORTHOGONAL POLYNOMIALS Proceedings of the International Conference Copyright 0 2007 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereoJ; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-270-643-0 ISBN-10 981-270-643-7

Printed in Singaporeby World Scientific Printers (S) Pte Ltd

Professor B. Aulbach

This page intentionally left blank

vii

PREFACE

The theory and application of difference equations, special functions and applications encompass a diverse variety of mathematical fields. This includes, but is not limited to, dynamical systems, special functions, orthogonal polynomials, numerical analysis, combinatorics, integrable systems, and applications to biology, physics and economics. This proceedings volume contains papers on these (and other) topics that were presented at the International Conference on Difference Equations and Applications held in Munich on July 25-30, 2005. Among others this conference brought together researchers from three societies and organizations: the International Society of Difference Equations (ISDE), Orthogonal Polynomials, Special Functions and Applications (OPSFA), and Symmetries and Integrability of Difference Equations (SIDE). The goal was to exchange ideas and explore new directions in these fields. This kind of collaborative effort is of paramount importance in promoting communication among the various groups using difference equations and special functions in their research. We hope that this volume will well serve junior and senior researchers in all fields related to difference equations, special functions and integrability.

The Editors

This page intentionally left blank

ix

Contents Preface

vii

Pascal Matrix, Classical Polynomials and Difference Equations L. Aceto and D. Tngiante On the Ruelle Zeta Function of an Expanding Interval Map J . F. Alves and J . L. Fachada Difference Schemes for the Singularly Perturbed Sobolev Equations G. M. Amiraliyev and I. G. Amiraliyeva On Global Periodicity of x,+2 F. Balibrea and A . Linero

= f(x,+1, 2,)

1

17

23 41

Logarithmic Order and Type of Indeterminate Moment Problems C. Berg and H. L. Pedersen (with an Appendix by W. Hayman)

51

A System of Biorthogonal Trigonometric Polynomials E. Berriochoa, A . Cachafeiro and J. Garcia-Amor

80

Quasi Monomiality and Linearization Coefficients for Sheffer Polynomial Sets H. Chaggara

90

d-orthogonal Polynomial Sets of Chebyshev Type Y. B e n Cheikh and N . B e n Romdhane

100

On Two Problems in Lacunary Polynomial Interpolation M. G. de Bruin

112

Semiclassical Linear F'unctionals of Class 2: The Symmetric Case A . M. Delgado and F. Marcelldn

122

A Myriad of Sierpinski Curve Julia Sets R. L. Devaney

131

Compulsory Asymptotic Behavior of Solutions of System of Three Discrete Equations J . Dabla% and I. R4ZiCkovd

149

Nonoscillatory Solutions for Emden-Fowler Type Difference Equations M. Cecchi, 2. Do&& M. Marini and I. VrkoC

159

X

The Comparative Index for Conjoined Bases of Symplectic Difference Systems J. V. Elyseeva

168

A Renaissance for a q-umbra1 Calculus T. Ernst

178

Fourth-order Bessel-type Special Functions: A Survey W. N . Everitt

189

Laplacians and the Cheeger Constants for Discrete Dynamical Systems S. Fernandes, C. Gra'cio and J. Sousa Ramos Algorithms Around Linear Differential Equations A . Fredet Stability of Differential-Difference Systems with Quadratic Right-Hand Sides D.Khysainov and I. Grytsay Characteristic Algebras of Discrete Equations I. Habibullin On the Asymptotic Behaviour of Solutions of Neuronic Difference Equations Y. Hamaya

213 224

241 249

258

Perturbation of Nonnegative Time Scale Quadratic Functionals R. Hilscher and V. R4.ZiEkova'

266

Monotonicity of Dynamical Systems and Their Discretizations 2. Horva'th

276

Construction of an Integral Manifold for a Linear Delay System K. R. Janglajew

286

Chaotic Behavior in a Two-dimensional Business Cycle Model C. Janua'rio, C. Gra'cio and J. Sousa Ramos

294

Multiscale Expansion of the Lattice sine-Gordon Equation X.-D. Ji, D.Levi and M. Petrera

305

Stability of Delay Difference and Differential Equations: Similarities and Distinctions M. M. Kipnis and I. S. Levitskaya

315

xi

Computer Algebra Methods for Orthogonal Polynomials W. Koepf Convergence to a Period-two Solution for a Class of Second Order Rational Difference Equations M. R. S. KulenoviC and 0. Merino Two Normal Ordering Problems and Certain Sheffer Polynomials W. Lang Existence of Solutions for Even Ordered Boundary Value Problems on a Time Scale J. Henderson and B. A . Lawrence Asymptotics and Zeros of Symmetrically Coherent Pairs of Hermite Type M. G. De Bruin, W. G. M. Groenevelt, F. Marcelldn, H. G. Meijer and J. J. Moreno-Balcdzar

325

344 354

369

378

Asymptotic Boundary Value Problems for Discrete Systems M. Marini, S. Matucci and P. Rehdk

394

The Limits of the Solutions of a Linear Delay Difference System H. Matsunaga

404

Riemann-Hilbert Problem for a Generalized Nikishin System A . F. Moreno

412

Inequalities and Turhians for Some Special Functions A . Laforgia and P. Natalini

422

Asymptotics in the Complex Plane of the Third Painlev6 Transcendent V. Yu. Novokshenov

432

Solutions of Dynamic Equations with Varying Time Scales B. A . Lawrence and R. W. Oberste-Vorth

452

Totally Discrete and Eulerian Time Scales K . J. Hall and R. W. Oberste-Vorth

462

A Generalization of the Discrete UC Heirarchy and Its Reductions Y. Ohta

471

xii

The Dichotomy Character of z,+1

=

PnXn+Ynxn-l

with Period-two Coefficients C. H. Gibbons, S. Kalabuiic' and C. B. Overdeep

480

Integral Comparison Theorems for Second Order Linear Dynamic Equations L. Erbe, A . Peterson and P. Reha'k

497

Nonoscillations in Odd Order Difference Systems of Mixed Type S. Pinelas

507

Superintegrability and Quasi-exact Solvability. The Anisotropic Oscillator G. S. Pogosyan

520

On the Ergodic and Special Properties of Generalized Boole Transformations A . K. Prykarpatsky and J. Feldman

527

Fock Representations for a Quadratic Commutation Relation C. Correia Ramos, N . Martins and J . Sousa Ramos

537

Chaotic Discrete Learning Systems M. Merces Ramos and P. Sarreira

547

On the Asymptotic Behavior of the Moments of Solutions of Stochastic Difference Equations J. Appleby, G. Berkolaiko and A. Rodkina

557

Orthogonal Polynomials and the Bezout Identity A . Ronveaux, A . Zarzo, I. Area and E. Godoy

566

Information Entropy of Gegenbauer Polynomials J. I. de Vicente, S. Gandy and J. Scinchez-Ruiz

579

Higher Genus Affine Lie Algebras of Krichever-Novikov Type M. Schlichenmaier

589

Asymptotic Trichotomy of Solutions of a Class of Even Order Nonlinear Neutral Difference Equations with Quasidifferences E. Schmeidel Critical Groups for Iterated Maps C. Correia Ramos, N . Martins, R. Severino and J. Sousa Ramos

600 610

xiii

Ideal Turbulence and Problems of Its Visualization A . N . Sharkovsky

617

Fine Structure of the Zeros of Orthogonal Polynomials: A Review B. Simon

636

On the Symmetries of Integrable Partial Difference Equations A . Tongas

654

Heun Functions versus Elliptic Functions G. Valent

664

Discrete PainlevQEquations for Recurrence Coefficients of Orthogonal Polynomials W. Van Assche

687

Orthogonal Polynomials on R+ and Birth-Death Processes with Killing P. Coolen-Schrijner and E. A . van D o o m

726

Computing Topological Invariants in Boundary Value Problems Reducible to Difference Equations R. Severino, A . Sharkovsky, J. Sousa Ramos and S. Vinagre

741

Abel’s Method on Summation by Parts and Bilateral Well-poised yJ~s-seriesIdentities W. C. Chu

752

+

(2 1)-dimensional Lattice Hierarchies Derived from Discrete Operator Zero Curvature Equation 2 . - N . Zhu

762

This page intentionally left blank

1

PASCAL MATRIX, CLASSICAL POLYNOMIALS AND DIFFERENCE EQUATIONS*

L. ACETO

Dipartimento d i Matematica Applicata “U. Dana”, Universitd d i Pisa, Via Diotisalvi Z1 I-56126 Pisa, Italy E-mail: 1.acetoOdma.unipi.it

D. TRIGIANTE Dipartimento d i Energetica “S. Stecco”, Universitb d i Firenze, Via C. Lombroso 6/17, 1-50134 Firenze, Italy E-mail: [email protected]

Although the Pascal matrix is one of the oldest in the history of Mathematics, owing to both its utility in many applications and its countless properties, it continues to create interest. In this paper we review some recent works on the Pascal matrix by focusing our attention on its relations with linear algebra, difference equations and classical polynomials, such as the Legendre, Bernestein and Laguerre polynomials.

1. Introduction The Pascal matrix is one of the most interesting number-patterns in the history of Mathematics. It was created long before 1654, the year in which Blaise Pascal wrote his Daite‘ du triangle arithme‘tique. In fact, as described by Boyer lo and Edwards 19, it can be attributed to the Chinese in about 1300. In Europe it was introduced by Bienewitz (P. Appianus) and Stifel in the sixteenth century. In the same period, it appears in Italy in the General Trattato di numeri e t misure written by Nicolb Tartaglia and for this reason it is sometimes known as “Tartaglia’s triangle”. It was Pascal, “This work is supported by the Italian M.I.U.R.

2

however, who studied carefully the properties of its entries and established their application to the theory of combinations and probability. In recent years this matrix, which has a lot of remarkable and deep information, has attracting the renewed attention of many experts not only in the field of pure mathematics but also in many different areas of applied mathematics such as statistics, numerical analysis, computer-aided design, and combinatorics (see, e.g., Refs. 1, 3-7, 26 and 27 to quote only few works). In many other papers the interest of the authors is devoted to algebraic aspects of this matrix and to some of its generalizations (see, e.g., Refs. 8, 9, 11, 13, 18, 20 and 28-30). Moreover, in some works the existing relations between the Pascal matrix and other known matrices such as Vandermonde, Stirling and Bernstein, are investigated. The Pascal matrix P is a lower triangular matrix whose elements are: 2315,23

It is worth to note that each entry represents the number of combinations of i objects taken j at a time. It is well known that (see, e.g., Refs. 2, 13)

(I.: :. 1

i.e., the Pascal matrix can be obtained as the exponential of H , sometimes called creation matrix, and defined by 0

H=

.

.. .

n-10

nxn

Nevertheless, these are not the only ways to define it. In fact, it can also be given as the solution, computed at t = 0, of

(1) the partial difference equation ( P A E ) : P i + l , j ( t ) = t P i , j ( t ) +Pi,j-l(t), P0,o =

1

P0,j

= 0,

j

> 0;

i,j

20

(2)

3

(2) the differential equation in RnX": d

--oo < t < 00, -P (t ) = H P ( t ) , (3) dt subject to the initial condition P(0) = I, with I denoting the identity matrix.

2. Relevant properties

The matrix P and, more generally, the time-dependet Pascal matrix P t , solutions both of the P A E (2) and of (3), have countless interesting properties. First of all, P is diagonally similar to its powers since it holds true that

P = D (t )-l P t D (t ),

t # 0,

with

D (t ) = diag(l,t, t 2 , .. . ,t"-'). Consequently,

P - l = D(-l) PD(-1). From PSPt = PS+tone obtains

c i

si-ktk-j

(:> (;)

= (s

+t y - j

(:> ,

a 2 j.

k=j

As an example, when s = t # 0 we get the combinatorial identity:

Consider the vector

( ( t )= (1,t, 2,... , t " - l ) T . It satisfies the differential equation in R" : d --03 -?J(t) = H y (t ), dt Then,

< t < 03.

(4)

<(t)= Pt((0) = P ((t - 1). Since ((0)

= eo,

the first unit basis vector in R", and ((1)

= e, we have

4

The last relation is the vector form of the well known

c (;) i

= 2i.

k=O

If the dimension of the matrix is infinite, i.e., 1 0 0 0 0 ~ 1 1 0 0 0 *

1 4 6 4 1 .

. . . . . .

provided that the convergence holds true, one has (see Ref. 18)

and

Consider now the shaft matrix

K=

(5)

. . 0

nxn

The shifted Pascal matrix is defined by

ps := e H + s K ,

{ (jir)

where s is a positive integer; its entries are

(Ps)ij =

for i2.i i , j =0,1,

otherwise

As proved in Ref. 3, the following relations hold true:

Ps = PS--1(Ii-K ) Ps-l

( I - K)Ps

PsK - KPS = KPSK.

..., n - 1.

5

The Pascal matrix can also be connected to other matrices associated with great names such as Fermat, Vandermonde and Stirling. 2.1. Connections with some known matrices

A strong connection exists between P and the Fermat matrix

l6

F , whose

entries are:

Note that Fij represents the number of combinations of (i+l)objects taken j at a time with repeats permitted. As proved in Ref. 11, the Cholesky factorization of F is given by

F = PPT. For this reason, sometimes F is called symmetric Pascal matrix. Obviously, F is definite positive and det(F) = 1 independently of the dimension. Moreover, by taking the diagonal elements of PPT from ( 1 ) and (7) it follows that

2 (r>2 (:) =

:= (i

+ l)cz,

k=O

where co = 1, cl,c 2 , . . . are the Catalan numbers 24. The Vandermonde matrix, defined by

W ( t )= ( s ( t ) P t ( t ) P2E(t) . . p n - W ) ) 7 satisfies W’(t)= H W ( t ) ,i.e., (3). Consequently,

W ( t )= Pt W ( 0 )

+

and then W ( t 1) = P W ( t ) .It follows that

+

P = W ( t 1)w(t)-l, v t Considering the identity

j=O

E R.

6

where Sj are the Stirling numbers of the second kind and t ( j ) the “factorial power” (i.e., t ( j ) = t(t - 1). . . (t - j l),do)= l),we have

+

s,o

0 ...

0

0

[ ( t )=

. 0

.

sy-1... ... s;:;

The matrix S is called Stirling matrix. It diagonalizes the matrix entries are the Lah numbers: 24

L,whose

c o o = 1, LOj = 0, LiO = 0,

i-1 i , j = 1 , 2,..., 72- 1.

0

otherwise

In fact,

S L s-l = D(-1). From (8) we get [ ( t )= D ( - l ) S v ( - t ) , or, equivalently,

v(t) = Lv(-% which is the vector form of the expression 24

ccij i

t(i)=

(-t) (j),

j=O

Moreover,

( S D f ) - l P ( s D f )= I

+K,

where D f = diag(O!, l!,2!, . . . , ( n - l)!)and K is defined according to (5). Hence, the normalized Stirling matrix ( S D f) gives an explicit similarity transformation that takes P into its Jordan canonical form. Four of the previous matrices are strictly related through the famous “LDU” decomposition:

W ( 0 )= S D f P T and, in general,

W ( t )= ( P t S ) D f P T .

7

Related to this factorization is the problem to express in different bases a polynomial p ( t ) interpolating the given data p ( t j ) = fj, on a uniform mesh t j = j , j = O , l , ..., n - l . L e t

One has

and then

P ( t ) = [<(t)lT([W4l-'f) = (W(o)-l<(t))Tf = (s-l<(t))'((PDf)-lf) = [Q(t)]'((PDp)-lf).

It is interesting to observe that the entries of vector W(O)-lJ(t) are the Lagrange polynomials and those of V ( t ) are the Newton ones, respectively. By the way, ( P D f ) - l applied to f generates the divided differences of the entries of the vector f . Hence, the change of basis follows by coupling appropriately the involved matrices and vectors. 3. Classical polynomials

Here we review the relations between P and families of classical polynomials 2,3, such as the Bernstein and Legendre polynomials which play an important role in surface reconstruction.

3.1. Bernstein and Legendre polynomials In the field of computer-aided design the choice of basis to represent curves and surfaces is often influenced by the efficiency of performing particular operations. Therefore, although the Bernstein form of a polynomial is the most used representation since it offers a valuable insight into its geometrical properties, in some cases, when, for example, an approximation procedure is involved, the use of orthogonal bases, such as the Legendre polynomials, is more appropriate. Recently the question of the convenience of using a particular basis, among the available ones, has been discussed (see, e.g., Ref. 21). In particular, the transformation between the Legendre and Bernstein bases has been analyzed.

8

Consider the matrix

B e ( t ):= PD(t)P-1.

(9)

Each entry is a Bernstein polynomial, namely,

The important property that Bernstein polynomials form a partition of unity, i.e.,

i=O

is easily derived from (9), i.e.,

Be(t)e= PD(t)P-'e = e. Moreover, the subdivision property

also follows from (9) since it can be checked that

Let us introduce now the matrices D(n) and R defined by:

The matrix

M =R-~HR is interesting both for its pattern and because it lays a role similar to the it matrix H . In fact, by posing B"(t) := ( e T B e ( t ) ) = (B,",By,. . . , satisfies the following differential equation:

4)

d dt

-B"(t) = M B " ( t ) .

9

As an example, for n = 6, the matrix M is 0 0 0 0

‘-5-1

0 0 0

5-3-2 0

4-1-3

0

0

0 0

3 1-4

0 0 0 2 \

0

3-5

0 0 0 0 1 5

Consequently, the matrix Q plays a role similar to that of the matrix P since

Bn(t)= eMtBn(0)E QtBn(0) (see (4)). Actually, Q is also algebraically similar to P. Namely, Q = R-lPR. Moreover, it has interesting properties. For example, denoting by J the anti-identity matrix, one has: (i) (ii) (iii) (iv) (v)

QTJD(-l) = P2; =J &J;

Q = (JPT)-l PT (JPT); B(n)(t+ 1) = Q B(”)(t); QTe = e.

Finally, denoting by p,””’(t) the Jucobi polynomials, as proved in Ref. 3 the matrix P-lB:(t) has 0

0

on the main diagonal the Legendre polynomials, i.e., P,“”’(t), i = O,l,. . . ,n - 1; on the sth lower diagonal, s = l , . . . , n - 1, the polynomials (-l)”Pi(4) ( t ) ,i = O,l,. . . ,n - 1 - s; on the sth upper diagonal the polynomials (1 - t)”P,!””’(t).

3.2. Bernoulli and Euler polynomials

The Bernoulli polynomials Bi ( t )satisfy the relations d

--Bi(t) = iBi-I(t), dt

6’

1 if

i=O

0 if

i>O

Bi(t)dt =

10

Introducing the vector b ( t ) = (Bo(t),B l ( t ) ,. . . , tions become:

d -b(t) = Hb(t), dt

the above rela-

b(t)dt = eo.

Consequently, b ( t ) = Pt b(O), where the entries of b(0) are the Bernoulli numbers. This relation, together with the integral condition given in (lo), leads to

1'

Ptdt b(0) := Lb(0) = eo.

The matrix

is itself a function of H and then commutes with all functions of H . In particular, PL-l = L-lP. Consequently,

b ( t ) = p t b ( ~= ) p t ~ - l e o= L-' Pteo = L-'C(t). By observing that

we also obtain

Ab(t)

b(t

+ 1) - b ( t ) = ( P - I)b(t)= ( P - I)L-lC(t) = HC(t).

Then, k

b(t

+ k + 1) - b ( t ) = H C C ( t + j ) . j=O

When t = 0, this is the vector form of the well known property: k

Bz(k

+ 1) - Bi(0) = 2 zjz-1. j=O

Consider now the Euler polynomials &(t) defined by

d ---Ei(t) dt

= 2&-1(t),

Ei(t

+ 1)+ Ei(t) = 2 t Z .

(11)

11

The vector e ( t ) = (Eo(t),E l ( t ) ,. . . ,E , - I ( ~ ) satisfies )~ (4) and

e(t

+ 1)+ e ( t ) = 2 E(t).

Consequently,

e ( t ) = Pt e(0). Moreover, ( P

+ I ) e ( t )= 2 c(t) and then

+~ ) - l ~ ( t ) , which express the fact that the matrix 2(P + I)-' transforms the basis of e(t)= 2 ( P

powers into the basis of Euler polynomials. The transformation between the Bernoulli and Euler polynomials can also be obtained. From (11) we get

e ( t )= 2 ( P Since L-' ( P + I ) = 2L-'

+H

+ 1 1 - l ~b(t).

(see Eq. (10) in Ref. 14), we have 1

b ( t ) = L-'e(t)

+ ZHe(t).

Furthermore, considering that (see, e.g., Eq. (11) in Ref. 25)

one has

~ ( 2 ) b ( t=) ~ - ' e ( 2t ) . 3.3. Hermite polynomials Such polynomials H i ( t ) satisfy the relations: d

-Hz(t) = 2iH&l(t). dt By introducing the vector h(t) = (Ho(t),H l ( t ) ,. . . ,H , - I ( ~ ) ) they ~ become: d

-h(t) = 2 H h ( t ) , dt whose solution is

h(t) = P%(O). Such relation, in particular, implies that h(t+l)=Ph(t+i).

12

3.4. Laguerre polynomials The generalized Laguerre polynomials Lj") ( t )satisfy the following relations (see, e.g., Ref. 22): a

k=O

By introducing the vector L ( a ) ( t )= ( L p ) ( t )L, p ) ( t ) ., . . ,~5:?,(t))~from the previous equation we obtain:

L(")(t)= P,D?lD(-l)<(t), with Pa the shifted Pascal matrix defined in (6). By using such relation it is easy to check that

L(,)(o)= P,D~'<(o)= P,eo. Moreover,

L(a-l)(t)= ( I - K)P,DJlJ(-t)

=

( I - K)L('Z)(t),

which express, in vector form, the known recurrence relations

Lj"-l)(t) = L,(,)(t)- Lj"(t). In particular, in correspondence of the value a = 0, the Laguerre polynomials are obtained, i.e.,

L y t ) = PD?lD(-l)<(t). 4. Matrices, difference equations and polynomials Consider the matrices

V=H

o

H,

where o denotes the Hadamard product, and n-l i=O

'Hi

Q=C(i!,Z. The entries of Q satisfy a partial difference equation much more difficult than (2). This time it involves three consecutive rows. Namely,

2k + 1

j-1

13

Let

where D is any diagonal matrix. The three term recurrence relation ( 1 2 ) among the entries of Q induces a similar recurrence relation among the polynomials, i.e., t+l 2

k2(t

f f k + l ( t ) = -ffk(t)

-

- 1)2

4(4P - 1)

f f k- 1

(t).

With initial conditions

f f o ( t )= 1 , the explicit form of

t+l f f l ( t )= 2 ’

f f k ( t ) iS

Dahlquist l7 already stressed the relation between f f k( t ) and the Legendre polynomials

In fact, for all k 1 0 ,we have

Hence, f f k ( t ) are proportional to the Legendre polynomials after the transformation of the left complex plane into the unit circle, which is more appropriate for discrete purpose. The induced P A E on the coefficients is Pk+l,j

P0,o

=

= 1,

PkJ-1

+Pk,j 2

- Pk-l,j-2 - 2Pk-1,j-1 4(4 - k - 2 )

1

Pl,O = P1,l = -, 2

whose explicit solution is (see (13)): 2

Pk,j = 02( k ) . (2k)! j

+Pk-l,j

(14)

14

It is interesting to note that a slight change in the initial conditions induces a significant change in the solution. Suppose that a k , j verifies (14) subject to the initial conditions

The solution is

Then, changing the initial conditions, a different class of polynomials is obtained:

They satisfy the following recurrence relation:

P o ( 4 = 0,

Pl(t) = t - 1.

It is worth to observe that the polynomials &(t) and ak(t) characterize the linear multistep methods for solving ordinary differential equations known in literature as TOMS l2 (acronym of Top Order Methods). By using properties of the polynomials derived from their recurrence relations in Ref. 4 a complete proof for the stability problem of such methods has been provided. Once more the use of the Pascal matrix simplifies the relation between the vectors a ( k )and /3(k)l whose entries are a k , j and &j,j = 0 , 1 , . . . k, respectively. In fact, = p-T

where now P has dimension k

c, =

{

c pT p(k),

+ 1 and C is the matrix whose entries are

( - ~ ) ~ - j - l / ( Z- j

) if

i >j i,j= O , l , . . . , k .

0

otherwise

The matrix P-TCPT also occurs in the definition of other classes of numerical methods for the solution of differential equations. ‘ , 1 2

15

5. Conclusions

We have reviewed some recent works on the Pascal matrix. It has been shown t h a t it is strictly related to other important matrices in applications as well as to classical polynomials. It arises also as a solution of differential and partial difference equations. In all cases, its use considerably simplifies both the relations among them and, consequently, the algebraic manipulations involved.

References 1. L. Aceto and D. Trigiante, On the A-stable methods in the GBDF class, Nonlinear Anal. Real World Appl., 3 (2002), 9-23. 2. L. Aceto and D. Trigiante, The matrices of Pascal and other greats, A m e r . Math. Monthly, 108 (2001), 232-245. 3. L. Aceto and D. Trigiante, The Pascal matrix and classical polynomials, R e n d . Circ. M a t . Palermo, (2) Suppl. 68 (2002), 219-228. 4. L. Aceto, R. Pandolfi and D. Trigiante, One parameter family of linear difference equations and the stability problem for the numerical solution of ODES, (in press on Advances in Difference Equations). 5. R. Aggarwala and M. P. Lamoureux, Inverting the Pascal matrix plus one, A m e r . Math. Monthly, 109 (2002), 371-377. 6. T. Arponen, A matrix approach to polynomials, Linear Algebra Appl., 359 (2003), 181-196. 7. T.Arponen, A matrix approach to polynomials 2, Linear Algebra Appl., 394 (2005), 257-276. 8. M. Bayat and H. Teimoori, Pascal k-eliminated functional matrix and its property, Linear Algebra Appl., 308 (2000), 65-75. 9. M. Bayat and H. Teimoori, The linear algebra of the generalized Pascal functional matrix, Linear Algebra Appl., 295 (1999), 81-89. 10. C. B. Boyer, A history of mathematics, John Wiley & Sons, Inc., New York, 1968. 11. R. Brawer and M. Pirovino, The linear algebra of the Pascal matrix, Linear Algebra Appl., 174 (1992), 13-23. 12. L. Brugnano and D. Trigiante, Solving differential problems by multistep initial and boundary value methods, Gordon & Breach Science Publishers, Amsterdam, 1998. 13. G. S. Call and D. J. Velleman, Pascal’s matrices, A m e r . M a t h . Monthly, 100 (1993), 372-376. 14. Gi-Sang Cheon, A note on the Bernoulli and Euler polynomials, Appl. Math. Lett., 16 (2003), 365-368. 15. Gi-Sang Cheon and Jin-So0 Kim, Stirling matrix via Pascal matrix, Linear Algebra Appl., 329 (2001), 49-59. 16. L. Comtet, Advanced combinatories, Reidel, Dordrecht, 1974. 17. G. Dahlquist, Convergence and stability in the numerical integration of

16

ordinary differential equations. Math. Scand., 4 (1956), 33-53. 18. A. Edelman and G. Strang, Pascal matrices, Amer. Math. Monthly, 111 (2004), 189-197. 19. A. W. F. Edwards, Pascal’s Arithmetical %angle, Oxford University Press, New York, 1987. 20. M. El-Mikkawy and Gi-Sang Cheon, A connection between a generalized Pascal matrix and the hypergeometric function, Appl. Math. Lett., 16 (2003), 1239-1243. 21. R. T. Farouki, Legendre-Bernstein basis transformations, J. Comput. Appl. Math., 119 (2000), 145-160. 22. I. S. Gradshteyn and I. M. Ryzhik, Tables of integrals, series and products, Academic Press, Inc., New York, 1980. 23. P. Maltais and T. A. Gulliver, Pascal matrices and Stirling numbers, Appl. Math. Lett., 11 (1998), 7-11. 24. J . Riordan, Combinatorial Identities, John Wiley & Sons, Inc., New York, 1968. 25. H. M. Srivastava and A. Pint&, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17 (2004), 375-380. 26. Z. Trzaska, On properties and structures of dynamic Pascal systems. Part I:

time domain analysis, Part 11: analysis in complex-frequency domain, Bull. of the Polish Academy of Sciences Technical Sciences, 42 (1994), 221-239. 27. Z. Trzaska, Time-varying Pascal systems, J. F4-ankh Inst. B, 332 (1995), 419-431. 28. Z. Z. Zhang, The linear algebra of the generalized Pascal matrix, Linear Algebra Appl., 250 (1997), 51-60. 29. Z. Z. Zhang and M. X. Liu, An extension of the generalized Pascal matrix and its algebraic properties, Linear Algebra Appl., 271 (1998), 169-177. 30. Z. Z. Zhang and T. Wang, Generalized Pascal matrix and recurrence sequences, Linear Algebra Appl., 283 (1998), 289-299.

17

ON THE RUELLE ZETA FUNCTION OF AN EXPANDING INTERVAL MAP

JOAO FERREIRA ALVES AND J. L. FACHADA DEPARTMENT OF MATHEMATICS, INSTITUTO SUPERIOR TECNICO, AV. ROVISCO PAIS 1, 1049-001 LISBOA, PORTUGAL. In the context of expanding piecewise monotone interval maps, we present an alternative proof of a main result of Baladi and Ruelle concerning weigthed zeta functions and wigthed kneading determinants.

1. Introduction Let I = [a,b] c R be a compact interval and C a finite set verifying { a , b } c C c I . A differentiable map

f :I\C-tI is called C-modal if If'(z)l > 1 for all z E I\C. one defines the nth iterate of f , by setting f1 = f

Thus, f" : I\Cn bY

+I

If n is a positive integer,

and fn+' = f o f", for n 2 1.

is a C,-modal map, where the finite set C, is given

C1 = C and Cn+l = Cn U f - " ( C ) . Notice that if f is a C- modal map on I , then the iterate f" has at most one fixed point in each connected component of I\Cn. Thus every f" has finitely many fixed points, and the formal power series

< = exp C n/l

~

n

with

N ( f")

= # {Z E

I\Cn : f" (x) = Z} ,

is called the Artin-Mazur2 zeta function of f. The previous definition is a particular case of a more general definition. Let f be a C- modal map on I , and g : I\C --f C such that g is constant on

18

each connected component of I\C. One defines the Ruelle zeta function4 of f , with weight 9, as the following invertible element of C [ [ z ] ](ring of formal power series, with complex coefficients, in the indeterminate z )

with

N,(f")

=

c

I>.(

9(z)9(f

...g(f"-'(4).

X E Fix( f ")

The notion of weighted kneading determinant was introduced by Baladi and Ruelle3 in a more general context, and plays a relevant role in the study of I,. Starting from a main relationship between this determinant and they were able to prove the following remarkable result:

c,

Theorem 1.1. Let f : I\C 4 I be a C-modal map, g : I\C 4 C a weight, and m = max{[g(z)I : z E I\C}. Then is meromorphic in ( 2 E C : It1 < m-I}.

cg

Our goal in this paper is to present a different proof of this result. In order to simplify the notations, we will assume for the remainder of the paper that the map f : I\C -+ I verifies lim f"(z) $! C and lim f"(z) $! C ,

x+c-

x+c+

for all c E C and n 2 1. In what follows, U denotes a (finite or infinite dimensional) vector space over C. The space of linear forms on U will be denoted by U * , and the space of all linear endomorphisms on U will be denoted by L ( U ) . If $ E L(U) and n is a nonnegative integer, the n t h iterate $" is defined recursively by $' = I ~ UE L(u), $" = $ o $"-' E L(u), for n 2 1.

Definition 1.1. A pair (cp, $) E L ( U )x L ( U ) has finite rank if $-cp has finite rank.

E

L(U)

Notice that if a pair (cp,$) has finite rank then the pair (cp",$") also has finite rank, for all n 2 0, and therefore the trace of cp" - $" is defined. Thus, we may define:

Definition 1.2. For any pair (cp,$) E L(U) x L(U) with finite rank, we define the determinant of (cp, $) as the following invertible element of C [ [ Z ] ]

19

Theorem 1.2. Let (cp,$) U*such that

E

L ( U ) x L ( U ) , u1,...,uq E U , and

a1

,...,aq E

i=l

Then

A(p,,,) = det(1- zM) holds in C [ [ z ] ] ,where I denotes the q x q identity matrix and

The previous result will be useful in the proof of our main result, from which Theorem 1.1is an immediate consequence.

Theorem 1.3. Iff : I\C -+ I is a C-modal map, g : I\C -+ C is a weight, and m = max{Ig(z)l : x E I\C}, then there exists a C-vector space, U , and (cp, $) E L ( U ) x L ( U ) with finite rank such that =A ),,: holds in C [ [ z ] ] , and A(q,+) converges for all IzI < m-'.

cg

2. Proof of Theorem 1.3

To prove Theorem 1.3 we introduce the following notation. I f f : I \ C -+ I is a C-modal map, define the sign E : I\C -+ {-l,l} by setting ~ ( x=) 1 if f is increasing in some neighborhood of z, and E ( Z ) = -1 otherwise. In what follows, if a : D C I -+ C is a map, we use the usual notation

a(z+) = lim a ( y ) and a(.-) y-x+

= lim a ( y ) . y-x-

Let I be a compact interval and U = S o ( I ; C ) ,the C-vector space whose basis is the set of all formal symbols x E I . Given a C-modal map f : I\C -+ I and a weight g : I \ C -+ C, one defines a pair (cp,+) E L(U) x L(U).

Definition 2.1. Let f : I\C -+ I be a C-modal map, and g : I\C -+ C a weight. Define cp : U -+ U as the unique linear morphism that verifies: cp(z) = E(x)g(x).f (z) if z E I\C, cp(x) = 0 if z E C (where E(x)g(z).f (x) means the product of the scalar E(x)g(x) E C by the vector f (z) E I c U ) .

20

Let f : I\C 4 I be a C-modal map, and g : I\C -+ C a weight. In order t o define the second linear endomorphism 1c, : U -+ U we introduce, for each c E C , the vectors v;, v z E U and the linear forms ,t?, ,,t?: E U* by setting:

v,

= f(c-)

E I c u,v:

P,(.)

=

{

= f(c+) E

E(C-)g(C-) 0

2

I

c u,

2c

x < c

and

for all x E I. Definition 2.2. Let f : I\C weight. Define 1c, : U -+ U by

1c, = 'p+

+I

be a C-modal map, and g : I\C

CP, @vv, + cs:

--+

Ca

@v:,

CEC

CEC

So, for any C-modal map f : I\C -+ I and any weight g : I\C -+ C, we obtain a pair of linear endomorphisms ('p, $) on U = & ( I ; C). From $ - 'p =

cPF

@ v,

+

CEC

cP:

and because C is finite, it follows immediately that and CEC

(1)

@ v,+,

CEC

CE

('p,1c,)

c

Alternatively, denoting the elements of C by a = c1 we may write

< c2 ... < cp-l < c,

with

and -e(ci+)g(ci+)

has finite rank

if f (ci+) > ci if f(s+) I ci

'

= b,

21

This equality is only a particular case of a more general one. Indeed, using a functorial argument', one shows that t r ( V - 'p") =

Wf"),

for all n 2 1. Therefore, from Definition 1.2 it follows that

So the proof of Theoreml.3 will follow, if we prove that holds in C[[z]]. A(,,,+) converges for all IzI < m-'. Put C = {q,..., .p} and define

+,

u2i = vi U2i-l

= vi ,azi = pa+ ,CYZi-1 = p i ,

for all i = 1,...,p . From (l),we have 2P

j=1

and from Theorem 1.2, it follows

A(v,+)= det(1- zM),

(3)

where the entry ( i , j ) of M is given "20

From the definitions of

'p,

ai, and u j , it follows immediately lQi('p" (Uj))l

I m"+',

for all n 2 0. Thus the entries of M converge for all IzI < m-l,which by (3) shows that A(,,+) converges for all IzI < m-', as desired.

22

Acknowledgments Partially supported by FCT/POCTI/FEDER

References 1. J. F. Alves and J. Sousa Ramos, Kneading theory: a functorial approach, Comm. Math. Phys. 204 (1999), 89-114. 2. M. Artin and B. Mazur, O n periodic points, Ann. of Math. (2) 81 (1965), 82-99.

3. V. Baladi and D. Ruelle, A n extension of the theorem of Milnor and Thurston on the zeta functions of interval maps, Ergodic Theory Dynam. Systems 14 (1994), 621-632.

4. D. Ruelle, Dynamical zeta functions and transfer operators. Notices Amer. Math. SOC.49,887-895 (2002).

23

DIFFERENCE SCHEMES FOR THE SINGULARLY PERTURBED SOBOLEV EQUATIONS

G. M. AMIRALIYEV AND I. G. AMIRALIYEVA Department of Mathematics Faculty of Science Yuzuncu Yzl University 65080 Van, TURKEY E-mail: gamirali20000yahoo.com

The present study is concerned with the numerical solution, using finite difference method of a one-dimensional initial-boundary value problem for a quasilinear Sobolev or pseudo-parabolic equation with initial jump. We have derived a method based on using finite elements with piecewise linear functions in space and with exponential functions in time and appropriate quadrature formulae with remainder term in integral form. In the initial layer, we introduce a special non-uniform mesh which is constructed by using estimates of derivatives of the exact solution and the analysis of the local truncation error. For the time integration we use the implicit rule. The fully discrete scheme is shown to be convergent of order 2 in space and of order one in time, uniformly in the singular perturbation parameter. Numerical results supporting the theory are presented.

Key words and phrases. Singular perturbation; Quasilinear problem; Difference scheme; Non-uniform mesh; Uniform convergence 2000 Subject Classification: 65M06, 65M15, 65M50

1. Introduction We consider in the rectangle Q = 0 U [O,T],R = (O,Z), perturbed initial-boundary value problem

u(0,t ) = u(Z,t ) = 0, t E (0, TI,

the singularly

(3)

24

where dU

L1[-]

at

a3u

= --

ax2at

+ -;au at

0 < E < 1 is a small positive parameter, a > 0 and /3 are given constants. p ( x ) and f ( x ,t ,u)are assumed to be sufficiently continuously differentiable functions in and Q x R, respectively. For small values of E , exponential initial layer appears along the line t = 0. Equations of this type arise in many areas of mechanics and physics. They are used to study heat conduction [7], homogeneous fluid flow in fissured rocks [4], shear in second order fluids [8, 19, 151 and other physical models. For a discussion of existence and uniqueness results of pseudoparabolic equations see, [9,171. Various numerical treatments of equations of this type in the regular cases have been considered in [13,10,11,2,14,18] (see also the references cited in them). The numerical investigation of singular perturbation cases has always been far from trivial because of the boundary layer behavior of the solution [l,31 (see, also monographs [16, 121). It is well known that difference schemes on a uniform mesh are not suitable to nonlinear singularly perturbed problems as a special fine mesh is required in boundary layer region and comparatively much coarser mesh elsewhere. Ideally, the mesh should be adapted to the features of the exact solution using an adaptive grid generation technique. This approach is now widely used for numerical solution of differential equations with steep, continuous solutions (see, for more recent works, e. g. monographs [16, 121). In the present paper we have derived a method based on using finite elements with piecewise linear and piecewise exponential basis functions and appropriate quadrature formulae with remainder term in integral form. In the initial layer, we introduce the non-uniform mesh, which is constructed by using the estimates of derivatives of the exact solution and the analysis of the local truncation error. This technique is similar to those in [5,6, 1, 21. The layout of the rest of this paper is as follows: in Section 2, we state some important properties of the exact solution. In Section 3, we describe the finite difference discretization and introduce the adaptive non-uniform grid. Section 4 deals with the error analysis for approximate solution. In section 5, we formulate the iterative algorithm for solving discrete problem and give the illustrative numerical resu1ts.A summary of the main conclusions puts an end to the paper.

25

2. Properties of the Solution of (1)-(3)

Here we give some useful asymptotic estimations of the exact solution of (1)-(3) that are needed in later sections. Henceforth, we denote by C a generic positive constant that is independent of E and of the mesh parameter. Such a subscripted constant is also independent of E and of the mesh parameters, but whose value is fixed. Let (v,W)O be the L2(R) inner product with norm llvll: = (z1,v)o. Also, H1(n) = {v : v,v’ E L2(R)} with norm 11~ 11; = 11~11: IIv’II: and llvJloo= max[o,l]Iv(x)I for v E C(n). In the future, we will assume for simplicity that 2 0.

+

2

Lemma 2.1. Let cp E C2[0,11 and the derivatives a s f asf af -( s = 1,2), axS auS at

are &-uniformly bounded on Q x R. Then, for the solution u ( x ,t ) of (1)-(3) the following estimates hold:

1-

5 CE-k,(2,t ) E Q, k = 0 , l ; s = 0 , 1 , 2

for any fixed 1 and T , and

provided that M0a-l-

27r

< 1,

where

Proof. Consider the integral identity

and taking into account that

dU (u-, u)o = 0

ax

(4)

26

after simple manipulation we have

where F ( z ,t ) = f(z,t ,0), and the bar indicates that the partial derivative is evaluated at in intermediate point called for by the mean value theorem. Using also the embedding inequality

we see that

and by inequality

consequently

27

From (8), by differential equality, it then follows that

where

Thus

This in turn, by virtue of embedding inequality

yields

Whence

which proves (4)for s = 0. Further part of the proof, one can proceed using similar technique described in [l,5, 61. 0 3. Discretization and Layer-adapted Mesh 3.1. Construction of the difference scheme

Let a set of mesh nodes that discretises Q be given by

w = WN x

WNo,

with WN

.., N - 1;h = l / N } , < ... < t N o = T , T j = tj - t j - l } ,

= {xi = ih, i = 1,2,

WNo =

(0 < t l

< t2

and

w& "No

= WN

u {XN

= WNo

= 1}, W N = WN

u {to = o},

w

u(20

= 0, X N = 1 )

= a N x WN,.

,

28

The nonuniform mesh C.JN~ will be specified at the end of the section. Define the following finite differences for any mesh function vi = v(zi) given on W N by

Given a function g notation

= gs = g(xi,tj) defined on a , we shall also use the

The approach of generating difference method is through the integral identity

29

where the basis functions

@ij (2,t )

have the form

@ijb,t)

= YJi(Z)+j(t),

with the usual piecewise linear functions for the space

< x < xi (zi+i - z ) / h ,xi < 2 < ~ i + i (Z- z i - i ) / h , xi-1

Cpi(X)

=

otherwise

and with the exponential functions

The parameter X j is given by

Using the appropriate interpolating quadrature rules with weight and remainder term in integral form, the fact that @:l) holds the problem

+(t) - c r + j ( t ) = 0, t j - 1 5 t

< tj, +j(tj)

=1

and the easily derived identity

in(9), as consistent with [l,2, 31, we obtain the relation for exact solution tU+f(Z,

t ,U ) + R

:= -EeUfrnx+&eUf--(Y'11Zx+Ps(U)+f(2,

1

S(u) = 3

(UU,

s

+

("2);)

The truncation error Ri has the form

R = (R(O)),+ R ( l ) ,

t ,U ) + R = 0, (10)

30

in which

Ki(x,E) = Ti(x - I ) - hM1(z- xi-i)(xi K ; ( x , I ) = Tl(. - El - Tlbi - 5), Tl(S)= s, s 2 0; Tl(S)= 0, s < 0.

- E),

Based on (lo), we propose the following difference scheme for approximating (1)-(3): ey

+ f(x,t ,Y ) := - - E o Y ~ +~ ~EOYf - a Y z x + P S ( Y )+ m,t , Y ) = 0,

Y ( 0 , t ) = Y ( l , t ) = 0, t E a&,

(14)

(16)

where O and S ( y ) are given by (11) and (12), respectively. It is easy t o see that the bilinear term in difference Eq. (14) has the analogous property as in differential case: ( S ( Y ) , Y ) O = 0.

31

3.2. The Mesh

According t o ( 5 ) , there is an initial layer in the neighborhood o f t = 0 of order ailel ln<hickness where a0 is given by (6). The difference scheme (14)-(16), in order to be &-uniformconvergent, we will use the layer adapted form of W N ~ .This is a special non-uniform mesh, which is condensed in the initial layer. The non-uniform mesh W N ~on the interval [0, T ]is formed by dividing the interval into two subintervals [0, u]and [u,TI, where

c = min

{g,

a i l &lnE1). [

(17)

The interval [O,a], which is inside the initial layer the solution contains large gradient. Here we use a fine mesh. In the interval [g,TI, that is away from the layer, we use a coarse, equidistant mesh. The corresponding mesh points are

tj E [O,u]: tj = - a ~ ' ~ l n [ l - ( l - & ) - -23. ],

NO

tj = -a;'&ln[l t j E [.,TI

j = 0, ..., N0/2, if u

< T/2, (18)

a T

- (1- e-*)X], j = 0, ...,N0/2, if u = T/2; NO

: t j = fJ+(j-N0/2)7,

j = No/2,

..., No,

T

= 2(T-u)/No.

(19) (20)

Here, without loss of generality, we assume that No is even. The special non-uniform mesh W N ~ given , by (18)-(20) denote by L J N ~ . 4. Uniform Error Estimates

Let z ( z , t ) = y ( z , t ) - u ( z , t ) , ( z , t ) E numerical solution satisfies

WN

x

W N ~ Then, .

& + f(z,t ,Y) - f(z,t ,u)= R, (z, t ) E

z ( z , o ) = 0, z E W N , z ( 0 ,t ) = z(Z, t ) = 0 , t E

the error in the

WN X L J N ~ ,

WN0,

where the truncation error R is given by (13).

4.1. Stability Bounds The discrete problem (14)-(16) enjoys the following stability property Lemma 4.1. The error function, z , satisfies llzlll

5 C m a llRll-1 t E '"NO 7

NO

(21)

32

provided that 1 -a-'lplzMo 2 where the constant MO is given by (7).

< 1,

Proof. From the identity

(kz)o + (fb, t ,Y) - f(z,t ,'111, z)o = (R,z)o after summation by parts, we get N

0

= (R,z)o.

(23)

By virtue of the easily derived identities

1 2 7 2 ZfZ= - ( z ) f + -zf, (7-

2

yyg

2

- '11210 = zzo

1 2

+

('11z)o -

- -[uz,z,)

2

Tj),

h2 2

-(uzzz)s,

+ -h2 (uzzz,zz] 6

33

h

I I141w,nl.l llzllp + 5 l141m,n 1.;

I

(S + t) 1141w,nIMl;

it follows from (23) that 1

1

p(ll.llp + ,Es(ll.l;)r

+!1.

-

IPI

(5 + ;)

Mo l.lp

2

I IlRll-1 l l ~ l l l *

(24) From this, under the assumption (22) we find for h

&e(Il&>r + KO 1:.

5 2 11RIl-1

5 112

ll.lll

7

with KO

9

= -(2a 12+9

- IPpkfO).

Consequently, (24) simplifies to EeSE

+ -62

KO

5 p, t E w N o ,

(25)

From (25), by difference analogue of differential inequality and discrete maximum principle, it follows the estimate

6 5 2~;' max p, WNO

which implies (21).

0

Lemma 4.2. The truncation error, R:, of the scheme (14)-(16) satisfies

llRll-l

I C (h2 + NG1) , t

E &N,,.

(26)

34

Proof. In view of the fact that I(R,z)ol

L

+ I(R(’),z)o

l(~?),z)ol

we obtain

(27) Furthermore, from the explicit expressions of, have

R ( l ) ,by using (4),we

Then, since u(2,t ) satisfies (5), the inequalities (28), (29) and (27) together imply

where a0 is the constant from (6). Further, Ri is estimated on [O,a]and [u, TI separately. We consider first the case u < T/2 and so n = -a,’& In&. In [a,T],which is outside the layer 5 C ( 5 - l e - F 5 1) by (5) and 7-j = T . Hereby we get from (30)

IIRjll-l 5 C ( h

+T), j

= N0/2

+ 1, ...,No.

Hence,

IIRq-l 5 C ( W 2 + 2TN,-1), j = N0/2

+ 1,...,No.

On the other hand, in the layer region [O,u],by (5), the inequality (30) becomes

35

Since

it then follows from (31) that

pql-l 5 c

(N-2

+ 4a;lN;l)

,j

= 1, ..., N0/2.

Now consider the case (T = T / 2 and so T / 2 < -a;'& ln&. Therefore, for t j E [O,(T] with (19),we deal with as in the case of the mesh region above. For t j E (0, TI, since r = 2(T - a)/No = T / N o ,

it follows from (30) that

IIRjll- 1


-

{ 12N-'

2 + (T + -eao )Nil}

,j

= N0/2

+ 1, ...,NO

which concludes the proof.

0

Lemma 4.1 and Lemma 4.2 yield the main result of the paper. Theorem 4.1. Let u ( x ,t ) be the solution of (1)-(3) and y i the solution of

(14)-(16). Then, under the hypotheses of Lemma 2.1 and Lemma 4.1 the following &-uniform estimate holds

36

5. Algorithm and Numerical Results

(a) We solve the problem (14)-(16) using the following quasilinearization technique

-&e&L + Ee$)

- ffyk) + P yg(n) (n-1)

{

yyo,

+

Y(n)Yo(n-1) - Y

(n-1)

t ) = y(")(Z, t ) = 0, t E Q N 0 ,

Y o(n-1)

(34)

t) where y(O)(z,t ) given (y(O)(O,t) = ~ ( ~ ) ( l=, 0). The linearized problem (32)-(34) is conveniently rewritten as

and we also supposed for simplicity that the function f is independent of u.For each j , the system (35)-(37) is tridiagonal system and is stable for

Ai > 0, Bi > 0, Ci > 0 Ci 2 Ai + Bi, 15 i 5 N - 1.

37

(b) In order to assess the performance of our method, we have applied it to the problem (1)-(3) with

f(z,t)= e-tsinnz, cp(z) = s i n m , a = 2, We take

a0

= 0.5, 1 = 1, T = 1.

(38)

= 1.7175 in (17). The initial guess is chosen as

As the exact solution is not known, we use the double mesh technique to estimate the errors and compute the experimental rate of convergence in our computed solution. That is, we compare the computed solutions with the solution on a mesh that is twice as fine in time. The error estimates obtained in this way are denoted by e?

=

y 1%

j,E,No -

I.

2j,~,2N0 'Ili

The convergence rate in No is then defined as = In (e2/eSNO) /ln2.

~ € 1 ~ 0

The resulting errors e? and the corresponding numbers pE>Noafter five iterations for particular values of E , N , NO are listed in the Tables 1-4. It can be observed that they are essentially in agreement with the theoretical analysis described above. Table 1. Errors e> N o and convergence rates p:' N o on W No = 16, N = 10.

N ~for

N = 10 No = 16

0.00217690 0.00217706 0.00217697

0.00121662 0.00123915 0.00123943 0.00123947

0.81084 0.81292 0.81271 0.81260

Table 2. Errors e z N o and convergence rates p:,No on W No = 32, N = 10. N = 10 No = 32

E

lo-' 1O-'

e:'NO

e:2No

p ~N o.

0.00121662 0.00123915 0.00123943 0.00123947

0.00065659 0.00067230 0.00067218 0.00067227

0.88982 0.88218 0.88276 0.88262

N ~ for

38 Table 3. Errors No = 16, N = 20.

and convergence rates ~

Table 4. Errors

and convergence rates

2 on ~WN" "for

on W

N ~for

1

1

Figure 1. Surface plot solution for NO = 32, N = 20,

E

=

In Fig. 1 we present a surface plot of the numerical approximation generated by the numerical method (14)-(16) applied to problem (1)-(3) with data (38) for No = 32, N = 20, E =

39

6. Conclusion An initial-boundary value problem for a quasilinear singularly perturbed Sobolev equation having a initial jump was considered. We constructed a numerical method for solving this problem, which generates &-uniformly convergent numerical approximations to the solution and its derivatives. The method comprises a special non-uniform mesh, which is fitted to the initial layer and constructed a priori in function of sizes of parameter E and the problem data. Nonlinear difference problem is solved by the present method after linearizing it by the Newton quasilinearization method. The main lines for the analysis of the uniform convergence carried out here can be used for the study of more complicated nonlinear problems.

References 1. G. M. Amiraliyev, On the numerical solution of the system of Boussinesque with boundary layers. USSR Modelling in Mechanics 3,3-14 (1988). 2. G. M. Amiraliyev, Difference method for the solution of one problem of the theory of dispersive waves. USSR Differential Equations 26, 2146-2154 (1990). 3. G. M. Amiraliyev, Y. D. Mamedov, Difference schemes on the uniform mesh for a singularly perturbed pseudo-parabolic equations, Tr.J. Math., 19, 207222, (1995).

4. G. Barenblatt, I. Zheltov and I. Kochina, Basic concept in the theory of seepage of homogeneous liquids in fissured rocks, J . Appl. Math. Mech., 24, 1286-1303, (1960). 5. I. P. Boglaev, Approximate solution of a nonlinear boundary value problem with a small parameter for the highest order derivative, USSR Comput. Math. Math. Phys., 25, 30-39, (1984). 6. I. P. Boglaev, Numerical method for a quasilinear parabolic equation with boundary layer, USSR Comput. Math. Math. Phys., 30, 716-726, (1990). 7. P. J. Chen and M. E. Gurtin, On a theory of heat condution involving two temperatures, 2. Angew. Math. Phys., 19,614-627, (1968). 8. B. D. Coleman and W. Noll, An approzimation theorem for functionals, with applications to continuum mechanics, Arch. National Mech. Anal., 6, 355370, (1960). 9. P. L. Davis, A quasilinear parabolic and related third order problem, J. Math. Anal. Appl., 49, 327-335, (1970). 10. R. E. Ewing, Numerical solution of Sobolev partial differential equataions, SIAM J. Numer. Anal., 12, 345-363, (1975). 11. R. E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equataions, SIAM J. Numer. Anal., 15, 1125-1150, (1978). 12. P. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan and G. I. Shishkin, Robust Computational Techniques for Boundary Layers, Charman &Hall/CRC, Boca Raton (2000).

40

13. W. H. Ford and T. W. Ting, Uniform error estimates for difference approximations to nonlinear pseudo-parabolic partial differential equations, SIAM J. Numer. Anal., 15, 155-169, (1974). 14. H. Gu, Characteristic finite element methods for non-linear Sobolev equations, Appl. Math. Comput., 102,51-62, (1999). 15. R. Huilgol, A second order fluids of the differential type, Internat. J . Nonlinear Mech., 3 , 471-482, (1968). 16. H. G . Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations, Springer Series in Computational Mathematics, Vol. 24, Springer, Berlin (1996). 17. R. E. Showalter, T. W. Ting, Pseudo-parabolic partial differential equations, SIAM J . Numer. Anal., 1, 1-26, (1970). 18. T. Sun and D. Yang, The finite difference streamline dihsion methods for Sobolev equations with convection-dominated term, Appl. Math. Comput., 125,325-345, (2002). 19. T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14,1-26, (1963).

41

ON GLOBAL PERIODICITY OF

zn+2

= f ( ~ , + i2,,)

F. BALIBREA AND A. LINER0 Uniuersidad d e Murcia Departamento de Matema'ticas Campus de Espinardo Apartado de Correos 4021 30100 Murcia (Spain) E-mail: balibreaaum. es, linerobaaum. es We study the existence of 3-cycles (global periodicity) for the difference equation xn+2 = f(xn+l,zn), where f is a continuous map from (0, w) x (0,w ) into (0, m) and the initial conditions are positive real numbers. We use strongly monotonicity properties of fiber maps obtained from f.

Key words and phrases. Difference equations; global periodicity; pcycle; fiber maps; monotonicity 2000 Subject Classification: 3 9 A ll

1. Introduction

Given the family of difference equations xn+k

= f (Xn+k-l, * ' '

> xn+l, Xn)

where f : Rk + R is a continuous map, and k is a positive integer (order of the equation), a strong property held by some of them is the gZobaZ periodicity which means there exists some (minimal) p E N verifying that for any initial condition ( X I ,... ,Xk) E Rk the sequence generated by the equation satisfies xn+p = xn for all n 2 1. Then we say that the equation is a p-cycle or simply f generates a p-cycle. For different classes of f's and different values of k E N this problem has been widely treated in the literature, see for example [6], [8], [ 5 ] , [3], 171, PI, [41. In particular for k = 2, xn+2

= f(xn+l,xn),

(1)

42

where it is assumed f : (0, m) x (0, m) + (0,m) is a continuous map, the problem of obtaining all f’s producing pcycles is not completely known. For p = 2 it is clear that the unique 2-cycle is given by 2,+2 = 2,. Our aim is to do some advances when p = 3. In particular we prove that if f separates the variables, that is, if f ( z , y ) = a ( z ) p ( y ) , then the unique 3-cycle is given by the family 5,+2 = -where c is a positive real constant. This result was announced in [2] without proof. Notice that this partially improves a result of [l, Th. 4.2 (E)] when k = 2 and another result of [7, Th. 1 (l).]. The paper is organized as follows. In Section 2 we give a general procedure to deal with a general map f(z,y), and obtain a sequence of preliminary results. In Section 3 we apply such procedure to solve the particular case f(z,y) = a ( z ) p ( y ) , and finally we state some pending questions. 2. Preliminary results for p

=3

Suppose that (1)is a 3-cycle. Let Then we have:

( 2 1 , ~ be) an

arbitrary initial condition.

23

= f(22,21),

24

= f(Z3,ZZ)= f(f(22,Z1),52) = 21,

25

= f ( 2 4 , 2 3 ) = f(21,f(22,m))= 22,

26

= f(25724) = f(z2,21) = 2 3 .

From these equalities we deduce that if Eq. (1) is a 3-cycle, then f must verify the functional equation

f(f(u,w), u ) = w for all u , w

> 0,

(2)

f(u,f(w, u ) )= w, for all u , w

> 0.

(3)

and

In particular, if u = w,from (2) and (3) we obtain

> 0,

(4)

f ( u , f(u,u)) = u, for all u > 0.

(5)

f(f(u,u),u) = u, for all u and

Throughout the paper, given a z fiber maps

> 0, we will denote by cpz and $=the

9.2, $z : (0,m) + (0,m)

43

defined by cp,(x) := f ( z , x ) , Ilfz(x):= f ( x , z ) , for all x

> 0.

(6)

From this definition and Eqs. (2)-(3) it follows that

$, o cp, = Idl(o,w)= cpr

$,

for all z

> 0.

(7) Then and cp, are inverse maps, and therefore they are symmetric with respect to the diagonal y = x. In the sequel the following set will play an important role o

7 = { u > 0 : f ( u , u )= u } . From now on, let f E C ( ( 0 ,oo) x (0, oo),(0, oo)) be a 3-cycle for (1). Now we are stating some properties of fiber maps associate to f . The first one is concerning the monotonicity of such fiber maps. Using (2) and (3) it is easy to prove that both cp, and t+bu are injective maps. Then: Lemma 2.1. I f f generates a 3-cycle, then:

(I) For any u,9, and $J,

are strictly monotone maps. (2) If w,w' are different positive real numbers,

cpw(x) # cpw'(x) f o r all x

> 0.

Let us denote by Fix(g) the set of fixed points of g E C((0,oo),(0,oo)). Lemma 2.2. For any u,

Fix(cp,) = Fix($,). Proof. If z E Fix(cp,), that is, p,(z) = z , by (7) we find z = $,((p,(z)) = $,(z), SO z E Fix(&). This proves the inclusion Fix(cp,) Fix(&). The other is similar.

Next result describes the asymptotic behaviour of fiber maps. Lemma 2.3. For each u > 0 , cp,((O, increasing with

lim cp,(z)

x+o+

or

(p,

= (0, oo),and either cp, is strictly

00))

= 0 and

lim cp,(x) = +oo,

x t w

is strictly decreasing with

lirn (p,(x) = +oo and

xto+

Similarly for qh,.

lim cp,(x) = 0.

X+oO

44

Proof. According t o (2) and (3), if we fix u > 0 we deduce that the image of the maps cpu and $u is exactly (O,+ca). The fact that cpu and $u are strictly monotone maps (see Lemma 2.1) finishes the proof.

Lemma 2.4. Let Eq. (1) be a 3-cycle. Then: (1) Either cpu and qU are strictly decreasing f o r all u,or both are strictly

increasing for all u. (2) If w,

> cpw(.) and $u(x) > $w(.) (in short, cpu > cp, and lClu > $,). cpU(.)

f o r all x > 0

Proof. From Lemma 2.2, Lemma 2.3, and (7) it is easy to see that cpu and $u have the same type of monotonicity for a given u > 0. Now suppose that there exist different values u,u'for which cpu is decreasing and cput is increasing. Then their graphics intersect (see Lemma 2.3) and there exists z > 0 with cpu(x) = put (x),in contradiction with Lemma 2.1. This proves Part (1). As a direct consequence of it, we obtain Part (2). 0

Lemma 2.5. If w E 3, then Fix(cp,) = Fix($,)

= {w}.

Proof. The equality Fix(cp,) = Fix($,) follows from Lemma 2.2. If cp, and $, are decreasing maps the statement is true. So, assume that cp, and ,$I are increasing maps (by Lemma 2.4 (1) all maps cpu, gU are strictly increasing). It is clear that {w} Fix(cp,). For the converse inclusion, consider a E Fix(cp,), a # w. So cp,(a) = f ( w , a ) = a and $,(a) = f(a, w) = a. Since (1) is a 3-cycle, by taking as initial conditions x1 = a, 22 = w, we obtain 23

= f(w, a) = a,

x4

= f ( a ,w) = a ,

x5 = f(a, a ) = x2 = w.

Hence, f ( a , a ) = w = On the other hand $ ~ ~ ( w=) f ( w , a ) = a. This implies that { a ,w} is a 2-periodic orbit of $Ia, which contradicts that $Ia is increasing. 0

Lemma 2.6. If cpu,$u

are strictly increasing maps for all u , then card(3)

5 1.

45 Proof. Reasoning by contradiction, suppose that there exist u , w E 7 , u < w. Firstly, we have that $,(ti) < $,(.I) = w because $, is increasing and w E 7 .By Lemma 2.5, Fix(cp,) = Fix($,) = { w } . This implies that either w > $,(x) > x for all x E (0, w ) or 0 < $,(x) < x for all x E (0, w ) . Suppose that w = $,(w) > $,(z) > x for all x E (0, w ) . Since cp, o I), = Idl(O,oo)(see (7)) and cp, is increasing, it follows x > cpw(x) for all x E (0, w ) . According to Lemma 2.1 the graphics of cp, and cp, does not intersect, and by Lemma 2.5 it is Fix(cp,) = Fix($,) = {u}. Hence cp,(x) > x for x > u.In particular, $,(u) = f ( u , w ) = cp,(w) > w = $,(w), in contradiction with the fact that $ , is increasing. Applying a similar reasoning to the case 0 < $,(x) < x < w for all x E (0, w ) , we obtain a new contradiction, and the proof finishes. 0 Proposition 2.1. For all u , cp,

and $, are strictly decreasing maps.

Proof. By Lemma 2.4, either cpz and $= are strictly decreasing for all z ,

or cpz and $z are strictly increasing for all z. Suppose that both cpz and $z are increasing maps for all z . From Lemma 2.6, card(7) 5 1. Let u !$ 7 ,u > 0. Then cp,(u) = $.J,(u)# u. Suppose that cp,(u) = $,(u) > u.Since $, is increasing it follows $,(cp,(u)) > $,(ti), so from (7) u > $,(u),a contradiction. Similarly, if cpu(u) = $,(u) < u,then $,((p,(u)) < $,(u),so u < $.J,(u),a new contradiction. fiom these contradictions we deduce that cpz and $= are strictly decreasing for all z > 0. n Consider the map h : ( 0 , ~ + ) ( 0 , ~ given ) by

h ( u ) = f(u,u). The following result states some useful properties of h.

(8)

Proposition 2.2. The map h verifies:

(1) h is a continuous strictly decreasing map. A s a consequence card(7) = 1. (2) lim,+o+ h ( x ) = + W and lim++ooh ( x ) = 0. (3) h(a,) = u,where {a,} := Fix(cp,) = Fix(&). (4) The inverse map h-' : ( 0 , ~ + ) ( 0 , ~ holds ) h k l ( u ) = a,, and

lim h-l(u) = +m,

u+o

lim h-'(u) = 0. U-++CO

46

(5) The n-iterate of h-l satisfies lim (h-l)"(u) = A,

n+m

where A is the unique fixed point of h, {A} := Fix(h). Proof. 1. The continuity of h is obvious. If u and Proposition 2.1 imply M u ) = f ( U , U ) = cpu(.)

> cpw(.)

< w then Lemma 2.4 (2)

> cpw(w) = f ( w , w )

=qw),

so h is strictly decreasing. Then h has exactly a fixed point, or equivalently card(F) = 1. 2. Since h is decreasing, there exists lim,,O+ h ( z ) =: 6 E R+ U {+GO}. Suppose that 6 < +co,and reach a contradiction. Considering values u < 6, by Lemma 2.4 (2) it follows cpu(u)> (PJ(u), u < 6. Consequently, Lemma 2.3 and the monotonicity of cpp give lim p,(u) = lim h(u) = 6 2 lim cps(u) = +co,

u-0

u+o

u+o

and we obtain a contradiction. Therefore, 6 = +m. It is clear there exists limz+m h ( z ) =: r] E R+ U (0). If r] > 0, then consider values u > r]. By Lemma 2.4 (2), cp,(u) > cpu(u),77 < u.From here, and using again Lemma 2.3, lim U+oO

= lim h(u)= r] U+oO

5 U+CC lim

= 0,

a contradiction. Therefore r] = 0. 3. Since (1) is a 3-cycle, if we take the initial conditions z1 = w , 2 2 = f ( u , w ) = &,(u),it is easily seen that f($~i(u),&,(u)) = w for all u > 0, where t,l~: is meant the second iterate of $J., In particular if u = a, we obtain w = f ($$(a,), t,hw(a,)) = f (a,, a,), as we claimed. 4. The existence and unicity of h-' follows from statements (1)and (2). Moreover h-' is also decreasing and then the limit values of the statement follow. To prove h-'(u) = a, we use (3). 5. Notice that the second iterates h2 and (h-l)' are inverse and strictly increasing maps. Moreover they have a unique fixed point, namely A. From part (2) it is easy to see h'(x) < II: if z < A, h'(x) > x if z > A, and consequently ( h - ' ) ' ( z ) > z if z < A, and (h-1)2(x) < z if z > A. Then limn+m((h-1)2)n(u) = A for any u > 0, and also h-'(limn,,((h-1)2)n(u)) = limn+CC((h-1)2n+1(u) = A. We conclude limn+.m(h-')n(u) = A. 0

47

Remark 2.1. From the point of view of the difference equation x,+2 = f(x,+l,x,) we have shown that if it is a 3-cycle, then it has exactly an equilibrium point which is exactly T = A, f ( A , A) = A. Notice that A is an attractor fixed point of h-l. 3. The particular case f ( z , y ) = u ( z ) p ( y ) In the sequel we will assume that f separates variables in the form

f ( Z , Y ) = 4.1

*

P(Y>,

where CJ, p : ( 0 ,co) + ( 0 ,co) are continuous maps. We will apply the results obtained in the above section concerning the properties of f and its fiber maps cp,, $, in order t o find all the 3-cycles having the form x,+2

.p

= +,+1)

( 4 .

(9)

Remember that h(u) = f ( u , u ) ,u > 0; {A} = Fix(h) = F ;cpzL(.) = f(u,.), $,(.) = f(.,u); {a,} = Fix(cp,) = Fix($u), h(a,) = u,h-l(u) = a,, u > 0. For the case f(x,y) = g ( x ) . p(y), we find cpu(z) = 4.1

.p(z>,

$zL(z)

= 4.1

. p(u>, h('Z1)= 4 . ).

In the following results we assume that (9) is a 3-cycle.

Lemma 3.1. The maps

CJ

and p are strictly decreasing, with

lim ~ ( w=)lim p ( w ) = +co,

w+o

W+O

lim ~ ( w=) lim p ( w ) = 0.

W++W

W-++CC

Lemma 3.2. For all u > 0 and n = 0 , 1 , 2 , . . . , it is

Proof. By (4)and f ( x , y ) = o(x)p(y) it is straightforward to check u = u ( u ) . p ( a ( u ) p ( u ) )and u = p ( u ) . o(a(u)p(u)).These equalities give =

$$#,

%

u > 0. Moreover

since a, = f(u,a,) = f ( u , h - l ( u ) ) = u(u)p(h-'(u)),and also a, = f ( a z L , u= ) f ( h - l ( u ) ,u)= c ~ ( h - l ( u ) ) p ( uIt) remains . to extend the property t o any positive integer n. We will get it by mathematical induction.

Lemma 3.3. Define g : (0,co) + ( 0 , ~ by) g ( x ) = # . T h e n for all x > 0 and for some K > 0.

g(x) = K

48

Proof. Take K := g(A). We are proving g(z) = K for any z Lemma 3.2, we have

# A. By

g(z) = g((h-')2n(z)>,for all z > 0, n = 0,1,2,. . .

Since g and (h-1)2nare continuous maps, and we know by Proposition 2.2 that limn.+oo(h-1)2n(z)= A for any z # A, we obtain g(z) = g(A) = K ,

as we claimed.

0

Theorem 3.1. Let a, p E C((0,co),(0, co)) and consider f ( z ,y) = a(z) . p ( y ) . Suppose that zn+2 = f(x,+l, z), is a 3-cycle. Then for some constant K > 0 we have

Proof. Suppose that the fiber maps (pu,$ju verify (pu = $ju for all u > 0. By Lemma 3.3, p(u) = C .) . ( a for some constant C > 0, and for all u > 0. On the other hand, since (pu = G,, u > 0, by (7) we find

. P ( $ j U ( X ) ) = c.4u) . 4 ( P u ( x ) ) c . a(u). a ( a ( u ) p ( z ) )= c . a ( u ) a(Ca(u)a(z))

z = vu(Gu(x)) = 4 ' z L )

=

*

so z=

Fix a value 3:

c ).(a

. u(Ca(u)a(z)), z, u > 0.

*

> 0, for instance x = 1. Then Eq. (10) is rewritten as 1

- a(Ca(l)a(u)),

--

c.).(a

u > 0.

Lemma 3.1 allows us to introduce the change of variable

s = Ca(u), s E

(0,oo).

Then Eq. (11) is transformed into 1

- = a ( b a ( l ) ) , 6 > 0.

s

Another change of variable, namely q = b a ( l ) ,gives

(10)

49

This proves that

) . ( a

K1

= -,u U

> 0,

where the constant is K1 = a(1).Since p(u) = C . a ( u ) for some constant C > 0 , and for all u > 0 , we also deduce

5

y) = for some positive constant where K2 = C . K1. Therefore f(z, K > 0 , as we tried to prove. Now suppose that cpu $ $u for some u > 0. Then there exists z E (0,m) for which cpu(z) # $u(z), that is a(u). p ( z ) # a (.). p(u), and consequently

4’1L)

-#-

42)

P(U) P(4’ which contradicts Lemma 3.3. Therefore, the case cpu $ $u is not allowed.

0

It is interesting to point out that if (1) is a 3-cycle and f separates the variables, that is, f(z,y) = ~ ( z.)p(y), then it must occur cpu z $u for all u > 0. It is an open problem to determine whether or not this property on the fiber maps cpu, $u remains true for any class of continuous map. A question remains unanswered, are the maps f(z,y) = the unique class providing 3-cycles of the type zn+l = f ( x n , z,-l)? As a partial advance, using the procedure of Section 2 it is easy to prove that if, for instance, f ( x , y ) = g(z* yo), with a,P E R,or f ( z , y ) = g ( z ) , then (1) is not a 3-cycle.

5

+

Bibliography 1. R.M. Abu-Saris, Q.M. Al-Hassan, “On global periodicity of difference equations”, J. Math. Anal. Appl. 283 (2003), 468-477. 2. F. Balibrea, A. Linero, “Some new results and open problems o n periodicity of difference equations”, to appear in Proceedings of ECIT’O4, in Grazer Math. Ber . 3. A. Cima, A. Gasull, F. Maiiosas, “On periodic rational difference equations of order k”, J. Difference Equ. Appl. 10 (2004), 549-559. 4. A. Cima, A. Gasull, V. Mariosa, “Global periodicity and complete integrability of discrete dynamical systems”, preprint, 2005. 5. M. Csornyei, M. Laczkovich, “Some periodic and non-periodic recursions”, Monatsh. Math. 132 (2001), 215-236. 6. V.L. Kocic, G. Ladas, “Global behavior of Nonlinear Difference Equations of Higher Order with Applications”, Mathematics and Its Applications 256, Kluwer Academic Publishers, Netherlands, 1993.

50

7. B.D. Mestel, “On globally periodic solutions of the difference equation x,+1 = f(z,)/z,-l”, J. Difference Equ. Appl. 9 (2003), 201-209. 8. Yu Zheng, “On periodic cycles of the Lyness equations”, Differential Equations Dynam. Systems 6 (1998), 319-324.

This paper has been partially supported by Grant 00684/PI/04 from Fundaci6n Seneca de la Comunidad Aut6noma de la Regi6n de Murcia and partially by Grant MTM 2005-03868 from Ministerio de Educaci6n y Ciencia of Spain and Fondos FEDER.

51

LOGARITHMIC ORDER AND TYPE OF INDETERMINATE MOMENT PROBLEMS

CHRISTIAN BERG AND HENRIK L. PEDERXEN* WITH AN APPENDIX BY WALTER HAYMAN We investigate a refined growth scale, logarithmic growth, for indeterminate moment problems of order zero. We show that the four entire functions appearing in the Nevanlinna parametrization have the same logarithmic order and type. In the appendix it is shown that the logarithmic indicator is constant.

2000 Mathematics Subject Classification: primary 44A60, secondary 30D15 Keywords: indeterminate moment problem, logarithmic order

1. Introduction and results This paper deals with the indeterminate moment problem on the real line. We are given a positive measure p on IR having moments of all orders and we assume that p is not determined by its moments. For details about the indeterminate moment problem see the monographs by Akhiezer' , by Shohat and Tamarkin26 or the survey paper by Berg3. Our notation follows that of Akhiezer'. In this indeterminate situation the solutions u to the moment problem form an infinite convex set V , which is compact in the vague topology. Nevanlinna has obtained a parametrization of V in terms of the so-called Pick functions. We recall that a holomorphic function cp defined in the upper half plane is called a Pick function if Scp(z) 2 0 for Sz > 0. The class of Pick functions is denoted by P. The Nevanlinna parametrization is the one-to-one correspondence uv H cp between V and P U {m} given by

*research supported by the carlsberg foundation

52

Here A, B , C and D are certain entire functions defined in terms of the orthonormal polynomials { P k } and the polynomials of the second kind {Qk} in the following way:

k=O

C ( z )= 1

+ z C P k ( O ) Q k ( z ) and k=O m

k=O

These functions are closely related due to the relation

A(z)D(z) B ( z ) C ( Z ) 1. Two other functions play a role, namely

) ~the maximal point mass of any solution to the We recall that l / ~ ( x is moment problem at the point x E R. The function q has a similar property when one considers the so-called shifted moment problem, cf. P e d e r ~ e n In Berg and Pedersen4 the entire functions A , B , C and D were shown to have the same order, type and indicator function. It was also shown that the logarithmically subharmonic functions p and q had that order, type and indicator. A result of M. Riesz states that each of the entire functions is of minimal exponential type and therefore the common order is a number between 0 and 1. The point of this paper is to investigate moment problems of order 0. The question arises if the growth of the four entire functions and p and q is also the same when one considers a refined growth scale for functions of order 0. We shall use a logarithmic scale, which has been used by other authors in connection with q-special functions. Several examples of indeterminate moment problems of order 0 have been investigated. The indeterminate moment problems within the socalled q-Askey scheme have been identified by Christiansenll. As examples of moment problems of order zero we mention in particular the moment

53

problems associated with the q-Meixner, q-Charlier, Al-Salam-Carlitz 11, q-Laguerre and Stieltjes-Wigert polynomials. Also the discrete q-Hermite 11, q-l-Meixner-Pollaczek, symmetric Al-Salam-Chihara I1 and continuous q-l-Hermite polynomials lead to moment problems of order zero. See Section 4. For an entire function f the quantity M ( f , r ) denotes the maximum modulus of f on the closed disk centered at the origin and of radius r. We recall that an entire function f is of order 0 if for any E > 0 there is TO > 0 such that logM(f,r) 5 T " ,

for r

2 TO.

The inequality log M ( f ,r ) 5 rEis thus true for r sufficiently large, and this we write as 1% M(f ,

5,

rE,

adopting a notation from Levin21. For an entire function f of order zero we define the logarithmic order P=Pf

as

p = i n f { a > 01 logM(f,r) 5,. (logr)"}. For non-constant f we must have p 2 1, by the usual proof of Liouville's theorem. It is easy to obtain that

When p < 00 we define the logarithmic type r = rf as

and it is readily found that

It is easily seen that if f (2) has logarithmic order p and logarithmic type r then so has the function f ( a z b) (for a # 0). Furthermore, the function f ( z )is~again of logarithmic order p but of logarithmic type nr, while f ( z n ) has logarithmic order p and logarithmic type r n p . It is also clear that if a transcendental entire function has logarithmic order equal to 1, then the logarithmic type must be infinite. For a polynomial of degree k 2 1 the logarithmic order is 1 and the type is k.

+

54

The indicator function for an entire function of finite logarithmic order p and finite logarithmic type is defined in the natural way as

However it turns out that the indicator of any entire function of finite logarithmic order and type is actually constant equal to the type. This fact can be deduced (at least when p 2 2) from results in a paper by Barry2 (see p. 469 in Barrys paper). M. Sodin has kindly informed us that the result can also be deduced from a result of Grishin13. In the Appendix we present a self-contained proof of this result by Walter Hayman. With these definitions we have the following result proving the conjecture 24.4.4 p. 651 in Ismail14. Because of the applications to the q-Askey scheme, Ismail called q-order, q-type and q-Phragmen-Lindelof-indicator what we have called logarithmic order, type and indicator, see Ismail14 p. 532.

Theorem 1.1. The functions A, B , C , D,p and q appearing in an indeterminate moment problem of order 0 have the same logarithmic order p 2 1. If p < co then they have the same logarithmic type. A n y combination A ( z ) t - C ( z ) and B ( z ) t - D ( z ) ) where t E IR U {co}, has also the same logarithmic order and type. The common logarithmic order and type of the functions of Theorem 1.1 are called the logarithmic order and type of the indeterminate Hamburger moment problem. The four entire functions occuring in the indeterminate Hamburger moment problem can be regarded as the entries of a certain 2 x 2 matrix of entire functions. This leads to the concept of a Nevanlinna matrix, which was introduced by Kreinlg, see also Akhiezerl. In Berg and Pedersen5 the common growth of the entries was investigated, and it was shown that all four entries have the same ordinary order and type.

Definition 1.1. An entire function N : CC --+SLz(C) of the form

is called a (real) Nevanlinna matrix if the entries are real transcendental entire functions and

55 If we consider the entire functions A , B , C and D from an indeterminate Hamburger moment problem, then the matrix

(i:;) -2;:)) defines a real Nevanlinna matrix, taking into account the relations (1) and

(3). Part of Theorem 1.1can be generalized to real Nevanlinna matrices. We have

Theorem 1.2. For any real Nevanlinna matrix of order zero

(t):

1

the entries A , B , C and D have the same logarithmic order and logarithmic type. This common order and type is also the logarithmic order and type of any of the functions At B , Ct D , where t E R.

+

+

Theorem 1.1 and 1.2 are proved in Section 2. It was shown in Bergweiler, Ishizaki and Yanagihara7 and in R a m i ~ ~ ~ that entire transcendental solutions of certain q-difference equations are of logarithmic order 2 and finite logarithmic type. Refined results about the zeros of such solutions are given in Bergweiler and Hayman6.

Remark 1.1. There is a notion of proximate or refined order for entire functions, originally introduced by Valiron28, see also Levin’s book22. In this general setup it is still true that the four entire functions in a Nevanlinna matrix have the same growth, due to the quite accurate estimates between the functions that we shall use in the proof of Theorem 1.2. Acknowledgement. The authors want to thank Mourad Ismail for the encouragement t o undertake the present investigation.

2. Proof of the main results

In this section we prove Theorems 1.1 and 1.2. The key to this is the following lemma.

Lemma 2.1. Let f and g be two transcendental entire functions such that f / g is a Pick function. Then f and g have the same logarithmic order and type.

56

Proof. We use the fact that any Pick function p admits an integral representation of the form

where a 2 0, b E R and T is a finite positive measure on the real line. The function f / g is a meromorphic Pick function, so from the integral representation we easily obtain

'(') - - a z + b - - b0 -zx g(z>

00

bn n=l (2 - an )an '

where a , bo 2 0 , b E R,{a,} is the set of nonzero poles, bn > 0, n 2 1 and C,"==, bn/a: < 00. From this series representation we see that

for some constant K and all z E C \ R. For IyI 2 1 this estimate gives us (with T = )1. lf(z)I

I K ( r 2+ l)M(g,r).

For IyI < 1, we get, since log If1 is subharmonic,

21,

loglf(z)l I -

log I f ( z

+ eit))ld t

I log K ( ( r + 1)2+ 1) -

1

2T

log Iy

+ sin t 1 dt + log M(g,r + 1).

If we combine this with the estimate for IyI 2 1, then we get log M ( f,r ) I K1

+ K2 log(?-+ 1)+ log M(g, + l), 7-

(4)

for suitable positive constants K1 and Kz. From this relation it follows that the logarithmic order pf of f is less than or equal to the logarithmic order pg of g: this is clear if pg = 00, so we may suppose that pg < 00. Let E > 0 be given. Then M(g, r ) I,.e ( l o g r ) P g + s ,

and hence, J/f(f ,

).

I,,eKl+K2 log(r+l)+(log(r+l))Ps+E <

-as

e(logr)Pg+z"

since pg 2 1. In this way we see that pf 5 ps. Clearly, the function -g/ f is also a Pick function, and so we get pg 5 p f . Therefore the two logarithmic orders must be identical. This common order

57

we denote by p. Assume now 1 < p < follows that Ki K2 log(r log M ( f , r ) 5 (log r>p (log

+

00.

+ 1)

From the relation (4) it also +

log M ( g ,r (log

+ 1)

This implies that the logarithmic type of f is less than or equal to the logarithmic type of g. Again, by considering -g/ f , we find that the two logarithmic types are equal. When p = 1 both f,g have logarithmic type

0

00.

Proof of Theorem 1.2. For any t E RU {m}, the meromorphic function

+ +

A(z)t B ( z ) C (z)t D ( z ) is a Pick function. Hence, by Lemma 2.1, the logarithmic order and the logarithmic type of the two functions At B and Ct D (for fixed t ) are identical. In particular (t = 00) the logarithmic order and type of A is the same as the logarithmic order and type of C , and similarly for B and D

+

+

(t = 0). In a real Nevanlinna matrix, the function D/C is also a Pick function, see e.g. Berg and Pedersen5, so the logarithmic order and type of D and C are also identical. For fixed t E R, the function

is thus also a meromorphic Pick function and therefore the logarithmic growth of Ct D is the same as the logarithmic growth of C. It is easy to see that also the matrix

+

(I;$) )

+

is a real Nevanlinna matrix. Therefore also B / A and hence (At B ) / A is a Pick function. Consequently, the logarithmic growth of At B is equal to the logarithmic growth of A. 0

+

Proof of Theorem 1.1. The assertions about the functions A , B , C and D follow from Theorem 1.2. We turn to the functions p and q. We claim that we also have p p = pq = p, where p is the common logarithmic order of the four entire functions. Indeed, it is enough to prove that p p = p ~ as, mentioned in Berg and Pedersen4. From the definition of D , (2), and the

58

Cauchy-Schwarz inequality we see M ( D ,r ) 5 p(O)rM(p,r ) so that p~ 5 p p . Furthermore, formula (22) in Berg and Pedersen4, stating M ( P , ~5) ~ M ( B ,r l ) M ( D , r l),yields p p 5 p ~ . We also obtain TD 5 rp and 27, 5 TB TD so that rp = r , the common logarithmic type. 0

+

+

+

3. Stieltjes moment problems

A Stieltjes moment problem may be determinate on the half-line, but indeterminate on the whole real line. One defines the quantity a as

It is a fact that a 5 0 and that the problem is indeterminate on the half-line (or in the sense of Stieltjes) if and only if a < 0, cf. Chihara" or Berg3. The set V+ = {u E V I supp(u) C [0, m)} of solutions to a Stieltjes moment problem, which is indeterminate in the sense of Stieltjes, can be parameterized via the one-to-one correspondence v, H u between V+ and S U { m} given by

where the functions P, Q , R and S can be defined as limits of convergents of the Stieltjes continued fraction. The parameter space S is the set of Stieltjes transforms, i.e. functions of the form

where a 2 0 and r is a positive measure on [O,m) such that the integral makes sense. This is the Krein parametrization of the solutions to an indeterminate Stieltjes moment problem, see Krein and Nudelman20 or Berg3. The functions P, Q , R and S are related to A , B , C and D as follows.

1 P ( z ) = A(-Z) - -C(-z), a

a

R ( z )= C(-z),

,

S(Z)= -D(-z).

Concerning these functions we have

Proposition 3.1. The entire functions P, Q , R and S all have the same logarithmic order and type as the indeterminate Hamburger moment problem.

59

Proof. This follows directly from the definitions of P , Q, R and S in terms 0 of linear combinations of A , B , C and D and Theorem 1.1. For a Stieltjes moment sequence {t,},LO one considers the corresponding symmetric Hamburger moment sequence given by { t o , 0, t l , 0 , t 2 , 0 , . . .}. There is a close connection between these two moment problems and relations between the entire functions in the two Nevanlinna parametrizations can be found in e.g. Chihara" or P e d e r ~ e n ~Let ~ . us just mention that the D-functions Ds and DH for the Stieltjes and Hamburger problems are connected by z D ~ ( z= ) Ds(z2). From this relation we easily obtain

Proposition 3.2. Let ps and rs denote the logarithmic order and type of an indeterminate Stieltjes moment problem and let p~ and T H denote the logarithmic order and type of the corresponding symmetric Hamburger moment pmblem. T h e n we have P H = PS,

TH

= Ts2''.

4. Examples

In this section we determine the logarithmic order and type of some indeterminate moment problems from the q-Askey scheme, which is discussed in Koekoek and Swarttouw". To do so we apply results from Section 5 below. The moment problem associated with the q-Meixner polynomials, which we denote as {M,(z 1;b , c ; q ) } , is indeterminate in the sense of Stieltjes. The four entire functions in the Krein parametrization have been computed in Theorem 1.3 in Christiansen". In particular the function Q is shown t o be given by

+

where b < l / q and c > 0. For the definition of basic hypergeometric series see Gasper and Rahman12 or Koekoek and Swarttouw". We denote the zeros of Q by {z,}, where 0 > x1 > x2 > . . .. By a result of Bergweiler and Hayman' there is a constant A > 0 such that

-Aq-2n as n-+oo, see Proposition 1.5 in Christiansenll for details. x,

N

(5)

Proposition 4.1. The indeterminate Stieltjes moment problem associated with the q-Meixner polynomials have logarithmic order equal t o 2 and logarithmic type equal to -1/(41og q ) .

60

Proof. We see from (5) and Proposition 5.6 that Q has logarithmic order equal to 2 and logarithmic type equal to -1/(4logq). Then the result follows from Proposition 3.1. 0 By specialization or taking limits of the parameters in the q-Meixner case we obtain: Corollary 4.1. The indeterminate Stieltjes moment problems associated with the q-Charlier, Al-Salam-Carlitz 11, q-Laguerre and Stieltjes- Wigert polynomials are all of logarithmic order 2 and logarithmic type -1/(4 logq). The Discrete q-Hermite I1 moment problem is symmetric and the corresponding Stieltjes moment problem is the q2-Laguerre moment problem with (Y = Applying Proposition 3.2 we get the following.

i.

Corollary 4.2. The indeterminate Hamburger moment problem associated with the Discrete q-Hermite 11 polynomials is of logarithmic order 2 and logarithmic type -1/(2 log q ) . The q-’-Hermite moment problem was treated in detail by Ismail and Masson15. The zeros of the function D are given explicitly as

1 , nE 2 Therefore the counting function satisfies 2, = - (qn - q-n)

n(r)

N

2 log r - logq’

- r

z.

+ 00.

From Proposition 5.6 we obtain:

Proposition 4.2. The indeterminate Hamburger moment problem associated with the Continuous q-l -Hermite polynomials has logarithmic order 2 and logarithmic type equal to -l/logq. 5. The logarithmic growth scale In this section we collect some facts about entire functions of finite logarithmic growth. Most of these facts can be found in the literature, but for the readers convenience we have included the proofs. For an entire function f with Taylor series M

n=O

61

the (ordinary) order is 0 if and only if log n

lim n'03

1%

= 0.

(&)

One can also express the logarithmic order and type in terms of the Taylor coefficients.

Proposition 5.1. For an entire function f (2) = C,"==, &zn of order 0 its logarithmic order p satisfies log n p = 1 limsup

+

n+m

Proof. We put

and we first show that p=-

P

p - 1'

Suppose that p is finite and let X logM(f,r)

> p. Then there exists ro > 1 such that

I (logr)',

for r 2 T O .

By applying the Cauchy estimates we find, for any n 2 0 and r 2 ro, log

5 (logr)'

- nlogr.

The function (p(r)= (logr)x - nlogr

(defined for r 2 1) attains its minimum for

which is bigger than TO for n 2 X(logro)x-l. For such n the minimum value over [TO,co[is

which is a negative quantity. It follows that for all sufficiently large n

62

so that

or

for all sufficiently large n. Hence

Since this holds for any X > p we must have p 2 p / ( p - 1). Notice that p = 00 if p = 1 and also that p = 00 if p = 1. Conversely, if p > 1 we choose v E (1,p ) and next no such that

1% 11% Icnll > Y , log n for all n 2 no. This implies 121 = r 2 1,

1 ~ <1 e-nu

for n

no-1

2 no. Then we have, for

00

n=O

n=no M

For given r so large that such that n1 2 no 1 and

+

(721-

- 1 < logr we choose n1, depending on

r,

I ) ~ - ’- 1 < logr 5 n7-l- 1.

(This is possible since Y > 1.) For n 2 nl we thus have logr 5 nv-’ - 1 so that logrn 5 nv - n, or rn 5 enYe-n. This yields M

00

n=n1

n=n1

C e-nYrn I C e-n

< 1.

For n E {no,.. . , n1 - 1) we have nv-l - 1 < logr so that

n < (logr

+ ql’(v-1).

63

Therefore

Y

n=no 00

< e(logr+l)''(u-l)

logr

C e-n n=l

Since v was an arbitrary number between 1 and p we conclude that f has logarithmic order 5 p / ( p - 1). We have therefore verified the relation (6) and from it we get 1 p=l+P-1

=1

log n + limsup loglloglcnll -1ogn 71-00

=

1

+ limsup n-+w

log n loglog

(*)

*

Remark 5.1. The logarithmic order is p'(2) in the notation of Shah and I ~ h a and q ~ the ~ p ( 2 , 2 ) order of Juneja, Kapoor and Bajpai". Proposition 5.2. For a n entire function f ( z ) = C,"==, &zn of logarithmic order p E (1,m) its logarithmic type 7 satisfies 7 =

( p - 1)P-1 PP

lim sup n-+m

n p-1.

64

Proof. Suppose that M ( f , r ) 5 eK(logr)Pfor r estimates we see that log)%)5 K(logr)P - nlogr,

>

From the Cauchy

TO.

r L TO.

The function r H K(1ogr)P - nlogr attains its minimum for r 2 1 when Kp(1ogr)p-' = n, i.e. when logr = ( n / ( K p ) ) A . For all sufficiently large n we must therefore have loglkl 5 (logr) (K(logr)P-l - n) =

)('K P

-

n

(: -

1)

,

so that

or

Since this holds for all sufficiently large n and K is an arbitrary number greater than the logarithmic type T we must have

72

( p - 1)P-1 PP

n lim sup n-m

p-1.

For the converse we argue as follows and put

Let

E

> 0 be given. We choose no such that for all n 2 no we have

I -(. which means that log Hence, for Iz( = r 2 1, n=O

+ E)-p--lnp-l. 1

n=no

n

65

When r is so large that 1

-(a+&)-P-'noP-'

L

+nolog?- > -no,

we choose the smallest integer nl = nl(r) > no such that for n

2 nl

+ E ) - P - ' n P - - l + nlogr 5 -n. This implies first of all that n1 - 1 < (a+E)(logr + l)f-', but to treat the 1

-(a

n's

between no and

n1 -

-

L

1 we look at the concave function of s

cp(s) =slog?.-

1

-

L

(a+E)-P-'Sp--l.

We find

+E)- _

P cp'(s) = logr - -(a

1 p-1

P-1

for

(c) f-1

s = so(r) =

1

Sb-l

=0

+

(a &)(lOgr)f-l.

Furthermore, cp(s) attains its maximum at s = so(r) and

We thus get

Therefore, for IzI

=r

sufficiently large,

and hence

Therefore the logarithmic type T satisfies

66

and letting

E -+

0, we see that

Remark 5.2. The logarithmic type is the T(2,2) type of Juneja, Kapoor and Bajpai17. Example 5.1. We let q E C and suppose that 0 < 141 hypergeometric series

<

1. The basic

defines an entire function of z when r 5 s and the parameters are such that none of the denominators become zero. We assume also that ,isis not a polynomial. is equal t o 2, as can be seen from PropoThe logarithmic order of sition 5.1. The logarithmic type is equal to

1 2(1+ s - r ) log l/lql' which follows from Proposition 5.2. In particular,

is of logarithmic order 2 and logarithmic type 1/(210g l/lql).

Example 5.2. We let q E C and suppose that 0 < IqI

< 1. Then

00

f ( z ) = cq"* zn n=O is (for a

> 1) of logarithmic order a and logarithmic type equal to r=

( a- 1)O-l aa

1 (log 1 / 1 q p - 1 *

67

(2) 00

n=O

is of logarithmic order 1 and infinite logarithmic type.

(3)

n=O

is of order zero, but its logarithmic order is co.

A transcendental entire function f of ordinary order less than 1 must have infinitely many zeros, which we label {a,} and number according to increasing order of magnitude and repeating each zero according to its multiplicity. We suppose that f(0) = 1 and from Hadamard’s factorization theorem, we get that

The growth of an entire function of ordinary order less than 1 is thus in principle determined by the zero distribution. We shall use the following quantities to describe this distribution. We define the usual zero counting function n ( r ) as

n ( r ) = # { n 2 1I Ian1

IT}.

We define the following quantities in terms of the zero counting function

N ( r )=

1‘

dt,

and

These quantities are related to M ( T )= M ( f ,T ) in the following way N ( T )I logM(r)

I N ( T )+ Q ( T )

(7)

T > 0. (See e.g. the relation (3.5.4) in Boasg.) If f is of (ordinary) order 0 we get from Hadamard’s first theorem that the convergence exponent of the zeros is also equal to 0. It means that we have

for

68

for all e > 0. In this situation we define the logarithmic convergence exponent p1 as

The following proposition expresses the logarithmic convergence exponent in terms of the logarithmic order of the zero counting function.

Proposition 5.3. We have log n(r> loglogr

p1 = limsup T+oo

Proof. First of all, we see by integration by parts that for a > 0 and r > r o > 1,

To ease notation we let

is bounded and hence If a > L we choose a E ( L ,a ) and notice that that lim,.--roon(r)/(logr)" = 0. Furthermore, since

the limit of the integral on the right-hand side of (8) (with a replaced by a ) as r + 00 is finite. Therefore

and consequently p1 5 a and thus we obtain that p1 Conversely, if a > p1 we have

5 L.

If we look again at (8) it means that

remains bounded as r

3 00,

(1% hence L 5 a. We conclude that p~ 2 L.

0

69

It is also possible to relate the logarithmic order and logarithmic convergence exponent. The proposition below is mentioned in the assumptions of Theorem 3.6.1 in Boasg in the special case where p = 2. Proposition 5.4. For an entire function of order 0 we have p = p1

+ 1.

To prove this proposition we give two lemmas. Lemma 5.1. Suppose that n ( r ) I const (logr)" for r Then, for r > 1,

> 1 and some a > 0.

N ( r ) 5 const (logr)@+l and (where [.] denotes the integer part)

where 0 5 S(r) 5 const (logr)@-[@l-l Proof. By definition we have N ( r ) 5 const

+

d t 5 const (log r)*

dt = const (log r)"+'

Concerning Q ( r ) we have

Here, by repeated integrations by parts,

= (logr)"

+ ar Jm T

where

dt

70

Since the exponent a - [a]- 1 is negative we find 6(r) 5 const (logr)a-[ul-l. R

Lemma 5.2. If, for some a > 0, log M(r) 5 const (log r)"+' then n(r) 5 const (log r)".

Proof. Since N ( r ) 5 logM(r) we have n(r) logr = n(r)

I'

< N(r2) 5 const (logr2)af1 5 const (logr)a+l.

t Hence n(r) 5 const (logr)".

Proof of Proposition 5.4. Suppose that ct! > p1. From Proposition 5.3 we have n ( r ) 5 (logr)" for all sufficiently large r . From (7) and Lemma 5.1 we thus have logM(r) 5 const (logr)"+', and therefore we see that p 5 a 1. From this we conclude that p 5 p1 1. On the other hand, if ,B > p then log M(r) 5 (log r)o for all sufficiently large r . By Lemma 5.2 we therefore have n (r) 5 const (logr)'-l, so that p1 5 ,B - 1. We have shown that pl 5 p - 1. It is also possible to relate the logarithmic type to the growth of the zero counting function. For an entire function of finite logarithmic order p > 1 we put

+

+

= K ( f ) = limsup

K

r-+m

n(r> (log

'

Proposition 5.5. For an entire function of finite logarithmic order p > 1 we have the following relation between the quantity K and the logarithmic type r:

r5 Proof. For any given E

K/p

5 er.

> 0 we choose ro > 1 such that n(r)

I ( K + E)(logr)P-',

for r 2 ro. Then we get N(r) = Const I

+ + €1 1 (log t (K

Const + -(log r)p. I K+E

P

)P- 1

dt

71 Since we have Q ( r ) 5 Const (1ogr)P-I we see that logM(f,r)

5 N ( r )+ Q ( r ) I Const

Therefore T 5 n/p. For E > 0 we have logM(f,r) 5 for any s > 1,

n ( r ) ( s- 1)logr 5

JI" "I"'

-d t

+(log + Const (log rlp-1. P +

(T

r>p

+ E)(logr)P for r 2

TO

and hence,

5 N ( r " )I logM(r") 5 (T + &)sP(logr)P

This gives

so that

It is easily found that the function p(s) = s P / ( s - l), s minimum for s = p / ( p - 1) and that the minimum is

>

1 attains its

Hence

Since ( p / ( p - 1))P-l 5 e we finally see that n 5 Tpe. We shall now see that some regularity.

T

0

= n/p provided that the zero distribution has

Proposition 5.6. Let f be an entire function of finite logarithmic order p > 1. Then the following are equivalent for r 4 00.

--

(a) n(r) X(1ogr)P-l. (ii) logM(r) $(logr)P.

Proof. Since the function Q ( r ) in (7) is O((1ogr)P-l) under each of the conditions (a),&), we have for A > 0 logM(r)

-x

-(logr)P P

N(r)

-x

-(logr)P. P

72

It is therefore enough to show that

n(.)

N

A(l0gr)P-'

H

A

N ( r ) N -(logr)P. P

We have

If n ( r )N A(1ogr)P-' then we choose TO such that

for r 2 T O , and this gives

5 Const + -(logr)j' &

P

Therefore

For s > 1 we have

Therefore, if N ( r ) N $(logr)" we find

= (SP - 1)

(a +

o(l))

,

and conclude that lim sup T'OO

4.1

< --.A S P - 1

(10gr)P-1 - p s - 1

If we let s tend to 1 we find IC 5 A. We next use the relation PTS

to find

x

SP-1

2 -p (s - 1)sp-1'

73

and therefore

6. Appendix: The Phragmh-Lindeliif indicator of some functions of order zero

This appendix was written by Walter Hayman during the “International Conference on Difference Equations, Special Functions and Applications” held in Munich, Germany in the period July 25 - July 30, 2005. We appreciate that Haymans result could be included in this appendix. Introduction and statement of results

Suppose that f(z) is a transcendental entire function. We write m ( r ) = inf If(z)I, IzI=r

M ( r ) = SUP lf(z)l, IzI=r

for the minimum and maximum modulus o f f respectively. Next we define a function P ( T ) to be of slow growth (s.g.) if positive nondecreasing in [0,co) and 9 ( 2 ~ ) 9 ( r ) , as r N

(9)

@(T)

is

-+ 03.

(10)

T + 00,

(11)

It follows immediately from (10) that

~ ( K TN )!P(T), as

whenever K > 1. For we may take K = 2P, for p = 1,2,. . ., and prove the result by induction on p , using (10). Since 9 is increasing, (11)then follows also for 2P < K < 2P+’. We can now state our results. Theorem 6.1. Suppose that with the above hypotheses,

Then

as r

4

00.

74

Corollary 6.1.

(14) Corollary 6.2.

h* (0) = lim sup log If T-00 @(r)

= a,

(15)

for 0 I 0 5 2 ~ .

S o m e preliminary results We assume from now on that f(0) = 1. Otherwise we apply our conclusions to f(z)/(czP),where p is a nonnegative integer and c a non-zero constant. This does not affect the relations (12), (13), (14) and (15). Next it follows from (11) that

@ ( r )= o ( r p ) , as r

(16)

-+00

whenever p is positive, cf. Theorem 1.3.1 in Bingham, Goldie and Teugels8. Now (12) shows that f has zero order. Thus by Hadamard’s theorem

where z, are the zeros of f . We deduce that, for IzI

= r,

Hence

We now have

Lemma 6.1. If n(r) denotes the number of zeros off in IzI

n ( r )= o(*(r)),

as

r

I r , then

+ m.

In fact we have by Jensen’s formula, cf. (7), if K

> 1 and E > 0,

75

using ( 1 1 ) and ( 1 2 ) . Hence n ( r )log K 5

lKr

+

5 (a E)P(r), r > rg.

This yields Lemma 6.1, since K can be chosen as large as we please. Lemma 6.2. For s

>0

We first put s / t = x so that our integral becomes

where we have put x

=

l / y , when x > 1. Also

This proves Lemma 6.2.

Lemma 6.3. We have

We can write the sum as

2r

log

(y 1-1 ) dn(t)

I)?(

=

[n(t)log 1 - r

=

-4%) log 3 + o

+ 2r

1: * t2 - r2

(19)

by Lemma 6.1. Also it follows from ( 1 1 ) that, for p = 1 , 2 , . . . and large r ,

t2

-

9(2r)2~/~ r2p+l

76

since for large r (and p by (10). We get

2 l ) , !P(2r2P) < 21/2!P(2r2P-1) < . . . < 2P/2P(2r)

Thus (19) and Lemma 6.1 yields Lemma 6.3.

Proof of the Theorem and its Corollaries We deduce from Lemma 6.3 that, for r

< t < 2r, we have

Thus

Again by Lemma 6.2

7r2

L -n(4r)

= o(!P(r)). (21) 2 Putting together (18), (20) and (21) we deduce (13) and the theorem is proved. To prove Corollary 6.1, we suppose given a positive E and then choose a large r , such that

log M ( r ) > (a- &)!P(T). In view of (11) and the fact that logM(r) increases with if r is sufficiently large

T,

we deduce that

logM(t) > (a! - 2&)!P(t), r 5 t 5 2r.

(22)

Next it follows from (13) that we can choose t , such that r 5 t 5 2r and logm(t)

> logM(t) - & ! P ( T ) 2 logM(t) - &!P(t),

if r is sufficiently large. On combining this with (22) we obtain

logm(t) > (a! - 3&)!P(t).

77

Since E is arbitrarily small we obtain

Since m(r)5 M ( r ) we have from (12)

This proves Corollary 6.1. Clearly for every 8 m(r)I If(reie)l I ~ ( r ) .

Thus (15) follows from (12) and (14) and Corollary 6.2 is proven. We remark that for 9 ( r ) we may take not only (logr)a, but (log r)" exp{(logr)P(log 1ogr)Y) etc. provided that /3 < 1. The conclusion (14) is clearly false if a = 00. We may take f(z) = e z , 9 ( r )= (logr)2. Then a = 00 in (12) and hlp(7T) = -00

and

References I. Akhiezer, T h e classical moment problem and some related questions in analysis, Oliver & boyd, Edinburgh, 1965. P. D. Barry, The minimum modulus of small integral and subharmonic functions, Proc. London Math. SOC.,12 (1962), 445 - 495. C. Berg, Indeterminate moment problems and the theory of entire functions, J. Comp. Appl. Math., 65 (1995), 27 - 55. C. Berg and H. L. Pedersen, On the order and type of the entire functions associated with an indeterminate Hamburger moment problem, Ark. Mat., 32 (1994), 1 - 11. C . Berg and H. L. Pedersen, Nevanlinna matrices of entire functions, Math. Nach., 171 (1995), 29 - 52. W. Bergweiler and W. Hayman, Zeros of solutions of a functional equation, Comput. Methods Funct. Theory, 3 (2003), 55 - 78. W. Bergweiler, K . Ishizaki and N. Yanagihara, Growth of meromorphic solutions of some functional equations I, Aequationes Math. 63 (2002), 140

1. N.

2.

3. 4.

5. 6.

7.

-

151.

78 8. N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular variation, Cambridge University Press, Cambridge 1987. 9. R. P. Boas, Entire functions, Academic Press, New York, 1954. 10. T. S. Chihara, Indeterminate symmetric moment problems, Math. Anal. Appl., 85 (1982), 331 -346. 11. J. S. Christiansen, Indeterminate moment problems within the Askey scheme, Ph.D. thesis, Department of Mathematics, University of Copenhagen (2004). 12. G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge 1990, second edition 2004. 13. A. F. Grishin, Uber Funktionen, die im Innern eines Winkels holomorph sind und dort nullte Ordnung haben. (Russian), Teor. Funkts., Funkts. Anal. Prilozh. 1 (1965), 41-56. 14. M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge 2005. 15. M. E. H. Ismail and D. R. Masson, q-Hermite polynomials, biorthogonal rational functions and q-beta integrals, Trans. Amer. Math. SOC. 346 (1994), 63- 116. 16. 0.P. Juneja, G. P. Kapoor and S. K. Bajpai, On the (p,q)-order and the lower ( p ,q)-order of an entire function, J . Reine Angeu. Math., 282 (1976), 53 - 67. 17. 0. P. Juneja, G. P. Kapoor and S. K. Bajpai, On the (p,q)-type and lower (p,q)-type of an entire function, J. Reine Angew. Math., 290 (1977), 180 190. 18. R. Koekoek and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its g-analogue, Report no. 98-17, TU-Delft, 1998. 19. M. G. Krein, On the indeterminate case in the boundary value problem for a Sturm-Liouville equation in the interval (O,oo),Izv. Akad. Nauk SSSR Ser. Mat., 16 (1952) (in Russian). 20. M. G. Krein and A. Nudelman, The Markou moment problem and extremal problems, American Mathematical Society, Providence, Rhode Island, 1977. 21. B. Ya. Levin, Lectures on entire functions American Mathematical Society, Providence, R.I., 1996. 22. B. Ya. Levin, Nullstellenuerteilung ganzer Funktionen Akademie Verlag Berlin, 1962. 23. H. L. Pedersen, The Nevanlinna matrix of entire functions associated with a shifted indeterminate Hamburger moment problem, Math. Scand., 74 (1994), 152 - 160. 24. H. L. Pedersen, Stieltjes moment problems and the Friedrichs extension of a positive definite operator, J . Approx. Theory, 83 (1995), 289 - 307. 25. J.-P. Ramis, About the growth of entire functions solutions of linear algebraic q-difference equations, Ann. Fac. Sci. Toulouse Math., Ser. 6 1 (1992), 53 - 94. 26. J. Shohat and J. Tamarkin, The problem of moments, American Mathematical Society, Providence Rhode Island, rev. ed., 1950. 27. S. N. Shah and M. Ishaq, On the maximum modulus and the coefficients of

79

an entire series, J. Indian Math. SOC.,16 (1952), 172 - 188. 28. G. Valiron, Lectures o n the General Theory of Integral Functions, Chelsea Publishing Company, New York, 1949. C. Berg, Department of Mathematics, University of Copenhagen, Universitetsparken 5, DK-2100, Copenhagen, Denmark. Email: [email protected] Fax: +45 3532 0704 H. L. Pedersen, Department of Natural Sciences, Royal Veterinary and Agricultural University, Thorvaldsensvej 40, DK-1871, Copenhagen, Denmark. Email: [email protected] Fax: +45 3528 2350

W. K. Hayman, Department of Mathematics, Imperial College London, South Kensington campus, London SW7 2AZ, UK. Fax: +44 (0)20 7594 8517

80

A SYSTEM OF BIORTHOGONAL TRIGONOMETRIC POLYNOMIALS

E L ~ A SBERRIOCHOA Universidad de Vigo, Facultad de Ciencias de Ourense, Campus As Lagoas, 32004 Ourense, Spain E-mail: esnaolaQuvigo. es

ALICIA CACHAFEIRO Universidad de Vigo, E T S I Industriales, Campus Lagoas Marcosende, 36310 Vigo, Spain E-mail: acachafeQuvigo.es JOSE GARCI’A-AMOR I. E.S. Torrente Ballester, Pontevedra, Spain E-mail: [email protected] In this paper we present a sequence of bi-orthogonal trigonometric polynomials, in the Szego’s sense, for which we study the usual topics in the orthogonal polynomial (OP) theory. In particular we obtain a new connection with OP on the unit circle and nice properties like: a five term recurrence relation with a Jacobi-type representation, a Favard’s type theorem and a Christoffel-Darboux type formula.

The research was supported by Ministerio de Educacih y Ciencia under grant number MTM2005- 1320. 1. Introduction

The bi-orthogonal systems of trigonometric polynomials is a classical topic introduced by Szeg6 in ‘. They can be considered as a generalization of the real trigonometric system, which has been extensively studied. In Szego studied the principal properties of these polynomials such as the connection with OP on the unit circle, the recurrence relation, properties about zeros and some applications to mechanical quadratures. In the present paper we are going to study a particular bi-orthogonal system with very useful and interesting properties. In a certain sense this is a more natural generalization of the classical trigonometric system. The organization of the paper is the following. In Section 2 we introduce

81

a sequence of bi-orthogonal trigonometric polynomials and we connect this system with a sequence of OP on the circle. We also obtain a five term recurrence relation which generalizes the classical three term recurrence relation satisfied by the O P on the interval. Some properties for the coefficients of this recurrence relation are obtained as well as the relations with the Verblunsky coefficients of the OP on the circle. As a consequence, a Christoffel-Darboux type formula and its confluent form are obtained. Finally, in Section 3 we obtain another interesting recurrence relation and we prove a Favard's type theorem concerning the existence of the measure of orthogonality. 2. A sequence of trigonometric polynomials

Let p be a finite positive Bore1 measure on I = [0,27~] with infinite points of increasing and let us consider the trigonometric system ordered as follows

{I,sine, cos 0, . . . ,sin no,cosn8,. . . }. We can apply the Gram-Schmidt method for obtaining an orthogonal basis {Pn(8)}of trigonometric polynomials, that is, a sequence satisfying

(pn(8), pm(8))fi=

/

2x

Pn(e)pm(e)dp(e)= knSn?rn,with kn

0

z 0, vn.

Throughout this paper we assume that the sequence is normalized in the following way. We want that the leading coefficient of these polynomials in sinno and cosn0 be 2l-" and Po(8) = 1. Hence

P2n-1(6J)= 2 1 - n sinne

+ ldt and P2,(8) = 21-n cosn0 + ldt,

where ldt denotes lower degree terms in the sense of the order. We are also going to consider the corresponding orthonormal sequence { p , ( 0 ) ) defined bY

where A , are the Gram determinants, for the ordered trigonometric system, defined in the usual way (see 2 ) . It is clear that in our conditions the sequences exist and they are unique. Notice that they are a bi-orthogonal sequence of trigonometric polynomials in the Szego's sense, (see 4 ) .

82

If we interpret the measure p as a measure on the unit circle T,we can consider the monic OP sequence related to p (MOPS(p)), { a n ( z ) } , and we can connect the bi-orthogonal trigonometric polynomials with the MOPS@). This is the aim of the following result.

Theorem 2.1. Let p be a finite positive Bore1 measure o n [0,27r] and let { Pn(0)} be the sequence of bi-orthogonal trigonometric polynomials. Let us consider the measure p as a measure o n T and let { a n ( z ) }be the MOPS(p). Then we have for z = ei9

Proof. Take into account that

On the other hand, we have the orthogonality properties

and it is also easy to obtain the expression for the norms. Finally, one can proceed in the same way for obtaining ( 2 ) and the corresponding norms. 0 Now it is clear that the orthonormal polynomials {pn(0)} can be computed

83

Theorem 2.2. Let {p,(13)} be the orthonormal trigonometric sequence. Then there exist three sequences of coeficients {an}n22, {bn}n20 and { c n } , ~ 1 such that f o r n 2 0 cosep,(e) = an+2pn+z(Q) + cn+lpn+l(e) + bnpn(8) + cnpn-l(Q) + anpn-2(41

(5)

1

with initial conditions p-2(8) = 0 , p-l(O) = 0 , po(B) = -7 andp1(8) = ,/-(sine mom2 -mf (sin 0, sine),

-

z),where mo

= (l,l),, ml =

. Moreover, it holds that

(l,sinO), and

+ c:+~+ b i + c i + a:

m2 =

< 1, Vn.

Proof. It is clear that we can write

k=O

k=O

and applying the orthogonality properties we obtain that u,,i = ai,,. Now taking into account the second expression we have u,,i = 0 if n > i 2, which implies

+

Since an-2,, = an,,-2 and un-1,, = an,,-1 we write, in order to simplify the notation, an,,-2 = a,, an,,-1 = c, and a,,, = b,, from which we get (5). Finally, applying (5) and the orthogonality properties we get

Remark 2.1. For each n the above recurrence relation can be written as

84

follows:

cos

e

.. .

.. .

.. .

.. .

.

.

.

.

.. .

.. .

.. .

.. .

............

............

.. .

.. .. .

.. .

.

.

an-1 %-I

f

0

+

.

bn-1

~n

~n

bn

an

0

.. .. .

~

\

It is clear that the matrix of coefficients is a five diagonal symmetric matrix. Like in the classical Szego connection we can relate the Verblunsky coefficients and the parameters of the five term recurrence relation. Next we give the corresponding formulas distinguishing between even and odd terms.

Theorem 2.3. Let {pn(B)} be the orthononnal sequence satisfying relation (5) and let {@,(z)} be the MOPS(p). Then the following relations hold

85

Proof. For computing

un+2

take into account that

In the even case, that is, if n = 2k we identify in ( 5 ) the coefficients of cos(k + l)O obtaining U 2 k + 2 . In the odd case, that is, if n = 2k - 1 we identify in (5) the coefficients of sin(k l)O obtaining uzk+l given by (7). For obtaining c, we use that c, = (cos8p,-1(8),p,(8))P. Taking into account relations (3) and (4) and distinguishing between even and odd terms we obtain:

+

CZn+l

= (coSep2n(e),P2n+l(e))P=

i 8J(1+ W@zn(O)))(l+?-R(@zn+z(O)))II @2n 1

X IlPIl @2n+l IIP

+

( ( z+ Z ) ( z - n @ 2 n ( Z ) z n K ( z - l ) ) ,z - ( n - 1 ) @ 2 n - l ( z )- z n - l G ( z - l ) ) P , from which it follows (11).We proceed in the same way to obtain CZ,, that is,

-i

'2n

((z

= ( c 0 s e ~ 2 ~ - 1 ( e ) ~ ~ 2 ~ ( e=) )8(1 ~ + %(@2n(0))) 11 Q z n

+ ;1) ( z - ( n - 1 ) @ 2 4 ) - z , - l G ( z - l ) ) ,

Z-n@2n(Z)

llP1l

@

X

2,-1

IIP

+znK(z-l))Pr

and applying the properties of the sequence { @ , ( z ) } we obtain (10). Finally, we get b, in the following way

and using the properties of the family { a n ( z ) }we get (8) and (9).

0

86

Corollary 2.1. Let { p n ( 6 ) } be the trigonometric orthogonal polynomial sequence satisfying relation (5) then (i) a, > 0 V n 2 2, b, E R Qn 2 0 and c, E R V n 2 1. (ii) If @,(O) E R, and we denote the cosine polynomial p2,(0) by r,(e), then for n 2 0

Proof. (i) It is immediate from (6), ( 7 ) , (8), (9), (10) and (11). (ii) Using (11) we obtain that the coefficients in relation ( 5 ) are czn+l = C 2 n - 1 = 0. (iii) Proceeding in the same way using (10) we get that the coefficients in relation ( 5 ) are c2, = c2,-2 = 0. 0

Remark 2.2. Relation (12) is the classical three term recurrence relation for O P on the interval and the coefficients {a,} and {b,} are related with the Verblunsky coefficients like in the Szego's relations, (see 5). Theorem 2.4. Let { p n ( 8 ) } be the sequence of trigonometric polynomials satisfying the five term recurrence relation (5) with sequences of coeficients {a,}, {b,} and { c , } . If we know p l ( e ) , @ 1 ( 0 ) , and the above sequences {an},22, {b,},lo and { C ~ } , > I , then we can obtain {@,(O)} for n 2 1. Proof. From p l ( e ) and @ 1 ( 0 )we get ' 3 ( @ 1 ( 0 ) )and 11 @ 1 Itll, and therefore we deduce % ( @ 2 ( 0 ) ) . Next, from c1 given by (11) we obtain S(@2(0). In order t o obtain the two next Verblunsky parameters @ 3 ( O ) and @4(0) we proceed as follows. Using the expressions of bl and b2 given by (9) and (8) we obtain %a3(0) and %€9(O) respectively. Now, taking into account the expressions of a3 ans c3 given by (7) and (11) we deduce R@4(O) and 3@4(0) respectively. If we proceed recursively in the same way, we obtain for every n 2 2, from b2,-1 and b2, given by (9) and (8), X@2,+1(0) and 3@2,+1(0) respectively. Finally, using the expressions of a2,+1 and c2,+1 we get %@2,+2(0) and S@2,+2 (0) respectively. 0

87

Another important topic in the theory of OP is the study of the kernel functions. In the next theorem we obtain a Christoffel-Darboux type formula and the corresponding confluent form. N

Theorem 2.5. Let us denote by KN(0,T) = x k = O p k ( 0 ) P k ( T ) the reproducing kernel for the ba-orthogonal system. T h e n it holds that

and we also have f o r equal arguments the confluent form given by:

Proof. The steps of the proof are the same as in the standard case. First we write the recurrence relation (5) multiplied by pn(.). Next we rewrite the same relation interchanging 0 by T and we substract one from the other. Now we sum these relations from 0 up to N and simplify obtaining (14). To obtain (15) we divide (14) by 0 - T and take limits. 0 Like in Szego’s paper 4 , and following the ideas given there, we can connect the kernels related to the sequences { p n ( 0 ) } and { Q n ( z ) } .

Corollary 2.2. If Kn(O,r) is the n-kernel function associated to the biorthogonal system and Sn(z,y ) as the n-kernel function associated with the

Proof. Take into account that ( y F ) n S 2 n ( ~y), has the reproducing property on z k z p k for IC = 0 , . . . ,n. 0

+

3. Favard’s type theorem

Another interesting recurrence relation can be obtained for the orthonormal trigonometric sequence. Indeed proceeding in the same way as in Theorem

88

2.2 one can obtain the following six term recurrence relations: 2n+2 2n+4 sinep2,(e) = 6k,2npk(e)and s i n e ~ ~ , + ~ (=e ) ~ i ~ , ~ ~ + ~ ~

C

C

k=2n-3

k=2n-1

rlo with 60,l = - > 0 and 6 0 , ~ = 0. (16) rll

Therefore the sequence of bi-orthogonal trigonometric polynomials satisfies relations ( 5 ) and (16). So our aim in this section is to study the inverse problem, obtaining what we call a Favard’s type theorem.

Theorem 3.1. Let { p n ( e ) } be a sequence of trigonometric polynomials satisfying (5) and (16) with their corresponding initial conditions. Then there is a unique measure o n [0,27r] such that {pn(8)} is the corresponding orthonormal sequence. Proof. First we construct a linear functional I, over the trigonometric polynomial space establishing its action over the basis {pn(0)}. We define L(pZ(0)) = 1 and L(pk(0)) = 0 for k 2 1. Using the recurrences (5), (16) and the linear character of L we obtain L(pn(B).l) = 0 Vn 2 1, L(pn(6)sine) = 0 Vn 2 2, and L(pn(0) cos 0)

=0

Vn

2 3.

(17)

Since we are going to proceed by induction, suppose that L(p,(B)cosjO) = O h > 2 j andL(p,(O)sinkB) = O V n > 2 k .

(18)

In order to prove the next step we are going to use the following wellknown trigonometric formulae cos(m 2)8 = 2 cos(m l)e cos 8 - cos me and sin(m 2)e = 2 sin(m 1)QcosO- sinme. So if n > 2 j 2

+

+

+

+ i)e) = 1

L(pn(e)cos(j

+

+

, ( ~ ~ ( e ~ ( 2-~cos(j ~ ~-ji)e)) e ~ =~ ~ e 2

2 ~ ( C2 O and if n

S cos ~ ~e p , ( e ) )

C r=-2

= ~ I , ( c o s ~ ~a r , n p n + r ( e ) ) = 0 ,

(19)

> 2 j +2

L(p,(e>sin(j

+ 1)e) = L ( p , ( 8 ) ( 2 ~ i n j O ~ 0-~sin(j 8 - 1)e)) = 2

C

~ I , ( ~ S ~ ~ ~ O C O= SO 2 ~~(,s(i O n 0) ) ar,npn+r(e)) = 0. r=-2

(20)

Therefore we have proved that

L(pn(B)cosje) = 0 Vn > 2 j and L ( p n ( 8 )sin Ice) = 0 Vn

2 2k,

(21)

89 for all j and all Ic. Hence using the linearity we obtain:

L ( p n ( 8 ) p m ( e ) )= 0 'dn # m. Finally we prove that L is positive definite (see l). By equating the leading - 22 a ~ ~ r ]hence 2~, terms in both sides of ( 5 ) for c0sOp2~-2(e)we have ~ 2 ~ =

+ w)

L(p;,(e)) = ( p z n ( e ) ( v cosne ~ ~ = v ~ ~ L ( cosne) P ~ ~= ( ~ ) V ~ (p2n(e)(2 ~ L cos(n - i)e case - cos(n - 2)e)) = V 2 n ~ ( c 0ePzn(e) s 2 cos(n - i)e) = 2 V 2 n a z n ~ ( p 2 n - z ( cos(n e) - i)e) = L(P2n-2(@)V2n-2cos(n - 1)e) = L(Pzn-2(WV2n-2 cos(n. - 110

+ 1dt)) =

~ ( & - ~ ( e ) )= 1 > 0. In a similar way it is easy t o prove that L(p?jn-l(e)) = 1. Hence our functional L defined in the space of trigonometric polynomials T , which is a dense subspace of C([O,27rl) with the norm (1 llm, is definite positive. Now we prove the continuity of L. If p ( e ) E T , since 11 p fp(8) 2 0 'do E [0,27r] and L is positive definite, then IL(p(0))l 5 L(1) 11 p llm. Hence L : T -+ R is a linear positive definite and bounded functional. Applying the Hahn-Banach theorem we have that there exists a unique linear bounded and positive extension L : C([O,27r1) + IR with the same norm. Finally, applying the Resz representation theorem (see 3, for linear and positive functionals we get that there exists a unique positive measure p such that L(f) = J?f(O)dp(O), 'df E C([O,27r]). Therefore we have

\lo

L ( P n ( % h ( e ) ) = q P n ( Q ) P m ( q ) = J?P,(e)Pm(e)dp(e)

=h n .

0

References 1. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. 2. P. Davis, Interpolation and approximation, Dover, 1975. 3. W. Rudin, Real and Complex analysis Tata McGraw-Hill, 2nd ed., 1979. 4. G. SzegG, O n bi-orthogonal systems of trigonometric polynomials, Magyar Tud. Alcad. Kutat6 Int. Kozl. 8 (1963), 255-273. 5. G. Szego, Orthogonal Polynomials, Amer. Math. SOC.Coll. Pub]., Vol. 23, 4th ed., Amer. Math. SOC.,Providence, RI, 1975.

90

QUASI MONOMIALITY AND LINEARIZATION COEFFICIENTS FOR SHEFFER POLYNOMIAL SETS

HAMZA CHAGGARA Institut Prkparatoire aux Etudes d’Ingknieur d e Monastir, Dkpartement de Prkparation en Math-Physique, 5019 Monastir, TUNISIE. e-mail: hamza. [email protected] n In this paper, we express explicitly the linearization coefficients between three polynomial sets of Sheffer type using their corresponding lowering operators. We obtain some well known results as particular cases. A Crofton-type formula is derived. As application, we give linearization coefficients for Gould-Hopper polynomials as well as some reduction formulae for Kamp6 de FBriet functions. K e y words. Linearization coefficients, Lowering operators, a-Appell polynomials, Sheffer polynomials, Crofton formula, Reduction formulae. 2000 M a t h e m a t i c s S u b j e c t Classification. 33C45, 41A10, 41A58.

1. Introduction Let P be the linear space of polynomials with complex coefficients. A polynomial sequence {P,},>o - in P is called a polynomial set (PS, for shorter) if and only if deg P, = n for all nonnegative integer n. Given three PSs {P,},~o, {Qn},20 and {%},20. The so-called general linearization problem between them asks to find the linearization coeficients Lij(lc) in the expression: i+j

Qi(x)Rj(z)=

C Lij(k)Pk(~).

(1)

k=O

Particular case of this problem, if P, = Qn = %, is the standard linearitation or Clebsh-Gordan-type problem. The computation of the linearization coefficients (LC, for shorter) plays an important role in many situations of pure and applied mathematics and also in physical and quantum chemical applications. In particular, the study of positivity conditions of Lij ( I c ) has received special attentions. This property has many important consequences. It gives rise to a convolution structure associated with the PS {P,},~o 7,28. Further interesting

91

applications were studied by Lasser 2 6 1 who observed a close relationship between the LC associated with orthogonal polynomials and the so-called polynomial hypergroup structures. The literature on this topic is extremely vast and a wide variety of methods, based on specific properties of the involved polynomials, have been devised for computing the LC L i j ( k ) either in closed form or by means of recursive relations (usually in k), exploiting for this purpose several of their specific properties: recurrence relation generating function , orthogonality weights and Rodrigue’s formula inversion formulae 11314 etc. A combinatorial approach to solve the linearization problems was also given in A general method, based on lowering operators 351 was developed in 31,32 to solve connection problems (expanding a polynomial f E P of degree n in terms of polynomials P,; m = 0,1,. . . ,n). This method has been already applied to solve standard linearization problem for basic and orthogonal Sheffer polynomials 33. In this paper, we deeply discuss the case where the involved PSs are of Sheffer type. The method depends on a simple manipulations of formal power series. The obtained LC are generally expressed in terms of terminating hypergeometric series. The outline of the paper is as follows. In Section 2, we give a result for general linearization problem which allows us to express explicitly the general linearization coefficients for Sheffer PSs. Some examples are given, in particular the standard LC for orthogonal Sheffer PSs are summarized in Table 1. In Section 3, we prove a Crofton-type formula and we give the LC for Appell polynomials. As application, we solve the linearization problem for Gould-Hopper polynomials. Particular cases are considered and used to obtain some reduction formulae for Kampk de Fkriet functions which, to the author’s knowledge, seem to be new. 1,2,3,27129,

8)17918124

22123,

4t15.

2. Linearization coefficients 2.1. A general result

Denote by A(-1) the space of operators cr acting on analytic functions that reduce the degree of every polynomial by exactly one and a(1) = 0. Definition 2.1. Let cr E A(-1) and let {P-},yo be a PS. {Pn}n>ois called a a-Appell PS if and only if

a ( P n ) = n P n - l , n = 1 , 2 ,....

(2)

92

Definition 2.2. Let a E A(-'). A PS {Bn},20 is called a sequence of basic polynomials for a if and only if (i) { B n } , ~ 0is a a-Appell PS. (ii) Bn(0)= bo,,, n = 0,1,. . .. In 35, it was shown that every a E A(-') has an unique sequence of basic polynomials.

Theorem 2.1. 35 Let {Pn},20 be a PS. Then there exist a unique a E A(-') and a unique power series A ( t ) = xEoantn,a0 # 0, such that {Pn}n>o is a u-Appell PS and A ( u ) ( B, ) = Pn,

n =O,1,.

. .,

(3)

where {Bn},>0 - is the sequence of basic polynomials f o r a . W e shall call {Pn},>0 - is a a-Appell PS of transfer power series A . An explicit expression of the general LC are given in 33 by use of the expression of the dual sequence of a given a-Appell PS {Pn}nloof transfer power series A 35.

Theorem 2.2. Let a E A(-1) and {Pn},20 be a a-Appell PS of transfer power series A . Then the general L C in (1) are given by 1 i , j = 0,1,. . . , and k = 0,1,. . . ,i+j, (4) L i j ( k ) = -akX(u)(Q&)(0), k! h

where A ( t ) = 1 . 4t) 2.2. Shener polynomials

Recall that a PS {Pn},>o is said t o be of Sheffer type A-zero (Sheffer PS, for shorter,) if and only if it has a generating function of the form 5,12:

where A and C are two formal power series: 00

00

A ( t )=

a0

k=O

It was shown in

#0

and

C ( t )= c c k t k f ' , q,# 0.

(6)

k=O

that a Sheffer PS generated by (5) is a-Appell PS of transfer power series A , where a = C*(D), D = and C* is the inverse of C ; i.e. C * ( C ( t ) = ) t and 34

&

93

C(C*(s))= s.

If C ( t )= t , we have (T = D.That corresponds to Appell polynomial sets 2 1 . Theorem 2.3. Let {P,},Lo,{ Q n } n 2 ~and {Rn}nio be three Sheffer PSs generated respectively by,

A1(t)e"cl(t),A2(t)e"C2(t) and A3(t)exC3(t). T h e n the generalized LC in (1) are given by

Proof. Denote by { B A 1 ) } n i {BA2)},>0 ~, and {BA3)}n20the respectively basic sequences associated to {Pn}nio, {Qn} and ni {Rn}nio. ~ P u t @(t) = = '&kaktn. According to (4)and the well known Leibnitz formula, we have

Next, we apply Theorem 2.3 to some well-known Sheffer PSs. 2.3. Generalized Hennite polynomials

Consider the PS generated by l o G(s,t ) = e-t3ext. The associated LC may be deduced from Theorem 2.3 with A ( t ) = eVt3 and C ( t ) = t. For this case

94

we have

Putting i = p + n , j = q+n and k double sum we obtain

=p + q -

n = i+j

-

3n in the previous

2.4. Modified Laguerre polynomials

The modified Laguerre polynomial set {LE-n}n2~is generated by 00

~ ( z , t=)

C ~ ; - " ( z ) =t ~(1 + t)* exp(-zt).

n=O

For the family { n ! L E - n } n / ~ we, have

+

1 (1 t ) P ( l + S)Y @k(tls)= (t k! ( l + t + s ) *

+ s)k

It follows

due to the double summation in the above expression, we recall the definition of the double hypergeometric series. The KampB de FBriet function FqP,::, a generalization of Appell series, is defined as follows 20,13:

where [apln= n y = l ( a j ) n . ... Then the LC for Modified Laguerre polynomials can be expressed as

2.5. Orthogonal Shefler polynomials

Application of (7) to orthogonal Sheffer PSs provides some identities which we collect in Table 1.

95 Table 1. Linearization coefficients for orthogonal Sheffer polynomial sets. Polynomial sets

Generating functions

Hermite

e-t2 p

t

*k(tr

&eZSt(s

L<j(i+j-k) 5 k 5 min(i,j)

8)

0

+t)k

(A) (i)

2nn!6k,2n

( 3( i l k !

Meixner- ((1 Pollaczeck

(26)k

+ 6t)' + t z ) - g arctan

(s+t+z6st)k k!(l-(l+62)st)V+k

x3F2

(i

(L) (i) k!

- k +q, i- k + l , j

+ j

k-1 - 5 , -2 g

-

k+l

3. Linearization coefficients for Appell PSs 3.1. General expression It was shown in 34 that every PS is quasi-monomial. That is to say, there exist a lowering operator u and a raising operator r , independent of n, such that

u(P,) = nPn-l

and

T(P,) = P,+1,

n = 1 , 2 , . ...

(10)

Lemma 3.1. (Croflon-type operational rule) Let A be a formal power series such that A ( 0 ) # 0 . Then, for every two power series f and g we have

Proof. Consider {P,},>o - a PS generated by A(t)eit = C,"=, Y t " . It was shown in 34 that {P,},~ois quasi-monomial under the action of 0

= D, T = X

+ A'(D)A^(D) and A ( D ) z n= P,.

96

Let f(z) = C,"==, anzn and g(z)= C,"==, bnzn be two formal power series. Then, by Cauchy product and sum manipulation, we have

If A(t) = ehtP,we discover the well known Crofton formula

ehDr(f.g)= f ( X

+ hrDr-')ehDrg.

6119

(12)

Theorem 3.1. Let {Pn},2o, {Qn}n?O and {&}n>o be three Appell PSS of transfer power series Al(t)= C,"==, %tn, A,(t) %tn and A3(t) = C,"=,2-t".Then the linearization coefficients in (1) are given b y

=C,"=o

(13)

or, equivalently,

Proof. According t o (4) and the well-known Leibnitz formula we have, k

&(k)

=

LA^l(D)C

k!

'

D P Q~~ k

I

- R~ p

x=o

C (i) (k - P )A^I(D)(Qi-pRj-k+p)( p=o k

=

(:)

p=o

x=O

That, by virtue of (11) leads t o (13). Now, according t o the fact that {Qn}+0- and {&},>o are two Appell PSs of transfer power series respectively Az(t) and A3(t),we have 31

97

Let A

{pn}nzobe the Appell PS of transfer power series A^l(t). Then we have

pr(x)= C;=o (:)G-rxnThe raising operator of

{pn}is T = X + A^'(D)A(D),so, h

h

rsPn = Pn+,.

(16)

By using (15) and (16) we get,

According t o (13)' we obtain,

Lij(k)=

2 6($(i)

(lc:p))

(3

(~)h-ncj-m~n+m-k~

n=O m=O

r:')

which, in view of the Vandermonde formula

=g(i)(nYk)'

',

x and y being complex numbers,

gives (14). Note that (14) coincides with Carlitz formula

l8

.

3.2. Application t o Gould-Hopper polynomials

Recall that the Gould-Hopper polynomials are generated by

c 00

exp(xt

+ htm) =

n=O

tn

g:(x, h)-, n!

lo

m E W.

So, { g ~ ( x , h ) } , ~iso an Appell PS with transfer power series A ( t ) exp(htm). It is also ( m - 1)-symmetric 1 6 . It follows that the linearization formula (1) is reduced to g y ( x , h 2 ) g y ( x ,h3) =

c

Lij(i

+j

=

- m k ) g z " + j - m k ( x hl). ,

mk
By virtue of (14) we obtain Lij(Z+j-mk) = X

(-hl)k(2 + j ) ! ( 2 + j - mlc)!

c

rm
+

& r

h s

(-1)'+"(i + j - m(r s))! (h*) (hJ s! ' (2 - r m ) ! ( j - sm)!(k- r - s)! r!

98

which can be expressed as

L&+j-mk)

=

(-hl)k(i

' j ) ! Fkg

(i + j - mk)!k!

-k : A(m, 4) A(m, ; -j); A(m, -i - j ) : -; -;

h2

h3

where A(m, A) abbreviates the array of m parameters * , j = 1 , . . . , m. Application of (7) and (19) t o some particular cases of Gould-Hopper polynomials provides some reduction formulae.

Case 1. For m = 1, the Gould-Hopper polynomials are reduced t o g n ( z , h ) = (x h)". So, the associated standard linearization coefficients are trivial. By comparing these coefficients t o (14), we obtain the reduction formula

+

Case 2. For h = -1 and m = 2, we obtain the Hermite polynomials

H,(x)= g:(22, -1). By comparing (19) and the LC in Table 1, we obtain the reduction formula . -i -d.-i,-S; . 2'i+j--I .' _._. 2 2 1,1 = Fi;: -n i+j

(

2 '

2

.

1

'

)

(2

i!j!(i + j - 2n)!2" - n ) ! ( j- n)!(i j ) ! '

+

Case 3. For h = -1 and m = 3, we obtain the PS generated by

G(x,t) = e P t 3 e x t .The associated LC are given by (8). On the other hand, the above coefficients can be derived from (19). It follows

References 1. A. RONVEAUX, M.N. HOUNKONNOU AND s. BELMEHDI, J . Phys. A : Math. Gen. 28, 4423 (1995). 2. A. RONVEAUX, Proc. Inter. Workshop on Orthogonal Polynomials in Mathematical Physics (LeganBs, Madrid, Spain) ( M. Alfano et al., ed.), 1996. 3. C. MARKETT, Constr. Appro%. 10, 317 (1994). 4. D. KIMA N D J. ZENG, Europ. J. Combinatorics 22, 313 (2001). 5. E.D. RAINVILLE, Special Functions, The Macmillan Company, New York, 1960. AND M. CARPANESE, J. Math. Anal. Appl. 227, 98 6. G. DATTOLI,A. TORRE (1997). 7. G . GASPER,Canad. J . Math. 22, 171 (1970).

99 A N D A. LUCHOW,Inter. J . Quantum Chemistry 48, 239 8. H. KLEINDIENST (1993). 9. H.M. SRIVASTAVA A N D H.L. MANOCHA, A treatise on Generating Functions, John Willey and Sons, New York, Chichester, Brisbane, Toronto, 1984. 10. H.W. GOULDA N D A.T. HOPPER,Duke Math. J . 29,51 (1962). 11. I. AREA,E. GODOY,A. RONVEAUX A N D A. ZARZO,J . Comput. Appl. Math. 133,151 (2001). 12. I.M. SHEFFER,Duke Math. J . 5,590 (1939). ~ F ~ R I E C. T , R. Acad. Sci. Paris 173,401 (1921). 13. J. K A M PDE 14. J. SANCHEZ-RUIZ A N D J.S. DEHESA,J . Comput. Appl. Math. 133, 579 (2001). 15. J. ZENG,Proc. London Math. SOC.65,1 (1992). 16. K. DOUAK,J. Comput. Appl. Math. 70,279 (1996). Amer. Math. Monthly 64,723 (1957). 17. L. CARLITZ, 18. L. CARLITZ,Collect. Math. 15,245 (1963). 19. M.W. CROFTON,Quart. J . Math. 16,323 (1879). 20. P. APPELLA N D J. K A M PDE ~ FERIET, Fonctions Hyperge'omktriques et Hypersphkriques: PolynGmes d ' H e m i t e Gauthier-Villars, Paris, 1926. 21. P. APPELL,Ann. Sci. Ecole Norm. Sup. 9,119 (1880). AND SANCHEZ22. P.L. ART&, J.S. DEHESA,A . MART~NEZ-FINKELSHTEIN RUIZ,J . Comput. Appl. Math. 99,15 (1998). 23. R. ALVAREZ-NODARSE, R . J . YANEz A N D J.S. DEHESA,J . Comput. Appl. Math. 89,171 (1998). 24. R. ASKEY,Orthogonal polynomials and Special functions, CBMS Regional Conference Series, SIAM, Philadelphia, 1975. 25. R. ASKEY,Studies in Applied Mathematics 6, Special Functions and Wave Propagation, SIAM, Philadelphia, 64 (1970). 26. R. LASSER,Rend. Math. Anal. 3,185 (1983). 27. R. LASSER,S I A M J. Math. Anal. 14,403 (1983). J . Math. Anal. Appl. 170, 158 (1992). 28. R. SZWARC, A N D A. RONVEAUX, Appl. Math. 24, 445 29. s. BELMEHDI,s. LEWANOWICZ (1995). 30. T.S. CHIHARA,A n introduction to orthogonal polynomials, Gordan and Breach, New York, London, Paris, 1978. 31. Y. BEN CHEIKHAND H. CHAGGARA, J . Comput. Appl. Math. 178,45 (2005). 32. Y. BEN CHEIKH AND H . CHAGGARA, J . Math. Anal. Appl., 319,665 (2006). 33. Y. BEN CHEIKH AND H. CHAGGARA, Int. Jour. Math. & Math. Sci., volume 2006, 1-15. 34. Y. BEN CHEIKH,Appl. Math. Comp. 141, 63 (2003). 35. Y. BEN CHEIKH,Georgian Math. J. 9,413 (2002).

100

d-ORTHOGONAL POLYNOMIAL SETS OF CHEBYSHEV TYPE

Y. BEN CHEIKH* Faculte' des Sciences de Monastir, Dkpartement de Mathkmatiques, 5019 Monastir, TUNISIE. E-mai1:youssef. bencheikhaplanet. tn

N. BEN ROMDHANE Institut Prkparatoire a m Etudes d 'Ingknieur de Monastir, Dkpartement de Pre'paration e n Math-Physique, 5019 Monastir, TUNISIE. E-mail: Neila. BenRomdhane @apeam.rnu. tn

In this paper, we characterize all d-orthogonal polynomial sets generated by funcW

A(t) = Pn(z)tn,where A and C are two formal power l--IC(t) n=O series satisfying A ( 0 ) = C'(0)= 1 and C(0) = 0. We, also, determine the corresponding &dimensional functional vector of the obtained polynomial sets and we produce the d-orthogonal Chebyshev polynomials of the first kind. Key words: Generating functions, d-orthogonal polynomials, &symmetric polynomials, associated polynomials, integral representations. Classification AMS: 42C05 33C45. tions of the type

1. Introduction and preliminary results

Multiple orthogonality is an extension of the concept of standard orthogonality. This notion has many applications in various domains of mathematics as analytic number theory, approximation theory, special functions theory, and spectral theory of operators. Some recent works were focused in the analysis of structural properties A convenient framework to discuss explicit examples of multiple orthogonal polynomials consists of considering a subclass of multiple orthogonal polynomials known as dorthogonal polynomials, and defined as follows. Let P be the linear space of polynomials with complex coefficients and 132,5,7917722y28.

*The corresponding author.

101 let P' be its algebraic dual. A polynomial sequence {P,},~o in P is called a polynomial set ( P S , for shorter) if and only if deg P, = n for n 2 1. We denote by (u, f ) the effect of the linear functional u E P' on the polyxn), n 2 0 the moments of u. nomial f E P and by (u),:= (u, Let {P,},~o be a P S in P. The corresponding dual sequence (u,),?~ is defined by: (u,,P,) = 6,, n, m = 0,1,.. . , 6,,m being the Kronecker symbol. Throughout this paper, we take by convention P-, = 0, n 2 1.

Definition 1.1. 10~20Let uo,u1, ... and ud-1 be d linear functionals and let {P,},?o be a P S in P. {P,},~o is called d-orthogonal polynomial set (d-OPS, for shorter) with respect to U = t(uO,ul,....,u d - 1 ) if it fulfills: m > n d k, n 2 0 , (Uk,P d , ) = 0, ( u k , p71pn d + k ) # 0, n 2 0 , for each k E {0,1, ...,d - 1). For the case d = 1, we meet the well known notion of orthogonality.

{

+

More attention was devoted to d-OPSs having certain additional properties, especially, to generalize some known characterization theorems for orthoge nal polynomials In this paper we investigate some d-OPSs related to Hahn operator 25: f(4x) - fb), IqI # 1, L,(f)(x) = (Q - 1). where f is such that the second member of this equality exists. This operator tends to the derivative operator D when q tends to 1. Let A(-1) be the space of linear operators cr acting on formal power series that reduce the degree of every polynomial by exactly one and cr(1) = 0. For cr E A('-'), a P S {P,},~o is called cr-Appell if cr(P,) = Pn-l, n 2 1. We consider the following problem: ( P ) Let cr E A(-') that commutates with L,. Find all PSs which are at the same t i m e d-OPSs and cr-Appell. Such characterization takes into account the fact that P S s which are obtainable from one another by a linear change of the variable are considered equivalent. This problem was partially solved by many authors. Carlitz and AlSalam treated the case cr = L, and d = l . The obtained result is the q-Al-Salam-Carlitz polynomials as a unique solution. Zaghouani extended this case for d positive integer. The limit case q = 1 and cr commutates with D was treated for d = 1 by Meixner *. The case q = 0 and d = 1 was considered by Ismail 18. The obtained polynomials are given by 4,14915932.

23t24

102

+

P,(z) = &(z) 2aU,-l(z) - cU,-2(2), n 2 0, where {U,},>o - is the Chebyshev PS of the second kind. The purpose of this paper is to extend the Ismail’s characterization to arbitrary positive integer d. We state the following main result. Theorem 1.1. Let a be an operator in A(-’) that commutates with Lo. Then the only d-orthogonal and a-Appell PSs are generated by

+

+

where U ( t ) = 1 a l t * * * + ad+1td+l, v ( t )= 1 + b2t2 + * . . + bd+1td+l and ad+l # bd+l # 0. Moreover P, satisfies

p,(z)

d + alV,”-,(x) +. .. + ad+lU,-d-l(z),

= ~,”(z)

n

2 0,

where {U,},2o is the Chebyshev d-OPS of the second kind 14(see (4)). The PSs in (1) were first encountered by L. Verde-Star l6 for some interpolation problems without involving the notion of d-orthogonality. This paper consists of five sections. Following the introduction and preliminaries necessary for the sequel, Section 2 is devoted to the proof of Theorem 1.1. In Section 3, we give special cases. Then, in Section 4, we state a (d 1)-order recurrence relation satisfied by the obtained polynomials and the d-dimensional functional for which the d-orthogonality holds. As application, we study the Chebyshev d-OPS of the second kind. Finally, in Section 5 , we produce the Chebyshev d-OPS of the first kind. In particular, for d = 2, we express the moments of the Chebyshev 2-OPS by means of integral representations.

+

2. Proof of the main result

To prove Theorem 1.1, we need the following results. Lemma 2.1. Let a be an operator in R-l that commutates with Lo. Then the following statements are equivalent.

where A and C are two formal power series ‘satisfying A(0) = C’(0) = 1 and C(O)= 0.

103

This lemma is a particular case of Corollary 3.2

29

with B ( t ) =

A.

Lemma 2.2. (Proposition 2.1 ') Let {Pn},20 be a PS generated b y

c 00

G ( x , t )=

Pn(x)tn and let X be the multiplicative operator b y x . Then

n=O

there exists an unique operator Zt E V(-') such that X G ( x ,t ) = Z t G ( x ,t ) , where V(-') is the space of linear operators acting on analytic functions that reduce the order of every analytic function by exactly one.

2 is called the transform operator of X Lemma 2.3.

31

Let {P,),lo

according to G ( x ,t ) .

be a polynomial set generated by 03

G ( x ,t ) = A(t)Go(z,t ) =

C P,(x)tn,

with Go(0,t ) = 1,

n=O

and let 2 be the transform of X according to G ( x , t ) . Then: {Pn}n>O is a d-OPS if and only i f

Z E VL;~),

where vs-'), r 2 2, is the space of linear operators r such that there exist r n )n20, k = 0, I , . . . , r - 1 satisfying: ~ & ~ ) a ~ #-0,' ) complex sequences ( a(k)

Proof of Theorem 1.1. According to Lemma 2.1, { P n } n 2 is ~ generated by

G ( x , t )=

Then

A(t)

1- x C ( t )l

with G ( x ,0 ) = 1.

, xG(x,t ) = G(X1t ) - A ( t ) G ( x 0) C(t)

It follows from Lemma 2.2 that the transform operator to G ( z , t )is given by

Z(1)=

1 - A( t )

C(t)

Xf =

2,of X

according

- A ( t ) f ' o ) .In particular, we have

C(t)

h

= q(t), X ( t )=

t t" c(t> = z ( t ) and X ( t " ) = A

C(t).

(2)

By virtue of Lemma 2.3, we deduce that { P n } n 2 is ~ a d-OPS if and only if q is a polynomial of degree d and z is a polynomial of degree ( d 1) with z(0) # 0. Finally, we have t z(t>- Q(t) A ( t ) = Z(t) - t q ( t )

+

44

z ( t ) - xt .

3. Special cases

In this section, we give some well known d-OPSs of type (1).

0

104

3.1. Case d=1.

The corresponding polynomials are given by Ismail 18. In particular, we have the Chebyshev polynomials of the first and the second kind, and the Allaway polynomials {Pn(x; S , T ) } { , > O-} 27. 3.2.

Case d=2.

Example 1. Chebyshev 2-OPS of the first kind 15. T h e associated sequence {P$?'}n2~ of { P n } n 2 with ~ respect to uo, the first vector of the dual sequence of {Pn}n>o, is defined by 21:

Pil)(x)=< u o , P n + l ( Z )

- Pn+1(6) >,

X-€

> 0.

Douak and Maroni determined all the 2-OPSs {Pn}n20 satisfying Pi1) = A P ; + , , n = 0, 1 , 2 , . . .. The obtained polynomials are generated n+ 1 bY 00 1- at2 - 2bt3 = Pn(z)tn, b # 0. (3) 1 at2 bt3 - xt n=O

+

C

+

These polynomials are particular case of (1). If we put a = 0 in (3), we obtain the Chebyshev 2-OPS of the first kind 15. Example 2. Chebyshev PS of the first kind in two variables. Ricci l9 generalized the Chebyshev PS of the first kind by considering polyu)},>o- generated by nomials in two variables {Tn(x, 00 1- ut2 2t3 = Tn(z, uy. 1 ut2 - t 3 - xt n=O If we consider u as a parameter, {Tn( ., U ) } ~ ? Ois a particular case of (3). +

3.3. Case d

+

+

c

21.

Example 1. Chebyshev d-OPS of the second kind 14. A d-OPS { P n } n 2 ~is called d-classical according t o Hahn property 26 if { Q n } n 2is ~ also a d-OPS, where Qn(x)= $PA+l(x). Douak and Maroni characterized all the d-classical and self-associated polynomials, Pi1) = p,, n 0, by solving a (d 1)-order recurrence relation. The obtained d-OPSs are particular case of (1). They are generated by

>

In particular, for second kind.

b2 =

. . . = bd

+

= 0 , we meet the Chebyshev d-OPS of the

105

Example 2. Chebyshev PS of the second kind in several variables 1 9 . Rcci generalized the Chebyshev PS of the second kind by considering polynomials of several variables {U$-''(ul, ...,U , - ~ ) } ~ ? Ogenerated by 00

If we consider u2,. . . andu,-l as parameters, we obtain a particular case of (4)corresponding to a ( r - 1)-OPS. 4. General properties

+

First, we give a (d 1)-order recurrence relation satisfied by the PSs in (1). Then we determine the d-dimensional functional for which the dorthogonality holds. As application, we apply the obtained result to the Chebyshev d-OPS of the second kind. 4.1. A ( d

+ 1)-order recurrence relation

Proposition 4.1. The PSs {Pn},>o given by ( 1 ) satisfy the following (d 1)-order recurrence relation:

+

d

Proof. By using ( 1 ) and (2), we have X ( l ) = x ( b k + l - ak+l)tk k=O

and X ( t n )

=

d+l 03 t n - ' ( x b i t i ) , n 2 1 . Then we substitute x P n ( x ) t n in n=O

i=O

xG(x,t ) = X t G ( x ,t ) and by identification we get the desired result.

+

Remark 4.1. This result generalizes the (d 1)-order recurrence relation satisfied by the Chebyshev d-OPS of the second kind l 4 and the third-order recurrence relation satisfied by the Chebyshev 2-OPS of the first15.

106

4.2. d -dimensional finctional In this subsection, we express explicitly the d-dimensional functional for which we have the d-orthogonality of the obtained d-OPSs (1). We state the following theorem by employing a method used in 30 for obtaining the dual sequence of a given PS.

Theorem 4.1. The PSs given by (1) are d-OPSs with respect to the ddimensional functional U = t ( u ~. .,. ,ud-l)defined, f o r f E P , by (uT,f) = r!

'0,T

+

Dn-1

trv

7 [( ,)'un]t=o

Lg (f)(o),

= 0, . . . 7

- 1.

I n particular, the moments are given by

To prove this theorem, we need the following lemmas.

Lemma 4.1.

30

Let { P n } n 2 ~be a d-OPS generated b y m

C

~(t)~(z= ~ ( t )pn(z)tn, )

(7)

n=O

where A, B and C are three formal power series such that A(O)C'(O)# 0 and C(0) = 0. Then the d-dimensional functional U = t ( u ~ l .. ,ud-l) . is given by 1 (C')' (UTlf)=-

[ 7 ( L 0 ) ( f ) ( t ) ]

I

f EP,

r=0717"'7d-11

r! A(C ) t=O where C* is the inverse of C , i.e C*(C(t))= t and C ( C * ( t )= ) t. Lemma 4.2. [Lagrange's expansion] Let f and $ be two functions of z analytic about the origin such that 4(0) # 0 , and let z = w$(z). Then

Proof of Theorem 4.1. Notice first that the PSs {Pn},>0 given by (1) are particular cases of (7) if we take A(t) =

44

-,

C(t)

=

t - and

v(t) 1 B ( t ) = -, Then, in order to determine the development of N(t) 1-t ZT in series, we use Lemma 4.2 with f ( z ) = - and $ ( z ) = v ( z ) , and we 4 2 )

replace, respectively, w and z by t and C * ( t )since C * ( t )= t v ( C * ( t ) ) . We get

107

A(C*1

c 7[(TI] O0

(C*>'

(t>= 60,r +

gn-1

trv

t n

tn.

2,

n=1

t=O

Finally, we use Lemma 4.1 to obtain the desired result. Next, we apply Theorem 4.1 to a particular case of (4). That is the Chebyshev d-OPS of the second kind {Un(.,d)}n>~ generated by l 4 m 1 = U,(Z,d ) t n , b # 0. 1 btd+l - xt n=O These polynomials were also studied in 3 .

C

+

Corollary 4.1. The moments of

{Un(.,d)}n>O

are given by

Proof. The polynomial set {Un(.,d)},>o- is a particular case of (1) with u(t)= 1 and v ( t ) = 1 b t d f l . Then, according to Theorem 4.1, we have l n ( 2 ~ = ~ r!n! ) ~ k=O (~)b*Dn-l[rtk((d+')+r-'+b(d+r+l)t(k+l)(d+l)fr-i L O .

+

It follows that ( u r ) n =

r!n

+ (d + r + 1)-n + l

Finally, by a simple calculus, we obtain the desired result.

0

For the particular case d = 2 and b = $, we have the following.

Corollary 4.2. The moments of the 2-dimensional functional U = t ( ~ ou1) , associated to {Un(., 2)},>0 - have the following integral representations

{

'2 + d D ) a 2LT11[(1 + J~_T;.T~)?

(u0)3p+i = 60,i(u1)3p+i = d1,i-

11[(1

- (1 - ~

D)*

- (1 - ~ ~ ~ 3 ) ~ 1 ~ 3

where i = 0 , 1 , 2 . Proof. According to (8) with d = 2, r = 0 and b = 4, -, (3p)! we have ( u o ) ~- ( - ) p Then, using the fact that - 27 p!(2p+ l)!' 2n (3p)! 1 1 2 2 ( 3 ~ 2) - 3(2p r ( p + - ) r ( p + -) = -and -3 3 fip!33p 2p+ 1 2p+ 1 we get

+

+ 1) 7

108

and under a linear transformation of the independent variable t, we find

11(1

that ( u o )=~'~d 3 - t2)p[(l+ t ) i - (1- t ) i ]t(1 - t 2 ) - g d t . 27r Finally, using the transformation t = d-, we obtain the first result. Similarly, by using = 2 ( 3 p + ~ ! , ~ $ 2 p + 2 ) , we obtain the second one.

&

Notice here that this result was first given by Douak and Maroni l4 where the authors used an approach based on the fact that the considered 2-OPS are 2-classical.

5. Chebyshev d-OPS of the first kind The Chebyshev PS of the first kind is the unique PS of type (l),orthogonal, symmetric and satisfying PL') = &PA+,. We start from these properties to produce the d-OPS of Chebyshev type of the first kind. A sequence {Pn},>0 - is called d-symmetric if it fulfills Pn(WkZ) = wzPn(z), where wk = exp(2ik7r/(d l)), n 2 0, k = 0 , 1 , 2 , . . .. When d = 1, the sequence {Pn},20is symmetric. Next, we determine all the d-symmetric PSs among those obtained in Theorem 1.1 and that satisfy Pi" = Qn, where Qn = &PA+,. Then we have

+

21332

Theorem 5.1. Let {Pn}n2~ be a PS given by (1). T h e n {Pn}ni~ is dsymmetric and satisfies PA'' = Q n if and only if it is generated by

c cy)

1 - dbtdS1 = Pn(x)tn,b # 0. 1 btd+l - xt n=O

+

(9)

To prove this result, we need the following Lemmas cy)

Lemma 5.1.

32

Let {Pn},20 be a PS generated by G ( x , t )= X P n ( z ) t n . n=O

T h e n {Pn}n2~ is d-symmetric if and only if G(Wkx,t ) = G(x,W k t ) , k 2 0.

109

Lemma 5.2.

Let {P,},Lo be a d-OPS. Then it satisfies

21914

d- 1 d-l-up

Po@) = 1, Pn(z) = (z - Pn-l)Pn-l@) - &-l-u

n-u-2(z),

n 2 1,

V=O

with 72 # 0. Moreover {Pil)}n>~ - is also a d-OPS and satisfies u=o

Proof of Theorem 5.1. From Lemma 5.1, {P,},~Ois d-symmetric if and 00 1 atd+l = Pn(z)tn, a # b # 0. only if it is generated by 1 b t d + l - zt n=O

+

+

1

Now, we determine all the sets {P,},~Owhich satisfy Pi1)= Qn. From (5), we deduce that {P,},~o satisfies Pn+1(2)= Xn+', 0 5 12 < d, Pd+1(2) = zPd(z) ( a - b)Po(z), Pn+d+l(z) = zPn+d(z) - bPn(z), n 2 1. By using Lemma 5.2, and since Pi1)= Qn, we have Qn+l(z) = zn+', 0 I n < d, Qn+d+l(Z) = zQn+d(z) - bQn(z), 71.20. Differentiating Pd+2 = zPd+l - bP1 and using PA+l = ( n l)Qn and Qd+l = xQd - bQo, we deduce that a = -db, which finishes the proof.0

+

+

The PS given by (9) is a natural extension of the Chebyshev PS of the first kind. In fact, when d = 1, we meet the Chebyshev OPS {T,},~o and when d = 2, the Chebyshev 2-OPS of the first kind 15. So, we call them Chebyshev d-OPS of the first kind and we denote them by (T,(.,d)},>o.Next, we deal with the corresponding d-dimensional functional.

Proposition 5.1. The Chebyshev d-OPS of the first kind generated by (9) has the following moments: 1

( U T = ~

(d

-&,iP( r!

+ 1)p + r ), n = (d+l)p+i,

i = 0,. . . ,d, r

= 0 , . . . , d-1.

(10)

Proof. The PS {Tn(.,d)}n>o - is a particular case of (1) with u(t) = 1 - dbtd+' and v(t) = 1 btd+'. Then, according t o Theorem 4.1, we have (u,),= %, n 2 1, where ( $)Iun = w,tp. By developing ( Q ) ' u " , we deduce that, for m = [p

+

+ 21,

110

Then, using a maple program, we obtain t h a t

d+r+2

bP

(ur)n= J r , i z

r'+ d+2 dr+d2 '+d 'f +r+r r3+3r2+4~d+4~dr+d2prr2~+3rp+~2pd~2d~+2pd+P2dzp2+dp~+d3p2

{

(d+4%2+7-(

(d+l)p+2+r P

d+l

P

) -k (. + (d;l)p-!l+r

)((d+l);+l+r)}.

Finally, by a simple calculus, we get the desired result. For the particular case d = 2 and b =

0

A, we have the following.

Corollary 5.1. The moments of the %dimensional functional U = t ( u ~u1) , associated t o {Tn(.,2)},3~ have the following integral representations

+ z3))-;[(l+ J

+ + (1- d m ) g ] z 3 P f 1 d z ,

(u0)n = 60,i-

(1 - z3))-G[(l JlX);(1 - J m ) ; ] z 3 p d z ,

(u1)n = 61,z-

(1 -

where n = 3 p

s ) g

+ i, i = 0,1,2.

This result was first given by Douak and Maroni l 5 where the authors used an approach based on the fact that the considered 2-OPSs are 2-classical. A (d 1)-order differential equation satisfied by the Chebyshev d-OPSs of the first and the second kind will be given in a forthcoming work.

+

References 1. A.I. Aptekarev, F. Marcellan and I.A. Rocha, J . Approx. Theory 90, 117 (1997). 2. A.I. Aptekarev, J . Comput. Appl. Math. 99,423 (1998). 3. A.I. Aptekarev, V. Kaliaguine, and J. Van Iseghem, Constr. Approx. 16,487

(2000). 4. A. Zaghouani, Georgian Math. J., to appear. 5. E.M. Nikishin and V.N. Sorokin, Trans. Math. Monographs 92.Amer. Math. SOC.Providence RI. 1991. 6. H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, John Willey and Sons, New York, Chichester, Brisbane, Toronto (1984). 7. J. Arvesii, J. Coussement, and W. Van Assche, J . Comput. Appl. Math. 153, 19 (2003). 8. J. Meixner, J. London Math. SOC.9,6 (1934). 9. J.M. Freeman, Congresseus Numerantium 48, 115 (1985). 10. J. Van Iseghem, Jr Comput. Appl. Math. 19,141 (1987) . 11. K. Douak, J . Comput. Appl. Math. 70,279 (1996). 12. K. Douak and P. Maroni, Analysis 12,71 (1992). 13. K. Douak and P. Maroni, J . Approx. Theory 82,177 (1995).

111 14. K. Douak and P. Maroni, Appl. Num. Math. 24, 23 (1997). 15. K. Douak and P. Maroni, Meth. Appl. Anal. 4, 404 (1997). 16. L. Verde-Star, Stud. Appl. Math. 88 153 (1993). 17. M.G. De Bruin, Lecture Notes in Math. 1171.Springer Verlag, 74 (1985). 18. M. Ismail, Bull. Inst. Politdin Iasi 20, 45 (1974). 19. P.E. Ricci, Rendiconti d i Matemtica 11,295 (1978). 20. P. Maroni, Ann. Fac. Sci. Toulouse 10, 105 (1989). 21. P. Maroni, Numer. Algorithms 3, 299 (1992). 22. V.A. Kalyagin and A. Ronveaux, J. Comput. Appl. Math. 67,207 (1996). 23. W.A. Al-Salam, Ann. Math. Pura App. 77,31 (1967). 24. W.A. Al-Salam and L.Carlitz, Math. Nachr. 30,47 (1965). 25. W. Hahn, Math. Nachr. 2, 4 (1949). 26. W. Hahn, Math. Zeit. 39, 634 (1935). 27. W.R. Allaway, Ph.D. Thesis, University of Alberta, Edmonton, (1972). 28. W. Van Assche and S.B. Yakubovich, Integ. Transf. Spec. Funct. 9, 229 (2000). 29. Y. Ben Cheikh, Appl. Math. Comput. 141,63 (2003). 30. Y. Ben Cheikh, Georgian Math. J. 9, 413 (2002). 31. Y. Ben Cheikh and A. Zaghouani, J . Comput. Appl. Math., to appear. 32. Y. Ben Cheikh and K . Douak, Bull. Belg. Math. SOC.7,107 (2000).

112

ON TWO PROBLEMS IN LACUNARY POLYNOMIAL INTERPOLATION

M.G. DE BRUIN Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 G A Delft, The Netherlands E-mail: [email protected] In Hermite interpolation the problem of finding a polynomial P of degree at most (q l ) n - 1 such that

+

4

and given-painvise has a unique solution (is regular) for arbitrary data distinct-interpolation nodes zi. When the orders of derivatives j are replaced by kj with 0 = ko < kl < . . . < k , (so called Henite-Birkhoff or lacunary interpolation), a general regularity property a s above does not hold. This paper will address two types of lacunary interpolation: there is one jump in the orders of derivatives only, either in the first (the case of (0, m,m+ 1,.. . ,m f r ) interpolation) or in the last entry (the case of (0,1,. . . ,r , r m ) interpolation).

+

1. Introduction

Interpolation with polynomials has been studied for quite a long time and the problem can be formulated in its most “general” form as given below. Let q 2 0 be a fixed integer; for each j with 0 5 j 5 q there are given (1) integers nj 2 1 and integers rnj with 0 = rno < ml < . . . < rnp, (2) distinct complex numbers (nodes) zf) (1 5 i 5 n j ) , (3) complex numbers (data) (1 5 i 5 n j ) .

4

Then find a polynomial P E I I N - ~N , degree 5 d ) satisfying

=

C4=onj

(IId:

polynomials with

If the solution is unique for arbitrary sets of data, the problem (1) is called regular; this is equivalent to the homogeneous problem (1) (put = 0 for

4

113

all i , j ) having the trivial solution P = 0 only. The main reference is the book Ref. 3; for a review paper see Ref. 4. The case q = 0 is Lagrange interpolation and the case mj = j , nj = n and z y ) = zi is usually referred to as Hermite interpolation. If in the case of “fixed” nodes for each order of derivative the mj are non-consecutive we have Hermite-Birlchoff interpolation. The situation where the nodes are different for different orders of derivative is called Pd-type interpolation (this type of problems originates in Ref. 1, where q = 1, mo = 0, ml = 1 and the nodes are connected with Legendre polynomials). Although there are rather general theorems on necessary and sufficient conditions for regularity, there are very few cases known where lacunary interpolation is regular. The most famous case is when the nodes are the n-th roots of unity (generated by the roots of zn - 1 = 0): under the simple conditions mo = 0, mj 5 n j problem (1) is regular (see Ref. 2). Recently (see Ref. 5) it was shown that for sufficiently large n the general lacunary PB1-type (0, ml, . . . ,m,) interpolation problem (0 = mo < ml < . . . < m,) is regular for nodes for the mjth derivative generated by the < aq. roots of zn -a: with 0 < a0 < a1 < In this paper two Hermite-Birkhoff interpolation problems will be studied where the “gap” is either between mo and ml or between m,-l and m,. The layout of the paper is as follows: in Secs. 2, 3 the two types of problems will be introduced and regularity results will be given, followed by the proofs in Secs. 4, 5. Finally a short list of references is given. 2. The case of ( 0 , 1,

...,T , T + m ) interpolation

Let in this section n 2 2 be an integer and consider n distinct complex interpolation nodes z l , . . . ,z,. Furthermore, r 2 0 and m >_ 1 are integers, c( (1 5 i 5 n, 0 5 j 5 r ) , di (1 5 i 5 n) are arbitrary complex data. Consider the interpolation problem of finding P E l - I ~ - l , N = ( r 2)” with

+

i

P(j)(Zi)=

4

P(r+m)(zi)= di

( I l i l n ,O<jsr), (1 5 i 5 n). ‘

(2)

Then we have the following results.

Theorem 2.1. For any n 2 2, r 2 0 , m 2 1, a necessary condition for regularity of problem ( 2 ) is

m + r 5 ( r + 1)n.

(3)

114

Theorem 2.2. For any n 2 2, r 2 0, problem (2) i s regular for m = 1 and m

+ r = ( r + 1)n.

(4)

Theorem 2.3. For n = 2, r 2 0 , problem ( 2 ) i s regular for 15m5r+2.

(5)

Remark 1. A result as in Theorem 2.3 for n 2 3 can only be formulated under extra conditions on the location of the interpolation nodes; conditions that are satisfied in the case of the roots of unity. For instance, adding an extra m-value to the result in (4)for n = 3 leads to: Theorem 2.4. For n = 3, r 1 0, problem ( 2 ) i s regular for m = 2 and m = 2r

+ 2 (i.e. m + r = n(r + 1 )

-

1)

(6)

if and only if

Remark 2. For n = 3, the Theorems 2.1, 2.2 and 2.4 lead to: necessary for regularity: 1 5 m 5 2r 3, regular for: m = 1, m = 2r 3, regular for: m = 2, m = 2r 2 under condition (7). This shows that the case r = 0 is completely covered by the previous theorems, while for r = 1 the (0,1,3) interpolation problern remains to be settled. Straightforward calculation of the determinant of the 9 x 9 system of linear equations for the coefficients of the interpolation polynomial again leads to condition (7):

+ +

+

Increasing the value of r again, we find extra conditions to ensure regularity for the full range of m-values [ l ,2r 31.

+

Remark 3. Increasing the number n of interpolation nodes also leads to an increasing number of conditions: they all appear in the form of homogeneous polynomials in the variables 2 1 , . . . , z, having to differ from zero.

115

3. The case of (O,m,m

+ 1 , . . . ,m +

T)

interpolation

Again n L 2 is an integer and the n complex interpolation nodes 21,. . . , z, are distinct. Furthermore, r 2 0 and m 2 1 are integers, (1 5 i 5 n, 0 5 j 5 r ) , di (1 5 i 5 n) are arbitrary complex data. Consider the interpolation problem of finding P E I I N - ~N , = ( r 2). with

4

+

i

(1 I i In), (1 5 i 5 n, o 5 j 5 r ) .

P ( z ~=) di

4

~ ( ~ + j ) (= z i )

(9)

Then we have the following results. Theorem 3.1. For a n y n regularity of problem (9) is

2

2, r

2

0, m 2 1, a necessary condition for

15m5n.

2 0, problem

Theorem 3.2. For a n y n 2 2, r

m

=

(10) (9) is regular for

1 and m = n.

(11)

Corollary 3.1. For n = 2, r 2 0, the problem (9) is regular for

l<m52.

(12)

Remark 4. For n = 3 the bounds on m are according to (10): 1 5 m 5 3, and for m = 1 and m = 3 the problem is regular for all r (Theorem 3.2), which leaves the case m = 2 to study. Introduce

+ + 23;

+~21.232 + 2 2 2 3 ; 03 := ~ 1 ~ 2 . ~ 3 . (13) Then we have regularity for the (0,2, . . . , 2 + r ) problem if and only if 01

:= 21

r = 0 : 3zi # r = 1: ti #

22

01

01

20:

:= ~

1

(1 5 i 5 3),

(1 5 i 5 3) and

30: - 27Otff2

r = 2 : zi

02

+ 68020;

-

29023

+ 52O:O3

-

234OlC2U3

+ 3510;

# 0,

# ~1 (1 5 i 5 3) and -

180f02

+ 49~20; - 34023 + 20O:O3

- 90CiU203

+ 1350;

#0

(14) From these conditions it is clear that the third roots of unity (generated by z3 - 1 = 0) lead to a regular problem as 01 = ~2 = 0, 03 = 1.

116 4. Proofs for Sec. 2

In this section the proofs for Theorems 2.1, 2.2, 2.3 and 2.4 will be given. Let the interpolation polynomial be given by N-1

k=O

+ 2 ) n and P satisfies (2) with homogeneous conditions

where N = ( r

P ( j ) ( z i )= 0 (1 5 i 5 n; 0 5 j 5 T and j = T

+ m).

(16)

Introduce the descending factorials [c]O:=1;

+

(17)

(->I),

[C]k:=C(C-l)"'(C--+l)

+

and assume that m T 2 ( r 1). = N - n,then the conditions (16) lead t o an N x N system of linear homogeneous equations for the unknown coefficients a k with matrix

where B is the ( N - n) x ( N - n) matrix that arises from ( 0 , 1 , .. . , r ) Hermite interpolation problem on the nodes zi, U is the n x ( N - n)matrix

Z,N-n+l

2 " -:

"- n]p,N--n"- n + l]lz;-n+l U=

"- n11.2;-n

"- n + 111z,N-n+l

"

"

-

n1,zy-n

-

n + 11,z;-n+1

... . .. *.

.

. ..

2,N-l

"- 1l1z,N-l "- 1112;-1 "- l],Z;-l

0 is an n x ( N - n) zero-matrix and C is the n x n matrix

'

(19)

117

After these preliminaries, we have: Proof of Theorems 2.1 and 2.2 Applying Laplace expansion on (18) using the rows containing 0 and C , we find det M

= det B

det C.

(21)

F o r m + r > ( r + l ) n + l =N-n+l,thefirstcolumnofCfrom(20)isthe zero column: det C = 0 and the problem is not regular, proving Theorem 2.1. For m T = (T 1)n = N - n, C reduces to the Vandermonde matrix on the n points zi, thus d e t C # 0. Because the Hermite interpolation problem connected with the matrix B is regular, det B # 0; with (21) we find det M # 0: the problem is regular. Finally, the case m = 1 leads to ordinary (0'1,. . . , T 1) Hermite interpolation: a regular problem. This proves Theorem 2.2.

+

+

+

For the other theorems in Sec. 3 a different approach is needed. Put

n n

A ( z )=

(Z -

i=l then the first set of conditions for P in (16) can be satisfied using

The final condition from (16) the leads to

Introduce the polynomials

satisfying

Aj(zi)=

2Z.i A'(zi) # 0

i =j .

Using Leibniz' rule the conditions (24) can be rewritten as

118

Proof of Theorem 2.3 Without loss of generality we can use m L 2 (the case m = 1 has been treated in Theorem 2.2). P u t n = 2, then

where ( j ,i) is either (1,2) or ( 2 , l ) . For n = 2 we have

Q ( z )=

Q:

+ Pz,

(29)

and the e values that contribute to the sum (27) reduce t o m-2 5 f2 5 m-1. In view of (28), we have to distinguish two cases: m - 1 5 r 1 and m - 1 > r 1, i.e. m 5 r 2 and m 2 r 3.

+

+

+

+

+

The case m 5 r 2. The sum in (27) consists of two terms only and inserting (28) we find after dividing out a factor[r lIm-2(zj - zi)r+l # O

+

r r

(

+ 3 - m ( r + 3 - m ) z l + ( m- l ) ( z -~ z2) + 3 - m (r + 3 - m ) z 2 ( m l)(zl - z2)) -

-

(;)

=

(4

The determinant of the matrix is (r+3-m){(r+3-m)+2(m-l)}(z~-z~) 0, leading t o regularity.

+

The case m 2 r 3. Now (27) either leads to one condition only for a, P ( m = r no conditions a t all ( m 2 r 4): the problem is not regular.

+

(30)

*

#

+ 3 ) or to 0

Proof of Theorem 2.4 For n = 3, the form of Q is

Q ( z )= a

+ Pz + y z 2 ,

(31)

and the l values that contribute t o the sum (27) reduce t o max(0, m - 3 ) 5 elm-1. The formulae for the Aj ( z ) ~ and + ~ the values of their derivatives a t the zi can again be given explicitly; because these values follow from each other by cyclic permutation of the indices, one set will be exhibited only. The case m = 2. For r 2 0: A1(z)r+lIZ1= (

~1 z ~ ) ' + ~ (z I23)r+l

DA1(z)'fllzl = ( T

+l)(zl

-

~ 2 ) " ( ~-1 Z3)T(2Z1- 2 2

(32) -

23)

119

The equations for the coefficients of Q from (31) are

where the clj follow for j = 1 , 2 , 3 from (32) after division by the non-zero factor A j ( ~ j ) ~ :

cjl = (r

+ l)Aj(zj); cj2 = Aj(zj) + (r + 1)zjA;zj); cj3 = 2zjAj(zj) + (r + l)z;AI(zj)

(34)

Explicitly calculating the determinant with a computer algebra system (e.g. MAPLE 9@),the value appears to be a non-zero constant multiplied by powers of the differences (zi - z j ) and the factor

Thus the problem is regular for m = 2 if and only if this factor (35) is different from zero: i.e. (7) has t o hold.

+

The case m = 2r 2. Starting point is (27), with summation running from max{m - 3,0} t o m - 1 because of (31). As r = 0 is covered above, we can assume r 2 1, thus m - 3 = 2r - 1 2 1 and we now need [I =

D2r-1A1(z)r+11z1 = [2r

+ [2r]2r-l{ (r + 1) (z,”+

El

=

+ 2]2r-1z;

2,”)

’D2rf1Al(z)T+llzl= [2r

-

[2r

+ 1 ] 2 r - - 1 ( ~+ l)(z2 + z3)z;

+ (r + 1)2z2z3}z1

+ 2]zr+1z1 - [2r + 1]2~+1(7-+ l)(z2 + z3),

(36) and the quantities &,vj resp. [ j that arise from (36) replacing the indices (1,2,3) by the permutation (2,3,1) resp. (3,1,2) on the right hand side. This time we find a 3 x 3 system as (33), but now with d i j (after dividing

120

out common factors

djl

# 0) in the matrix given by

+ (2r + 1 ) ~ j ; = + 2(2r +

= [ j ; dj2 = [ j ~ j

4 3

r1.j"

1)VjZj

+ 2r(2r + 1)Cj.

Using MAPLE 9@, we again find condition (7) for regularity.

(37)

0

Proof of (8) The conditions (16) give a 9 x 9 system of linear homogeneous equations for the coefficients of the interpolating polynomial (15), with matrix of coefficients { a i j }given by aij = zt-' a2+3,j

ai+S,j

(1 5 2 5 3; 1 5 j 5 9), j-1

5 2 5 3; 1 5 j 5 9), = [ j - 1 ] 3 ~ ; - ~(1 I i 5 3; 1 5 j 5 9).

= [j - l]lZi

(1

(38)

The first entry of the rows 1, 2 and 3 are zero, just as the first, second and third entry of rows 7, 8 and 9. With the aid of Maple 9@ condition (8) follows. 5. Proofs for Sec. 3 Using the same method as in the first part of Sec. 4 to find the matrix (18) and assuming m 2 n, we now find the matrix of coefficients t o be

with V the n x n Vandermonde matrix on the points zi,U is an n x ( N - n ) matrix 2s"

,I"+' . . . p

- 1

(40)

0 is an ( N - n) x n zero matrix and L is an ( N - n ) x ( N - n ) matrix with elements & given by Pi+3k,j

+ j l r n + k Z in + j - m - k

= [n

(1 5 i 5 3; 0 5 k 5 r ; 1 5 j 5 N ) . (41)

Proof of Theorems 3.1 and 3.2 Applying Laplace expansion on (39) using the columns containing V and 0, we find det M = det V det L.

(42)

121

For m 2 n + l , the first column of L from (41) is the zero column: det L = 0 and the problem is not regular, proving Theorem 3.1. For m = n, L reduces t o a matrix with elements ~ i given j by yi+3k,j =

+

[n

j-1-k j]n+kZi

(1 I i

I 3; O 5 k I r ; 1 I j L N ) .

(43)

+

Thus the first, second, . . . , ( r 1)-rst 3-tuple of rows starts with 0, 1, . . . , r columns of zeros. This is exactly the matrix for the (0,1,. . . ,r ) Hermite problem where the unknown interpolation polynomial is introduced in the “manipulated” form

c

N-n-1

P*(Z)=

+

[n

(44)

k]kbkZk,

k=O

instead of using b k for coefficients. This does not interfere with regularity of the problem, because the multiplicators [n k ] k are all different from zero. That this manipulated, but regular, Hermite problem has indeed the same matrix as that with coefficients (43), follows from

+

k =j

+

+

and [n k ] k [ k ] j = [TI k ] k + j . Thus d e t M = d e t V det L # 0 and the problem is regular. Finally, the case m = 1 leads t o ordinary (0,1, . . . ,r 1) Hermite interpolation: a regular problem. This proves Theorem 3.2. 0

+

Proof of Remark 3 Calculate the determinants for (39) explicitly with Maple 9@.

0

References 1. L.G. PB1, Analysis Math. 1, 197-205 (1975). 2. A.S. Cavaretta, A. Sharma and R.S. Varga, Trans. Amer. Math. SOC.259, 621-628 (1980). 3. G.G. Lorentz, S.D. Riemenschneider and K. Jetter, Birkhofl Interpolation, Addison Wesley Publ.Co., Mass. USA, 1983. 4. M.G. de Bruin, Nonlinear Analysis 47, 1931-1940 (2001). 5. M.G. de Bruin and A. Sharma, Comp. Meth. and Function Theory 3, 305-323 (2003).

122

SEMICLASSICAL LINEAR FUNCTIONALS OF CLASS 2: THE SYMMETRIC CASE*

A. M. DELGADO+& F. MARCELLAN$ Universidad Carlos 111 de Madrid Departamento de Matemciticas Avenida de la Universidad, 30 2891 1 Legane's (Madrid)-SPAIN

We describe the family of symmetric semiclassical linear functionals of class s = 2. In this way, some known linear functionals appear as well as a new one. We focus our attention in the integral representation of them using the corresponding Pearson equation.

1. Introduction

Semiclassical linear functionals associated with weight functions were considered by J. Shohat [13] in the framework of the analysis of orthogonal polynomials satisfying second order linear differential equations with polynomial coefficients (holonomic equations). Later on, P. Maroni has extensively studied such a kind of linear functionals with a special emphasis on the structural properties of such linear functionals. For instance, [Ill constitutes a relevant survey on the subject. The classification of semiclassical linear functionals according to some criteria of optimal information from the so-called Pearson equation, i.e., a first order linear differential distributional equation satisfied by the linear functional, plays a central role in the constructive theory of such linear functionals. S. Belmehdi in [2] gives a complete description of all semiclassical linear functionals of class 1 using such an approach. The class of a *This work is supported by Direccidn General de Investigacidn, Ministerio de Ciencia y Tecnologia of Spain, under grant BFM 2003-06335-CO3-02 and INTAS Research Network NeCCA INTAS 03-31-6637. t E-mail: adelgadommath.uc3m. es *Email: [email protected]. es

123

linear functional provides upper bounds of the degrees of the polynomial coefficients of the holonomic equation mentioned above. The aim of our contribution is the description of symmetric semiclassical linear functionals of class 2. We will give the integral representation of all possible situations. In a work in progress, we are studying the connection of this question with some extension of symmetric coherent pairs of measures. They appear in the analysis of polynomials orthogonal with respect to a Sobolev inner product associated with a 2D-vector of symmetric measures.

2. Some definitions and preliminary results Denote by P the linear space of polynomials with real coefficients and by P, the subspace of polynomials with degree at most n. Let u be a linear functional defined on P and p E P. Then, the linear functional p u is defined by ( p , q ) = ( u , p q ) , q E P.This linear functional is said to be the left multiplication of a linear functional by a polynomial. The derivative of the linear functional u is the linear functional Du defined by ( D u , p ) = -(%P’), P E P. The linear functional u is said to be regular (or quasi-definite according to [3]) if there exists a sequence of polynomials Pn, n E N, such that i) deg P, = n, V n E N, ii) (u, PnPm)= k,6,,,, kn # 0, V n ,m E N. If k, > 0 for all n E N,then u is positive-definite. A regular linear functional u is said to be semiclassical if there exist two polynomials q3 and +, with deg+ 2 1, satisfying

+

D[4u] +u = 0.

(1) If u is a regular linear functional, then one can assume the pair ($,+) is admissible. This means that, if $(z) = atxt . . . and +(x) = b,zT . . . , then one of the following conditions holds: i) t - 1 # T , ii) if t - 1 = T , then nat # b, for all n E N. Thus, the class of u (see [ll])is defined as

+

s=

+

min max(degq4 - 2,degq - l}, (+>+)EX

+

being X = {($, +) admissible and D[$u] +u = 0). It can be shown that the admissible pair (q3,+) that minimizes the class of u is unique, setting the polynomial $ to be monic. Given c E @, we consider the linear application Bc : P 4 P by (B,p)(z) = m, I-c p E P. Using this application, we define the linear functional (x-

124

c)-lu by ((x - c)-lu,p) = (u,Ocp),p E JP'. In general, for any polynomial q E JP', the linear functional q-lu is defined by

where L ( z ; p )denotes the interpolatory polynomial of p in the zeros of q , taking into account the multiplicity. For instance, if q(x) = (x - c ) ~then ,

Notice that (x- c ) ( x- c ) - h = u , meanwhile (x- c)-'(x - c)u = u - uO&, being ZLO the first moment of u and S, the Dirac linear functional defined by ( L P ) = P ( 4 . The linear functional u is said to be symmetric if the corresponding moments of odd order vanish, i.e., ( u ,xZn+')= 0 , n 2 0 (see, for example, [ 3 ] ) . Moreover, the symmetric linear functional u is said to be the symmetrized of the linear functional ii if ( u , x Z n = ) (ii,xn).Notice that if u is regular and symmetric, then ii is regular [lo], but the converse is not true. This means that if the linear functional ii is regular, then the symmetrized u may not be regular. Indeed, a necessary and sufficient condition is the regularity of the linear functional xii [ 3 ] . With respect to the symmetrization process and the class of the semiclassical linear functionals, the next result was proved in [l].For convenience, in terms of the polynomial coefficients for the Pearson equation it reads (see [4]):

Lemma 2.1. Let ii be a semiclassical linear functional of class 5 satisfying D[$ii]+&i = 0. If u i s the symmetrized of ii then, u i s again a semiclassical linear functional of class s satisfying (1). Furthermore,

(i) if $ ( O ) = 0 and $ ( O )

+ 2&0)

=

0, then

s = 25, with

d(x)

=

= x"%)(x2> + 2(804)(x2)1, ( e o & ( x 2 ) , and (ii) if $ ( O ) = 0 and $ ( O ) 2&0) # 0, then s = 25 1, with 4 ( x ) = x(eo$)(x2>, and $(x) = 2 4 ( x 2 ) , (iii) if $(O) # 0, then s = 25 3, with $(x) = x$(x2), and q-J(x)= 2 [ - 4 ( 2 ) xC"?j(xc")].

+

+

+

+

As a consequence, ( a ) if s is odd, the polynomials 4 and $ in (1) are, respectively, odd and even functions,

125

(b) i f s is even, the polynomials 4 and $ in (1) are, respectively, even and odd functions. For any non zero real number a , we define the linear functional u, := h,u by (h,u,p) := (u, hap),where, h,p(z) :=p ( a z ) for all p E P. It is clear that using this linear change of variable, the semiclassical character of the linear functional u remains valid for u,. In fact, if u is the symmetrized of 6,then u, is the symmetrized of h,26. Moreover, if (1) holds for u,then u, satisfies the Pearson equation D[+,ua] $,u, = 0 with

+

4,(z)

= ath,-l+(z), I/J,(x)= at-'ha-I$(x),

(2)

where t = deg 4. We conclude this section with some useful remarks. Let v and w be two regular linear functionals with u = w do. Then, V = 5 60, where V and ul are the corresponding symmetrized linear functionals for v and w respectively. Let u and u be regular linear functionals such that w = XU. Then, the symmetrized linear functionals are related by fi = x - ~ G M60. The aim of this contribution is to describe the set of symmetric semiclassical linear functionals of class 2 from the analysis of the corresponding Pearson differential equation.

+

+

+

3. Construction of symmetric semiclassical linear functionals of class 2

In [1,4] we can find the classification of the symmetric semiclassical linear functionals of class 1. Just for completion of our presentation, we summarize here this result. 1. Generalized Gegenbauer linear functional, 1

(u,p) = llp(r)/zl"(l with a

> -1, a # 0, and X > -1/2. +(z) = z(z2 - l),

-

x2)x-1/2dz,

Moreover, u satisfies (1) with

$(z) = -(a

+ 2X + 2)22 + a + 1.

This is the symmetrized of the classical Jacobi linear functional shifted to the interval ( 0 , l ) . 2. Symmetrized Bessel linear functional, is the solution of the Pearson equation [X3U]' -

+

( ( a 3 ) 2 + 4)"

= 0.

126

3. Hermite-Chihara linear functional:

with a

> -1,

(Y

# 0.

Moreover, u satisfies (1) with

$(x) = 2x2 - ( a

$(x) = 2 ,

+ 1).

This is the symmetrized of the classical Laguerre linear functional. Since we want to study the symmetric semiclassical linear functionals of class 2, then we are in cases (i) and (b) of Lemma 2.1. Thus, three different situations appear in terms of the polynomials involved in the Pearson equation (1) for u: 1. $(x) = x4 Ax2 B , $(x) = Cx3 Dz,with ICI ID1 # 0. 2. $(x) = x2 B , $(z) = Cx3 Dx,with C # 0. 3. $(x) = 1, $(z) = Cx3 Dx,with C # 0. Moreover, if u is the symmetrized of the linear functional ii then, from Lemma 2.1 we get 1. $(z) = x3 Ax2 B z , &x) = v x- with ICI ID1 # 0. 2. $(x) = x2 B z , q ( x ) = - $, with ICI # 0. 3. $(x) = z, q(z) = g x 2 g x - with ICI # 0. We analyze each case according to the zeros of the polynomial $. 1. We write $(x) = x(x - ( Y ~ ) ( zaz), $(z) = y2x2 ylx yo, with yo = -ala2/2, and we define the linear functional d, := ha-lii. Then, from (2) it holds ($,aa)' = 0 with

+ +

+

+ +

+

+

+

9% 8%'+ v x+ '

z,

+

+

+

+

i,

+

+

+ qaiia

= u-3$(ax) = z(z

u-1(Y1)(z - a - 1 4 , &(s) = a-2$(ux) = 72x2 y1u-12 you-2.

&(x)

1.a. Assume a1 then (3) becomes

$,(x)

#

a2

and

5

-

+

1 ~ ~ 2 1Taking .

+

(3)

a = a1 and X = a 2 / ( ~ 1 ,

= x(5 - l ) ( x - A),

$a(S) =

-(p

f

+ 5/2)z2 + (7 + 3/2 + X(p + 3 / 2 ) ) ~- X/2,

where, for convenience, we have renamed the coefficients y1 = a1 (y+3/2)+ a2(p 3/2)) and 7 2 = -(p y 5/2) in the polynomial (we will proceed in a similar way in the cases in below). By integration of the corresponding differential equation, we have

+

+ +

(Q,,p)=

I'

p(z)z-1'2(1 -.)@(A

4

- z)Ydz,

127

with /3 > -1 and 1x1 > 1. Then, by symmetrization we have that u is, up to a dilatation, the linear functional given by 1

( % P ) = s_1P(2)lA - 221y(l- 22)Pd& with y u with

# 0, ,B > -1,

+(2) =

and

(z2- 1)(x2 - A),

1x1 > 1. Moreover, from Lemma 2.1, (1) holds for 11,(2)= - 2 4 1

+ y + A(1 + p) - (2 + p + y)x2).

This is a generalized Gegenbauer linear functional (see [6]). 1.b. Assume a1 = 0 and a2 # 0. In this case we choose a = a2 in ( 3 ) to get

&(.)

= x2(2- l), &(.)

= -(a

+ p + 3)x2+ ( p + 2)2.

Then, 0, = Kx-lJa>p+l+ M & , where J a i f l + l is the classical Jacobi linear functional shifted to the interval [0,1], satisfying the differential equation (2(2-

l)Ja>p+l)r

This means

+ ( p+ 2

1

-

( a + p + 3 ) z ) J a J + l = 0.

1

(Ga7P)

P(z)@(l - x)ad2

0

+ NP(o),

with a,,O > -1. Then, up to a factor multiplication, the symmetric linear functional u is given by symmetrization,

being a,p > -1, and p # 0. Finally, the corresponding polynomials for u in (I) are, using (i) in Lemma 2.1, +(2) = 2 ( 2 2 -

l ) , $(2) = 2(3

4, 11,

+ 2p - (2a + 2p + 5)22).

This is a symmetrized Jacobi-type linear functional (see Ref. 8), that is a particular case of the so-called Koornwinder linear functionals [7]. 1.c. Assume a1 = a2 # 0. We choose a = a1, then ( 3 ) becomes &(z) = X(X

-

1)2,& ( s )

=

-(p

+ 5/2)x2 + (7 - ,O

-

1)” - 1/2.

Thus, the corresponding weight function w satisfies

(4a(+Wr d a ( . ) W ( . )

-

1cla(4 -

&(.)

1/2 I P + 2 2 2-1

+-( 2 -71 ) 2 *

128

Then,

if y > 0, and the symmetrized

ZL

is defined by

being y > 0. Finally, the corresponding polynomials q5,$ for u in (1) are, using (i) in Lemma 2.1,

4(2) = (z2- 1)2,$ ( 2 ) = 2 4 2 + p + y - (2 + p).". If y = 0, then we are in a particular case of (4) with ,L? = -3/4 (see [ S ] ) . 1.d. Assume a1 = a2 = 0. Thus, choosing a = -y1/2, (3) becomes &(.)

= 2 3 , &(s) = - 2 z ( a z

+ 1).

+

Then, iia = Kz-'B" Mbo, being B" the classical Bessel linear functional solution of the Pearson equation (,2,,),

-

2(az

+ 1)B" = 0.

Thus, the polynomials in (1) for u are $(2)= 24, $(2) =

(1 - 4 4 2 3 - 42.

See [9] for more details. 2. We have the polynomials $(z) = x ( z + B ) and ?It(.) = y~2~+ylz+yo, with yo = -B/2. Defining the linear functional ii, := ha-iii, by (2) it holds (&ii,)' = 0 with

+ qaiia

+

&(.) = a-2q5(az) = 2(z a-lB), &(z) = a-l$(az) = yza22 y12 2.a. Assuming B

4,(x) with X

with /3 factor,

# 0.

+

# 0, we choose a = -B

= ~ ( -2I),

qa(.)

2

+ you-1.

(5)

in (5) to get

= AX - (A

+ ,L? + 3/2)2 + 1/2,

Then,

> -1. The symmetrized linear functional u is, up to a constant

129

with

p > -1, and the polynomials in the differential equation (1) for u are $(z) = z2 - 1, $J(z)= 2z(Xz2 - x - p

-

1).

As a particular case of these functionals, we have the truncated Hermite polynomials, associated with the weight function e-xzX[-N,N]. w e refer the readers to Ref. [5]. 2.b. Assume B = 0. We choose a = yF1 and (5) becomes &(z) = 2 2 , &(z) = z

2

-

+

( a 2)z,

+

from where we deduce that 21, = K X - ~ C ~ + 'Md(0) where La+' is the classical Laguerre linear functional satisfying

(ZP+l)/ Thus, (iia,p) =

+ (z

/

-

( a + 2 ) ) P + l = 0.

ca

p(z)zae-zdz

0

with a

+N ~ ( o ) ,

> -1. Then, the symmetrized linear functional u is defined by J-00

with a

> -1, and the polynomials 4, in (1) corresponding to u are $J

4(z) = 2 2 , +(.)

= 42x2 - 2 a

-

3).

This is a generalized Hermite-type linear functional (see [9]). 3. Taking ii, := h,-Iii, by (2) it holds (&iia)' $aiia = 0 with

+

&(z) = a-l&(az) = z,

4,(z)

= $ ( a x ) = y2a2z2

Taking a such that y2a2 = 2, we get $,(z) we deduce

= 2z2

+ ylaz

-

1/2.

+ Xz - 1/2, from where

Finally, the corresponding symmetric linear functional is,

J-CO

satisfying (1) with

+ 2xz.

$(z) = 1, $(z) = 4z3

This is known as a Freud linear functional (see Ref. [12]).

130

References 1. J. Arves6, J. Atia, F. Marcellan, O n semiclassical linear functionals: the symmetric companion, Commun. Anal. Theory Contin. Fract. 10 (2002) 1329. 2. S.Belmehdi, On semi-classical linear functionals of class s = 1 . Classification and integral representations, Indag. Mathem. N. S. 3 (1992) 253-275. 3. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978. 4. A. Ghressi, L. Kheriji, Some D-semiclassical linear f o r m s in connection with perturbation-symmetrization, submitted. 5. H. F. King, M. Dupuis, Numerical integration using R y s polynomials, J. Computational Phys. 21 (1976) 144-165. 6. V. P. Konoplev, On orthogonal plynomials with weight functions vanishing or becoming infinite at isolated points of the interval of orthogonality, Soviet Math. Dokl. 2 (1961) 1538-1541. 7. T. H. Koornwinder, Orthogonal polynomials with weight function (1 - ~ ) ~ ( l + x ) +~m6(z + 1)+ N 6 ( z - l),Canad. Math. Bull. 27(2) (1984) 205-214. 8. L. L. Littlejohn, On the classification of differential equations having orthogonal polynomials solutions, Ann. Mat. Pura et Appl. 138 (1984) 35-53. 9. F. Marcellan, P. Maroni, Sur l'adjonction d'une masse de Dirac Ci une forme re'gulidre et semi-classique, Ann. Mat. Pura Appl. (4) 162 (1992) 1-22. 10. P. Maroni, Sur la de'composition quadratique d'une suite de polyn6mes orthogonaua: I, Riv. Mat. Pura Appl. 6 (1990) 19-53. 11. P. Maroni, Une the'orie alge'brique des polyn6mes orthogonaux. Application aux polyn6mes orthogonaux semi-classiques, in Orthogonal polynomials and their applications (Erice, 1990), C. Brezinski et al. Editors, IMACS Ann. Comput. Appl. Math. 9 Baltzer, Base1 (1991) 95-130. 12. P. Nevai, Orthogonal polynomials associated with exp(-z4).Proc. Canad. Math. Soc. 3 (1983) 263-285. 13. J. Shohat, A diflerential equation for orthogonal polynomials, Duke Math. J. 5 (1939) 401-417.

131

A MYRIAD OF SIERPINSKI CURVE JULIA SETS

ROBERT L. DEVANEY* Dept. of Mathematics Boston University Boston, M A 02215, USA E-mail: bob9bu. e d u

This paper is a survey of the numerous Sierpinski curve Julia sets that arise in the family of rational maps given by

F x ( z ) = zn

x +where n 2 2, d 2 1. zd

While these Julia sets are all the same from a topological point of view (they are all homeomorphic), the dynamics on these sets are almost always very different in the sense that no two maps are topologically conjugate.

1. Introduction

The Sierpinski carpet fractal shown in Figure 1 is one of the most important planar, compact, connected sets for several reasons. First of all, it is a universal plane continuum in the sense that it contains a homeomorphic copy of any planar, one-dimensional, compact, connected set. (Here we mean one topological dimension, not Hausdorff or fractal or any other dimension.) For example, the complicated curve shown in Figure 2 may be homeomorphically deformed so that it sits inside the Sierpinski carpet fractal. Secondly, there is a topological characterization of this set: any planar set that is compact, connected, nowhere dense, locally connected, and has the property that any pair of complementary domains are bounded by disjoint simple closed curves is homeomorphic to the Sierpinski carpet". Any set that is homeomorphic to the Sierpinski carpet is called a Sierpinski

curve. In recent years, we have shown that Sierpinski curves arise as the Julia sets of certain complex rational functions in a variety of different ways. In *Partially supported by NSF grant 0205779

132

Figure 1. The Sierpinski carpet.

this paper we shall describe several of these possibilities. What is interesting here is that, while these Julia sets are always the same from a topological point of view, the dynamics on these sets are often quite different. For simplicity, we shall restrict attention t o functions of the form

Fx(z)=zn+-

x Zd

wheren>2,d>l

although many other families of rational maps exhibit similar types of Julia sets. We assume here that n 2 2 so that the point a t 00 is a superattracting fixed point. Hence there is an immediate basin of attraction of 00 which we denote by Bx. As we show below, even though the maps in this family have high degree, there really is (up to symmetry) only one “free” critical orbit. As is well known in complex dynamics, the behavior of this critical orbit plays a principal role in determining both the topology of and the dynamics on the Julia sets of these maps. In particular, if the critical orbit eventually enters Bx, we have the following result7. Theorem (The Escape Trichotomy). If the free critical orbit remains bounded, then the Julia set of FA is a connected set. However, (1)

If the critical value lies in Bx, then the Julia set of FA is a Cantor set;

133

Figure 2. This curve (containing the Sierpinski gasket, the Koch curve, the topologists' sine curve, and many other things) may be homeomorphically realigned to fit into the Sierpinski carpet.

(2) I f the free critical orbit enters Bx at the second iteration, then the Julia set is a Cantor set of simple closed curves; (3) If the free critical orbit enters Bx at the third or higher iteration, then the Julia set is a Sierpinski curve. Since there is only one free critical orbit, the A-plane is therefore the natural parameter plane for these families. In Figure 3, we have plotted these planes in the cases n = d = 2 and n = d = 3. The black points in this picture correspond t o parameter values for which the free critical orbit does not escape to 00. The white regions in this picture represent A-values for which the critical orbit tends to 00. The exterior region corresponds to parameter values for which the Julia set is a Cantor set; we call this set the Cantor set locus. The small white region in the center of the picture corresponds t o parameter values for which the Julia set is a Cantor set of simple closed curves. We call this region the McMullen domain. We remark that it is known" that McMullen domains exist if and only if l / n + l / d < 1. All other white regions in this picture correspond t o parameters for which

134

the free critical orbit escapes after three or more iterations so the Julia set is a Sierpinski curve. These regions are called Sierpinski holes. It is known4 and16 that there are infinitely many such regions in each parameter plane. Hence the Julia set of FA is a connected set for all A-values except those in the Cantor set locus and the McMullen domain. So we call this set of parameters the connectedness locus.

Figure 3.

The parameter planes for the cases n = 3 and n = 4.

It turns out that there are many other ways that the Julia set of FA can be a Sierpinski curve besides case 3 in the Escape Trichotomy. The goal of this paper is to give a sketch of the proof of the following results: Theorem. Let wx denote the image of one of the free critical points of FA. (1) If F,k(wx) lands o n a repelling periodic point of FA that does not lie in the boundary of Bx, then the Julia set is a Sierpinski curve; (2) If X lies in the main cardioid of certain "buried" Mandelbrot sets

with base period k in the parameter plane, then again the Julia set of FA is a Sierpinski curve; (3) If, as in the Escape Trichotomy, F,k(wx) with k > 1 is the first point in the orbit of vx that lies in Bx, then the Julia set of FA is a Sierpinski curve. The dynamical distinctions between these cases are given by:

135

Corollary. If X I and A2 are any parameter values arising from different cases of this result, then the dynamics on the corresponding Julia sets are not topologically conjugate to one another. Even if these parameter values arise from the same case, but the corresponding k-values are different, then again the dynamics of the two maps are not topologically conjugate. This result follows immediately from the fact that, in each case, the behavior of the critical orbits is very different, but a conjugacy between two such maps must preserve this behavior. 2. Preliminaries

In the dynamical plane, the object of principal interest is the Julia set of FA which we denote by J ( F x ) . The Julia set is the set of points at which the family of iterates of F A , {F,”}, fails to be a normal family in the sense of Montel. It is known that J ( F x ) is also the closure of the set of repelling periodic points for FA as well as the boundary of the set of points whose orbits escape to 00 under iteration of FA. As a consequence, J ( F x ) is the regime where FA behaves chaotically. The complement of the Julia set is known as the Fatou set. Here the dynamical behavior is quite tame. Note that Bx and all of its preimages must lie in the Fatou set. Since the point at 00 is a superattracting fked point for FA, it is well known13 that FA is conjugate to z H zn in a neighborhood of 00 in Bx. There is also a pole of order d for FA at the origin, so there is a neighborhood of 0 that is mapped into Bx by FA. If this neighborhood is disjoint from Bx, then we denote the preimage of Bx that contains 0 by Tx. So the only preimages of Bx are Bx and Tx. We call Tx the trap door since any orbit that eventually enters the immediate basin of 00 must “fall through” Tx enroute to Bx. One computes easily that there are n d critical points for FA and that all of them assume the form wkcx where cx is one of the critical points and w = exp(2ri/(n d)). We call these points the free critical points. Similarly, the critical values vx are arranged symmetrically with respect to z H w z , though there need not be n d of them. There are n d prepoles at the points (-X)ll(n+d). The proof of the following Proposition is straightforward.

+

+

+

Proposition (Dynamical Symmetry). Suppose w Then F x ( w z ) = w”Fx(z).

+

=

exp(2ri/(n

+ d)).

As a consequence of this result, the orbits of points of the form wjz all behave “symmetrically” under iteration of FA. For example, if F j ( z ) + 00,

136

then F;(wkz) also tends to 00 for each k. If F i ( z ) tends to an attracting cycle, then so does F;(wkz). We remark, however, that the cycles involved may be different depending on k and, indeed, they may even have different periods. Nonetheless, all points lying on these attracting cycles are of the form w j z o for some zo E C. For example, when n = 2,d = 1, there are parameters for which some of the critical points tend to an attracting fixed point zo on the real line, whereas wzo and w2z0 lie on an attracting 2-cycle which attracts other critical points. 3. Escape Sierpinski Curve Julia Sets

In this section we assume that the third or higher iterate of the critical point is the first that enters Bx. We then give a complete proof that, in this case, the Julia set is a Sierpinski curve. In later sections we merely sketch the somewhat similar proofs that the Julia sets in other cases are Sierpinski curves. In Figure 4, we display four Julia sets drawn from the family where n = d = 2. All of these sets are Sierpinski curves, and all have different dynamics since the number of iterates it takes for the critical orbit to reach Bx is different in each case. We first observe that, unlike most connected Julia sets of polynomials, for A-values in the connectedness locus, the complement of the closure of Bx consists of a single open component. We do not have Julia sets in this case like the well-known Douady rabbit or basilica Julia sets from quadratic dynamics.

Proposition. Suppose that the free critical orbit tends to m, but the critical values do not lie in Bx u Tx. Then the set C - 5 has a single open, connected component. Proof: Suppose first that C-% has more than one connected component. Let WObe the component of C - 5 that contains the origin. Note that all of Tx must lie in WO.We claim that at least one of the prepoles also lies in WO.Suppose this is not the case. By symmetry, all of the prepoles either lie in the same component of the Fatou set or else they all lie in distinct components. In the latter case, this means that each Fatou component containing a prepole is mapped one-to-one onto WO.Therefore there must be n d of these components. Now there are no critical points in the Julia set by assumption, so every point in the boundary of WOhas n d preimages, one in each of the boundaries of these components. But there are also d preimages of any such point in the boundary of the trap door

+

+

137

X

= -1/16

X = -1/4

X

= -0.01

x = -0.001

Figure 4. The Julia sets for various values of X when n = d = 2.

which is contained inside 6and mapped to the boundary of BA. Since the boundary of TAcannot equal the boundary of WO,this yields too many preimages for any point in the boundary of WO.Therefore all of the prepoles lie in the same component of the Fatou set, and this component must surround the origin and separate Bx from K. This, however, is impossible, since the boundary of WOis contained in the boundary of BA. Hence one and therefore all n d of the prepoles lie in WOand so FA is n d to 1 on WO.Therefore all of the preimages of points in WOmust also lie in WO. Now suppose that there is a second component Wl in There are no points in W1 that map into Wo. Consider a point on the boundary of W1

+

+

@-z.

138

that does not also lie on the boundary of WOand choose a neighborhood of this point that does not meet WO.By Montel’s Theorem, the forward images of this neighborhood map over points in WO.But this cannot happen, since all preimages of points in WOlie in WO.This proves that Wl does not exist.

0 Proposition. The Julia set of FA is compact, connected, locally connected, and nowhere dense. Proof: Since we are assuming that all of the critical orbits eventually enter the basin of m, the Fatou set consists of the union of Bx and all of its preimages. Hence we have that the Julia set is given by C-UFrI (Bx).That is, J ( F x ) is @ with countably many disjoint, open, simply connected sets removed. Hence J ( F x ) is compact and connected. Since J ( F x ) # @, J ( F x ) cannot contain any open sets, so J ( F x ) is also nowhere dense13. Finally, since the critical orbits all tend to co and hence do not lie in or accumulate on J ( F x ) , standard arguments show that J ( F x ) is locally connected13. 0 Thus we have shown that J ( F x ) possesses four of the five defining properties of a Sierpinski curve. It suffices t o show that the boundaries of the complementary domains are bounded by simple closed curves that are disjoint. This is a little more difficult. Proposition. The boundary of Bx, d B x , as well as all of the preimages of Bx are simple closed curves. These boundary curves are pairwise disjoint. Proof: By the previous Proposition, J ( F x ) is locally connected, so it follows that d B x is also locally connected. Now recall that, near 00, FA is analytically conjugate to z H zn. That is, there exists an analytic homeomorphism qjx : Bx 4 - rn, where D is the open unit disk in the plane. The map $A satisfies

r

h 0 FA(.)

=

The preimage under of the straight ray with argument 8 in - is called the external ray of angle 8 and denoted by r(8).Since the boundary of Bx is locally connected, it is known15 that all of the external rays land at a point in the boundary of Bx. Thus, to show that this boundary is a simple closed curve, it suffices t o prove that no two external rays land at the same point. that To see this, first recall that WOdenotes the component of C contains the origin, and that WOis both connected and simply connected. Suppose that there exists p E dBx such that $ti) and Y ( t 2 ) both land on

139

p . Since these rays together with the point p form a Jordan curve, we have that WOlies entirely within one of the two open components created by this Jordan curve. Let y(t1, t z ) denote the union of all of the external rays whose angles lie between tl and t 2 (where we assume that the angle between these two rays is smaller than n). Without loss of generality, assume that WOis such that WOn y(t1, t z ) = 0 (so WOis “outside” the sector y(t1, t z ) between y(t1) and y(t2)). We claim that there exist positive integers q and m such that the region

+

and neither of the external rays q/m nor ( q l ) / m land on ~ W O If .this were not possible, then all rays in y(tl, t z ) would land at p . This gives a contradiction because the set of angles 8 E R/Zsuch that the landing point of the ray with angle 8 is p has measure OI3. So suppose we have such q and m. As above, let y(q/m,( q l ) / m ) denote the union of the external rays contained between q/m and (q+l)/m. After m iterations y ( q / m ,(q+l)/m) is mapped over all of Bx. In particular, if the external ray of angle 8 lands on a(C then there is a ray of angle 4 E y ( q / m , ( q l ) / m )whose image under FY is ~ ( 8 ) .Since q!~ E T(q/m,(q+l)/m)we have that y(4) does not land on aW0. Hence there exists a neighborhood N+ of y(4) such that N+ n WOis empty. However, since F Y ( y ( 4 ) )lands on the boundary of WOwe know that F Y ( N + )n WO is not empty. This is a contradiction since points not in WOnever enter WO. Hence, we can never have two rays landing at the same point on aBx, implying that aBx is a simple closed curve. It follows that all of the preimages of Bx are also bounded by simple closed curves. We claim that no two of these curves can intersect. To see this, suppose first that there exists a point zo E dBx n aTx. Then there exists an external ray y in Bx landing at 20. In Tx, there also exists a preimage, 7 , of an external ray that connects 0 to Z O. But the images of 7 and y are the same external ray, and so it follows that zo is a critical point. But this contradicts our assumption that all critical orbits tend to 00. So d B x and aTx are disjoint simple closed curves. Now suppose that two earlier preimages of aBx intersect, say one preimage in FLn(aBx)and one in Fr”(8Bx). If n # m, then by mapping these preimages forward, we see that dBx and 8Tx also meet, which cannot happen. If n = m, then this intersection point must again be a critical point, so this cannot occur either.

+

+

x),

140

0 This shows that the Julia set is a Sierpinski curve when it takes three or more iterations for the critical orbit to enter Bx. To see that there are infinitely many such sets that are dynamically distinct, note that the number of iterations that it takes for the critical orbit to enter Bx is a dynamical invariant: a conjugacy between any two such sets must map the invariant boundaries of the basin of 00 to each other. Hence the jth preimages of the basins must be mapped to each other by the conjugacy. But the only preimages of the basin on which the maps are not one-to-one are those that contain the critical points. Hence, in order for these maps to be conjugate, the critical orbits must all take the same number of iterations to enter the basin.

4. Buried Sierpinski Curves

In this section, we discuss an infinite collection of dynamically distinct Sierpinski curve Julia sets for the family FA where the Fatou components are quite different than those described in previous sections. Instead of being just a single superattracting basin at 00 and its preimages, the Fatou set in these examples consists of a collection of finite attracting basins together with the basin at 00 as well as all of their preimages. As before, the dynamics on these Julia sets are all distinct from one another as well as from those mentioned above, but again, all of these Julia sets are homeomorphic.

Figure 5.

The Julia sets for F x ( z ) = z2

+ X/z

where X = -0.327 and X = -0.5066.

141

For simplicity, we shall restrict attention in this section to the special family Fx(z) = z2 A/z with A E W-.The examples we give arise in all of the other families, though their construction is a little more complicated in the general case. In Figure 5, we display the Julia set of FA when X = -0.327. For this map, there are superattracting basins of period 3 and period 6 together with the basin at 03. We also display the case where A = -0.5066 for which there are three different superattracting basins of period 4 together with the basin at 00. The basins of the finite cycles in these pictures are displayed in black. For A real and negative, the graph of Fx(2) shows that there is a positive real fixed point for FA which we denote by p ( A ) . Also, .(A) = (X/2)'I3 is a critical point on the real line and

+

3 .(A) = -A213 2213 is the corresponding critical value. Let A* = -16/27. Straightforward calculations show that p ( A * ) = 4/3 and p(A*) is repelling. Furthermore, the real critical point .(A*) = -2/3 is pre-fixed, i.e., FA. (.(A*)) = 4/3 = p(A*). For A-values slightly less negative than A*, the real critical value lies to the left of p ( A ) and hence subsequent points on the orbit of the critical value begin to decrease. Graphical iteration shows that there is a sequence of A-values tending to A* for which the critical orbit decreases along the positive axis and then, at the next iteration, lands back at .(A). See Figure 6. Thus, for these A-values, we have a superattracting cycle. Straightforward analysis6 of the real dynamics of these functions shows:

Theorem. There i s a decreasing sequence A, E R- for n 2 3 with A, -+ A* = -16/27 and having the property that FA, has a superattracting cycle of period n given by xj(A,) = Fx,(zj-l(An)), where (1) ZO(A,) = %,(An) = .(A,), and (2) 20 < 0 < 2,-1 < 2,-2 < . . . < 21 = .(A,)

< p(A,).

Now fix a particular parameter value X = A, for which FA has a superattracting periodic point xo lying in R- as described in the previous theorem. We say that a basin of attraction of FA is buried if the boundary of this basin is disjoint from the boundaries of all other basins of attraction (including Bx). Note that, if the basin of one point on an attracting cycle is buried, then so too are all forward and backward images of this basin, so

142

Figure 6. The graphs of Fx(x)= x 2

+ X/x where X = Xq and X = A.,

the entire basin of the cycle is buried. As in the previous section, standard arguments show that these Julia sets are compact, connected, locally connected, and nowhere dense. We need only show that the all of the basins of attraction are bounded by disjoint simple closed curves. The case of the basin at 00 follows as in the previous section. For the finite basins, a different argument is necessary. We refer to6 for the details. We have:

Theorem. For the sequence ofparameter values A, E R-, all of the basins of FA, are buried and so J ( F x ) is a Sierpinski curve. As discussed earlier, any two Sierpinski curves are homeomorphic. Hence J(Fx,) is topologically equivalent to J(Fx,) for any n and rn. However, each of these Julia sets is dynamically distinct from the others since the periods of the superattracting cycles are different. 5. Structurally Unstable Sierpinski Curves In this section we turn our attention to the case where the critical orbit eventually lands on a repelling periodic point of FA that does not lie in aBx. Here again the Julia set is a Sierpinski curve and the dynamical behavior on this set is very different from the previous cases. For simplicity, we shall restrict attention in this section to the case n = d = 2, i.e., the family

x

F~(z) = z2+ Z

143

and specifically to the case when X is negative. In Figure 7 we display the parameter plane for this family. By the results of McMullenll, there is

Figure 7. The parameter plane for the family z2

+ X/z2.

no McMullen domain for this family. Indeed, the large white region near the center of this picture is a Sierpinski hole. If we examine the parameter plane along the negative real axis, it appears that there are infinitely many Sierpinski holes. In Figure 8, we display several magnifications of the parameter plane along the negative real axis. In each case we see a large Sierpinski hole flanked by a pair of smaller Serpinski holes which, in turn, are each flanked by a pair of even smaller Sierpinski holes. It appears that the parameter plane along the negative real axis consists of a Cantor set where the removed intervals are the intersections of R- with Sierpinski holes. Indeed, this is the case. In fact, along the negative real axis we actually have a Cantor necklace. To define a Cantor necklace, we let r denote the Cantor middle thirds set in the unit interval [0,1]. We regard this interval as a subset of the real axis in the plane. For each open interval of length 1/3" removed from the unit interval in the construction of I?, we replace this interval by an open disk of diameter 1/3" centered at the midpoint of the removed interval. Thus the boundary of this open disk meets the Cantor set a t the two endpoints of the removed interval. We call the resulting set the Cantor middle-thirds necklace. See Figure 9. Any set homeomorphic t o the Cantor middle-thirds

144

+

Figure 8. Two magnifications of the parameter plane for the family z2 X/z2 along the negative real axis. In the first image, -0.4 5 ReX 5 -0.06 and, in the second, -0.2 5 Re X 5 -0.15

necklace is then called a Cantor necklace. We do not include the boundary of the open disks in the Cantor necklace for the following technical reason: it is sometimes difficult in practice to verify that these bounding curves in the parameter plane are simple closed curves.

Figure 9.

The Cantor middle-thirds necklace.

Cantor necklaces also appear in the dynamical plane. To construct such sets when X E EX-, we first observe that the intersection of J ( F x ) and the real line is an invariant Cantor set that we call r x . To see this, note that the graph of the real function FA shows that FA maps the interval [ - p x , p ~ ] in two-to-one fashion over itself, where p~ is the fixed point for FA on the positive real axis and on the boundary of Bx. See Figure 10. The fact that such a Cantor set exists follows easily in the case where IF{(x)I > 1 for all x E [ - p x , p x ] . Unfortunately, this is not always the case, since, as X + 0,

145

the graph of FA approaches the graph of x2. Nonetheless, techniques from complex dynamics involving the Poincark metric and similar to those used in the case of z 2 c give this result’. Well known facts from dynamics also show that this Cantor set varies continuously as X varies and that the set of repelling periodic points in rx is dense in r x . In standard fashion we may also associate an itinerary s(A) = ( ~ 0 ~ 1 ~.)2consisting . . of 0’s and 1’s to each point in FA. Finally, the open intervals in the complement of in [ - p x , p x ] contain points that eventually map into Bx. These intervals lie in simply connected open sets that are preimages of Bx, and so the union of these disks together with l?x produces the Cantor necklace.

+

Figure 10. The graph of FA on the real line for X door on the real axis.

< 0.

The points fq bound the trap

Now consider the second iterate of the critical points. All four of these critical points lie off the real line, but a straightforward calculation shows that the second images of the critical points are all the same and are given by 4X 1/4. Let G(X) = 4X 1/4. So G(X) E IR when X is negative and G(X) decreases as X decreases along this axis. A straightforward computation shows that we have G(0) > 0 but G(X) < -px for X sufficiently negative. It follows that there is at least one A-value for which G(X) = .(A) for each possible itinerary. In fact, it is known6 that there is a unique such X on the real line for each itinerary and that, in fact, this set of A-values is a Cantor set in parameter plane. This is the Cantor set portion of the Cantor necklace in the parameter plane. Now choose a A-value for which G(X) lands on a periodic point in that is not equal to px (or an eventually periodic point that does not land on

+

+

146

P A ) . These are the “buried” cycles in the Cantor set since they do not lie on

the boundary of any complementary interval. For such a A-value, the Julia set is also a Sierpinski curve. Indeed, the only components of the Fatou set are the basin at 00 and all of its preimages since all of the critical orbits are eventually periodic. As above, these preimages are again bounded by disjoint simple closed curves. Moreover the Julia set is compact, connected, nowhere dense, and locally connected (this last property follows since we are in the analogue of the Misiurewicz point case for rational maps17). Notice that in any neighborhood of such a A-value, we have infinitely many other parameters for which the critical orbit also lands on a periodic or eventually periodic point in the Cantor set that does not lie in 8Bx and whose period is different from the original cycle. Hence the Julia sets for these parameters also are Sierpinski curves, and the dynamics on them are always different provided that the periods of the cycles are different. In addition, there are infinitely many intervals that lie in the complement of in any such neighborhood, so these yield infinitely many dynamically distinct escape Sierpinski curve Julia sets in this neighborhood as well. We have shown:

Theorem. Suppose that the critical orbit of FA lands on a point in the Cantor set l?A that is periodic or eventually periodic and that this orbit does not lie in 8Bx. Then the Julia set of FA i s a Sierpinski curve. Moreover, in any neighborhood of A E R-, there are infinitely many other parameter values whose Julia sets are Sierpinski curves of this type as well as infinitely many other escape Sierpinski curves. All of these parameters have dynamical behavior that i s different from that of FA. Since we have such vastly different dynamical behavior in any neighborhood of such a A-value, such a map is structurally unstable at that point. As a remark, if the critical orbit eventually lands on the fixed point P A , then the Julia set is what we call a “hybrid” Sierpinski curve. The only difference between this type of set and a Sierpinski curve is that infinitely many of the complementary domains have boundaries that now touch at exactly two points, while infinitely many others have boundaries that are disjoint from all the other bounding curves. 6. Final Comments and Conjectures

While we have shown that there exist infinitely many dynamically different types of Sierpinski curve Julia sets in these families, much more remains to be done. Here are some open problems and conjectures.

147

We have shown that if two maps have critical orbits that escape to 00, then their Julia sets are Sierpinski curves. If the escape times of these critical orbits are different for these two parameters, then these maps are not conjugate on thier Julia sets. So the question is: what happens if we have two such maps whose escpae times are the same?

Conjecture. Excluding any pair of symmetrically located Sierpinski holes in the parameter plane, the maps corresponding to parameter values drawn from different Sierpinski holes are always dynamically diflerent.

To prove this result, one needs to find a dynamic invariant for these maps that is different from the escape time. We anticipate that the invariant Cantor necklaces described in the previous section will play a role in this. In the previous section, we discussed the case where the critical orbit eventually lands on a repelling cycle in the Cantor set FA. But it is known that there is a unique parameter value for which the critical orbit lands on any point in this Cantor set. For example, there are uncountably many parameters for which the critical orbit lands on an orbit which never cycles. For these maps, the Julia set is again a Sierpinski curve. But here we no longer have the period of the cycle on which the critical orbit lands as a dynamical invariant. Nonetheless, we expect that any two such parameters will have distinct dynamics. f o r which Conjecture. Suppose the critical orbit lands on a point in the itinerary is neither periodic nor eventually periodic. Then the dynamics of FA is distinct from any other parameter value for which the critical orbit lands o n a point in A'I with a diiferent itinerary. Besides invariant sets that are Cantor sets, we have shown recently that there are parameter values for which the Julia sets contain a Cantor set of invariant circles. This is not the McMullen domain case, as the critical values of these maps do not lie in TA. Rather, for these parameters, the Julia set is connected. We suspect that, just as in the previous section, one can find unique parameter values for which the critical orbit lands on a particular point in this invariant set. This would produce a similar set in the parameter plane.

Conjecture. There is a Cantor set of simple closed curves in the parameter plane f o r which the critical orbits eventually land o n the invariant set of circles in dynamical plane. For these parameters, the Julia sets are again Sierpinski curves, and we conjecture that all of these maps (excluding symmetric cases) have distinct dynamics.

148 We expect t h a t the curves alluded to in this conjecture would actually contain many of t h e parameters in the Cantor set produced in t h e previous section. T h i s would produce a much huger array of dynamically distinct Sierpinski curve Julia sets.

References 1. Blanchard, P., Devaney, R. L., Look, D. M., Seal, P., and Shapiro, Y. Sier-

pinski Curve Julia Sets and Singular Perturbations of Complex Polynomials. Ergodic Theory and Dynamical Systems 25 (2005), 1047-1055. 2. Devaney, R. L. Baby Mandelbrot Sets Adorned with Halos. To appear in Complex Dynamics: Twenty Five Years After the Appearance of the Mandelbrot Set, Amer. Math. SOC. 3. Devaney, R. L. Structure of the McMullen Domain in the Parameter Planes for Rational Maps. Fundamenta Mathematicae 185 (2005), 267-285. 4. Devaney, R. L. The McMullen Domain: Satellite Mandelbrot Sets and Sierpinski Holes. To appear. 5. Devaney, R. L. and Marotta, S. The McMullen Domain: Rings Around the Boundary. To appear in Trans. Amer. Math. SOC. 6. Devaney, R. L. and Look, D. M, A Criterion for Sierpinski Curve Julia Sets. To appear in Topology Proceedings. 7. Devaney, R. L., Look, D. M, and Uminsky, D. The Escape Trichotomy for Singularly Perturbed Rational Maps. To appear in Indiana Univ. Math. J. 8. Douady, A. and Hubbard, J. Etude Dynamique des PolynGmes Complexes. Publ. Math. D’Orsay (1984). 9. Douady, A. and Hubbard, J. On the Dynamics of Polynomial-like Mappings. A n n . Sci. E N S Paris 18 (1985), 287-343. 10. Mane, R., Sad, P., and Sullivan, D. On the Dynamics of Rational Maps. Ann. Sci. E N S Paris 16 (1983), 193-217. 11. McMullen, C. Automorphisms of Rational Maps. Holomorphic Functions and Moduli. Vol. 1. Math. Sci. Res. Inst. Publ. 10. Springer, New York, 1988. 12. McMullen, C. The Classification of Conformal Dynamical Systems. Current Developments in Mathematics. Int’l. Press, Cambridge, MA, (1995) 323-360. 13. Milnor, J. Dynamics in One Complex Variable. Vieweg, 1999. 14. Milnor, J. and Tan Lei. A “Sierpinski Carpet” as Julia Set. Appendix F in Geometry and Dynamics of Quadratic Rational Maps. Experiment. Math. 2 (1993), 37-83. 15. Petersen, C. and Ryd, G. Convergence of Rational Rays in Parameter Spaces, The Mandelbrot set: Theme and Variations, London Mathematical Society, Lecture Note Series 274, Cambridge University Press, 161-172, 2000. 16. Roesch, P. On Captures for the Family f x ( z ) = z2 X/z2. To appear. 17. Yongcheng, Y. On the Julia Set of Semi-hyperbolic Rational Maps. Chinese Journal of Contemporary Mathematics. 20 (1999), 469-476. 18. Whyburn, G. T. Topological Characterization of the Sierpinski Curve. Fundamenta Mathematicae 45 (1958), 320-324.

+

149

COMPULSORY ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SYSTEM OF THREE DISCRETE EQUATIONS

JOSEF DIBL~K Department of Mathematics, Faculty of Electrical Engineering and communication, Brno University of Technology, Technickd 8, 616 00 Brno, Czech Republic E-mail: [email protected] IRENA R~JZICKOVA Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, Technickd 8, 616 00 Brno, Czech Republic E-mail: [email protected] This contribution is devoted to a discussion of the asymptotic behavior of solutions of systems of three difference equations. We try to connect two techniques the so called retract type technique and Liapunov type approach. We show that under appropriate conditions containing conditions typical for the retract technique approach, and conditions typical for the Liapunov type approach, there exists at least one solution of the system considered the graph of which stays in a prescribed domain.

1. Introduction

In this contribution we investigate the asymptotic behavior for k the solutions of the system of three difference equations

(k)= f l (k,u1 (k),uZ(k), u3 (k)), AuZ (k) = f Z ( k ,u1 (k),uZ (k),213 (k)), Au3(k) = f3 (k,u1 ( k ) ,u2 (k),u3 (k))

+

00

of

Aul

+

(1)

with f l , f z , f 3 : N ( a ) x R3 + R, Aui(k) = ui(k 1) - ui(k),i = 1 , 2 , 3 , where k E N ( a ) = { a , a 1,.. .}, a E W is fixed and N = {0,1,. . .}. We recall some known facts. The solution of system (1)is defined as an infinite sequence of number vectors

+

+

+

{ u ( a ) , u ( a l ) , u ( a 2), . . . ,u(a

+ n ) ,. .. }

150

with u = (ul,uZ,u3) such that for any k E N ( a ) , equalities (1) hold. The existence and uniqueness of the solution of system (1) with a prescribed initial condition (ul(a),UZ(a),u3(a)) = ('@,u$,u$) E R3

(2)

on N ( a ) is obvious. The sequence {(k,.l(k),u2(k),u~(k))},k E N ( a ) , is called the graph of the solution (ul,u2,u3)= (ul(k),u2(k),u~(k)) for k E N ( a ) of initial problem ( l ) , (2). If for every fixed k E N ( a ) the right hand sides fi(k,u1,u ~u3), , i = 1,2,3, are continuous with respect to their arguments ul,u2,u3,then the solution of initial problem (l),(2) depends continuously on the initial data. 1.1. Description of the Problem Considered and Auxiliary

Notions Let us define a set R

c N ( a ) x R3 as R

:=

u R(k)

k E N (a)

where

R ( k ) := ((k,ul,uZ,u3): k

E N(a),Ui E

R,bi(k)< U i < Ci(k),i = 1,2,3}

with bi,ci : N ( a ) + R, i = 1,2,3, being auxiliary functions such that bi(k) < ci(lc) for each k E N ( a ) . Such set 52 is called a polyfucial set. Our aim is to find sufficient conditions with respect to the right-hand side of system (1) which guarantee the existence of at least one solution u ( k ) = (u;(k),u;(k),u:(k)), k E N ( a ) satisfying

(k,.;(k),.f(k),.~(kc))

E

R(k),

(3)

for every k E N ( a ). Since the asymptotic behavior of a solution satisfying (3) is in some sense conditioned by the sets R(k), we call the asymptotic behavior of such solutions compulsory asymptotic behavior. 1.2. Preliminary In paper [2] the above described problem is solved via Liapunov type technique. Here we will combine this technique with the retract type technique which was used in paper [l].Before we start, recall some basic notions that will be used.

151

1.2.1. Consequent point

A point M f ( k , u l , u 2 , u g ) is called the first consequent point of a point M = ( k * ,uT, u3,us)E N ( a ) x IR3 (and we write Mf = C [ M ] )if k = k"

+1

and

ui = u5

+ fi(Ic*,u;,uf,u;)

for

i

=

1,2,3.

1.2.2. Liapunov type polyfacial set We say that a polyfacial set R is Liapunov type with respect to the discrete system (1) if

bi(k

+ 1) < ui + f i ( k ,

u 1 , u 2 , u3)

< Ci(k

+ 1)

for every i = 1,2,3 and every ( k ,u)E R. The geometrical meaning of this property is this: If a point M = ( k ,u1,u2,u g ) lies inside the set R(k), then its first consequent point C [ M ] stays inside R(k 1). In this contribution we will deal with sets that need not be of Liapunov type, but they will have a similar property. We say that a polyfacial set R is Liapunov type with respect to the j - t h variabZe with respect to the discrete system (1) if

+

( k , u )E R + b j ( I c + l )



The geometrical meaning is that if M = ( k ,u 1 , u2,u g ) E R(k), then the ujcoordinate of its first consequent point stays between b j ( k + 1)and c j ( k + l ) , meanwhile the other coordinates of C [ M ]may be arbitrary. 1.2.3. Points of strict egress and their geometrical sense An important role in the application of the retract type technique is played by the so called strict egress points. Before we define these points, let us describe the boundary of the set R in detail. As one can easily see,

j=1,2,3

j=1,2,3

with

nj, := { ( k , u ): k

E

5 ui 5 Ci(k),i

N ( U ) , U j = bj(k),bi(k)

= 1 , 2 , 3 , i# j }

and

Rb

:= { ( k , u ): Ic E

N ( a ) , u j = c j ( k ) , b i ( k ) 5 ui 5 ci(Ic),i = 1 , 2 , 3 , i f j } .

In accordance with [l 1, a point ( k , u l , u 2 , u 3 ) E a R is a point of the type of strict egress for the polyfacial set R with respect to discrete system (1) if a n d o n l y i f f o r s o m e j c {1,2,3} uj = b j ( k )

and

f j ( k 7 U 1 , u 2 , u 3 )< b j ( k

+ 1) - b j ( k ) ,

uj = c j ( k )

and

fj(k,ul,u2,u3) > cj(k

+ 1) - c j ( k ) .

or

Geometrically these inequalities mean the following: If a point M = M ( k ,211,212, us)E dR is a point of the type of strict egress, then the first consequent point C [ M ]@ n ( k 1).

+

1.2.4. Retract and retraction

If A C B are any two sets of a topological space and T : B + A is a continuous mapping from B onto A such that ~ ( p=) p for every p E A , then T is said t o be a retraction of B onto A. If there exists a retraction of B onto A , A is called a retract of B . 2. Main Result

Let R be a polyfacial set. In this part we will solve our problem supposing that for one fixed j E { 1,2,3} the corresponding sets R k and 0; consist of points of the type of strict egress and that the set R is of Liapunov type with respect to both remaining variables. Theorem 2.1. Let b i ( k ) , c i ( k ) , bi(k) < c i ( k ) , i = 1 , 2 , 3 be realfunctions defined on N ( a ) and let f i : N ( a ) x IR3 + IR, i = 1 , 2 , 3 be functions that are continuous with respect to their last three arguments. Suppose that for a fixed j E {1,2,3} all the points of the sets R;, R& are points of strict egress, i.e., if ( k ,u1, u2, u3) E R&, then

fj(k,ul,u2,u3) < bj(k

+ 1) - b j ( k ) ,

(4)

and if ( k ,u1, u2, u3) E R&, then f j ( k , u l , u 2 , u 3 ) > Cj(k+ 1) - c j ( k ) .

(5)

Further suppose the set 52 is of the Liapunov type with respect to the i-th variable f o r i = 1 , 2 , 3 ,i # j , i.e. for every ( k ,ul,u2,213) E R

bi(k

+ 1) < ui + f i ( k ,

211, u2,u3)

< ci(k + 1).

153

Then there exists a solution u = (u;(Ic),u;(Ic),u;(Ic)) of system (1) satisfying the inequalities

for every Ic E N ( a ) .

2.1. Proof of the Main Result The proof will be performed by a contradiction. We will suppose that there exists no solution satisfying inequalities (6) for every k E N ( a ) . Under this supposition we prove that there exists a continuous mapping (a retraction) of a closed interval onto its both endpoints which is, by known facts, impossible. 2.1.1. Specification of the general scheme of the proof

Without the loss of generality we may suppose that the j in Theorem 2.1 are of the type of is equal to 1, i.e. all the points of the sets and strict egress. Each solution of system (1) is uniquely determined by the chosen initial condition

For the following considerations, let u; and u: with u: E (bi(a),ci(a)), i = 2,3, be chosen arbitrarily but fixed. Now the solution of (1) is given just by the choice of u?,we can write

Define the closed interval

Hereafter we show that, under the supposition that there exists no solution satisfying inequalities ( 6 ) , there exists a retraction R (which will be a composition of two auxiliary mappings) of the set B := I onto the set A := d I = { b l ( a ) , q ( a ) } .This contradiction will prove our result.

154

2.1.2. Auxiliary mapping R1

Let a point GI E I be fixed. The initial condition ul(a) = GI defines (in the case iil E i n t l ) the corresponding solution u = u(k,G1) = (ui(k, G I ) , u2(k, GI), ~ ( kGI)) , which accordingly to our supposition does not satisfy inequalities (6) for every k E N ( a ) . That means that there exists an index s E N ( a ) such that u(k,G1) E R(k) for a 5 k < s and u(s,Gl) $ R(s). This is true for 61 E d l , too. Indeed, if Gl E d I , i.e. GI = bl(a) or GI = cl(a), then s = a, because u(a,G1) $ R(a). Further, because of the strict egress property of R b and R&, either ui(a 1,Gil) < bl(a 1) or u l ( a 1 , G l ) > cl(a 1). If GI E int I, then, as the set R is of the Liapunov type with respect to the second a third variable and u(s - 1,Gl) E R(s - l ) ,

+

+

+

b2(S) b3(S)

+

< u2(s,G1) < < u3(S,Gl) <

c2(s)

c3(s)

and the validity of inequalities (6) has to be violated in the ul-coordinate. Now, two cases are possible: either u(s,Gl) $ R(s) or u(s,Gl) E dR(s). In the first case ul(s,iil) < bl(s) or ul(s,Gl) > cl(s). In the second case ul(s,Gl) = b l ( s ) or ul(s,iil) = c1(s) and, due to (4)and (5), u l ( s + l , G l ) < b l ( s 1) or u1(s 1 , G l ) > c1(s 1). Unfortunately, for the next consideration the index s (the first index for which the graph of the solution is out of 0) would be of little use. What we will need is the last index for which the graph of the solution stays inside We will denote this index as k* and will call it the leaving index. Obviously, k* = s - 1 if u ( s , 61) 4 n(s> and k* = s if u ( s , 61) E dR(s). We could write k* = k* (GI) but we will mostly omit the argument 61, unless it is necessary. From the above considerations it follows that

+

+

+

n.

and w(k*

+ 1,G)< bl(k* + 1)

or ul(k*

+ 1 , ~ 1>) cl(k* + 1)

Now we construct the auxiliary mapping R1: I R x Iw: For k E N ( a ) denote Vk,k+l the convex hull of the set

((k,bl(k)),(k,Cl(k)),(k+

1 , b l ( k + l ) ) , ( k + L C l ( k + 1)))

in Rz . (Recall that the convex hull of a set of points is the smallest convex set that includes the given set of points.) Obviously, Vk,k+l is for every

155

k E N ( a ) a trapezoid. Define R1 (GI) as the point of intersection of the line segment defined by its end points (k*,u1(k*,ih)), (k* l , u l ( k * 1 , G l ) ) with the boundary of Vk*,p+1 (see Figure 1). The mapping R1 is well defined on I , because a line segment with one end point inside of a convex set or on its boundary and the other end point outside of this set has to have just one intersection with the boundary of that set.

+

a

a+l

k*-1

k*

+

k*+1

t

Figure 1. Construction of the mapping R1

Prove that the mapping R1 is continuous. The point R1(G1)= (t(Gl),U(Gl)) lies either on the line segment with the end points ( k * ,b ~ ( k * ) (k* ), 1, bl(k* 1))or on the line segment with the end points (k*,cl(k*)), (k* l ,cl(k * 1)). Consider the second case (the first is analogical). The boundary line segment is described by the equation

+ +

+ +

U = c(k*) + (c(k*

+ 1) - c(k*))(t- k * ) ,

t E [k*,k* + 11

and the line segment joining the points (k*,u1(k*,G1)), (k* 1,Gl)) by the equation

(7)

+ l,ul(k* +

Solving equations (7) and (8) with respect to t and U we get the coor-

156

dinates of the point

R1(GI):

V(G1)= Cl(k*) +

(u1@*,Gd

Cl(k*

- Cl(k*))(Cl(k*

+ 1)- q(k*))

+ 1) - W(k* + 1,Gl)+ul(k*,G1)- C1(k*)

Let {v,}:=~ be any sequence with {v,} E I such that v, -+ 61.We will show that Rl(v,)-+ Rl(G1). Because of the continuity of the functions fi,

i = 1,2,3, ul(k,v,) -+ ul(k,Gl) for every k E N ( a ) .

(9)

We have to consider two cases:

I) (k*,u(k*,Gl)) E R(k*), i.e. bl(k*) < ul(k*,Gl)< cl(k*) 11) (k*,u(k*,Gl)) E aR(k*), i.e. ul(k*,~1) = cl(k*)

+

+ 1). I) In this case also ul(k*,vn)< cl(k*) and ul(k* + l,~,)> s(k* + 1)

Recall that in both cases ul(k* 1,Gl)> cl(k*

for n sufficiently large. That means that the leaving index k*(vn) is the same as k* given by Gl and thus

t ( w n ) = k* +

Ul(k*,%) - Cl(k*)

Cl(k*

+ 1)-u1(k* + Lv,)

+ul(k*,vn) - Cl(k*) '

The desired convergence Rl(v,)-+ Rl(G1)is easy to see due to (9). 11) Suppose k* > a (the special case k* = a will be studied later). In this case there can be u1(k*, v,) 5 cl(k*) for some members of the sequence {v,} and ul(k*,vn)> cl(k*) for the others. Without the loss of generality, we can suppose that there are two infinite subsequences {vk,,} and {vl,,} such that

u~(k*,vk,,) L cl(k*) and ul(k*+ 1 , ~> c~l ( k)* + 1) and u1(k*,W')> Cl(k*).

For the subsequence {vkn} the situation is the same as in the case I) and the convergence is guaranteed.

157

As for the subsequence {q,,}, the leaving index k*(q,) is different from k* given by i i 1 because ( k * ,ul(lc*,VZ,)) is already out of 2.For n sufficiently large, k*(VZ,) = k* - 1

becauseul(k*-llG1) < c l ( k * - l ) and thus a l s o u l ( k * - l , v ~ , ) < c l ( k * - 1 ) . Hence, the value of the mapping R1 for 211, is t(Vln)

U(Vln)

= k* - 1

Ul(k* - l , V l , ) - Cl(k* - 1) + Cl(k*)- Ul(k*,Vl,) + Ul(k* - l , V l , ) - Cl(k* - 1) '

= c(k* - 1 )

- l , w z n ) - Cl(k* - l ) ) ( C l ( k * ) Cl(k* - 1)) + (Ul(k* Cl(k*)- u l ( k * , v z n ) +Ul(k* - l , V l , ) -c1(k* - 1 ) '

Due t o (9), ui(k*,vl,) -+ u l ( k * , i i l ) = c1(k*) and thus

t(wzn) U(Vl,)

+ k* - 1 + Ul(k* - l,Wl,)

- Cl(k* - 1)

Ul(k* - 1,VZ,) - Cl(k* - 1)

+ c(k* - 1)+ (Ul(k* - l , V l n )

= k*,

- Cl(k* - l ) ) ( C l ( k * )- Cl(k* - 1))

Ul(k* - l , V l n ) - Cl(k* - 1)

Simplifying the last expression we get U(vl,) -+ c l ( k * ) . The convergence is proved, because Rl(Gl) = ( k * , c l ( k * ) ) . We have shown that Rl(vk,) + Rl(G1) and Rl(q,) -+ Rl(ii1) and thus Rl(v,) + Rl(ii1). Finally, let us consider the situation 11) when k* = a. This case does not cause any problem, too. If iil = c l ( a ) , then w, = u l ( k * , v n ) < q ( k * ) for all n E N and ul(lc* l , ~ , > ) cl(k* 1 ) for n sufficiently large and thus we have the situation described in I).

+

+

2.1.3. Auxiliary mapping R:! When we unite all the sets get an unbounded set

Vk,k+l

v :=

defined in the previous paragraph, we

u

Vk,k+l,

kEN(a)

the boundary of which consists of three parts:

av = VA u VB u VC, where VA is the vertical line segment connecting the points ( a , b l ( a ) ) and ( a , c l ( a ) ) ,VB is the polygonal line connecting the nodes ( k , b l ( k ) ) , k E N ( a ) , and VC is the polygonal line connecting the nodes ( k , c l ( k ) ) , k E N(a).

158

As we have seen, for 51 E I , Rl(ii1) E VBU Vc. Define RZ : VBU Vc --+ {bl ( a ) ,c1 ( a ) } as bl(a) if P E VB,

Rz(P) =

c l ( a ) if P E Vc,

The mapping Rz is obviously continuous. 2.1.4. Resulting mapping R and its properties

Define R := R2 o R1. Due to construction we have

R(bi(a))= bi(a), R(ci(a))= ci(a), and R ( I ) = 8 I . The mapping R is continuous because of the continuity of the partial mappings R1 and R2, so it is the sought contraction of I onto a I . But such a contraction cannot exist and thus we get a contradiction and the proof is complete.

Acknowledgments The first author was supported by the Grant 201/04/0580 of Czech Grant Agency (Prague) and by the Council of Czech Government MSM 00216 30503. The second author was supported by the Grant No 1/0026/03 of the Grant Agency of Slovak Republic (VEGA).

References 1. D I B L ~JK . , Anti-Lyapunov method for systems of discrete equations, Nonlinear Anal., Theory Methods Appl., 57 (2004), 1043-1057. 2. D I B L ~J., K Asymptotic behaviour of solutions of systems of discrete equations via Liapunov type technique, Comput. Math. Appl. 45 (2003), 1041-1057.

159

NONOSCILLATORY SOLUTIONS FOR EMDEN-FOWLER TYPE DIFFERENCE EQUATIONS

M. CECCHI

Depart. of Electronics and Telecom., University of Florence Via S. Marta 3, 50139 Florence, Italy E-mail: [email protected]

z. DOSLA* Depart. of Mathematics, Masaryk University Jana'Ekovo na'm. Za, 66295 Bmo, Czech Republic E-mail: [email protected] M. MARINI Depart. of Electronics and Telecom., University of Florence Via S. Marta 3, 50139 Firenze, Italy E-mail: [email protected] I. VRKOC Mathematical Institute of the Academy of Sciences Zitna' 25, Prague 115 67, Czech Republic E-mail: [email protected]

Using certain summation inequalities, the coexistence of various types of nonoscillatory solutions for an Emden-Fowler type difference equation is investigated. Discrepancies between discrete and continuous cases are pointed out as well.

1. Introduction

Consider the Emden-Fowler type difference equation

A (anIA% l%n A%) = b, Iz,+1 I%n &+I

7

(1)

where a = {u,}, b = {b,} are positive real sequences for n 2 1 and Q > 0, ,B > 0. Equation (1) has been widely investigated in the literature (see, *Supported by the Grant Agency of the Czech Academy of Sciences, grant 201/04/0580

160

e.g., Refs. 1, 2, 5, 6, 10, 11, 12 and references therein). Our aim here is to extend to equation (1) some recent results obtained by the authors in Ref. 7 for the half-linear equation

A (an IAzn lawAzn) = bn 1zn+1laSgnzn+1.

(2) More precisely, the coexistence of possible types of nonoscillatory solutions of (1) is examined by using certain summation inequalities stated in Ref. 9. Some discrepancies between the continuous and discrete cases are illustrated and open problems are proposed. For the sake of simplicity, we restrict our study to solutions z = { z n } for which X k > 0 for large k. It is easy to show (see, e.g., Lemma 1 in Ref. 4) that any nontrivial solution of (1) is monotone for large k, and belongs to one of the two classes

M+ = {z solution of (1) : z k > 0 , Azk > o for large k}, M- = {z solution of (1) : X k > 0 , Azk < 0 for k 2 1). Both classes are nonempty: clearly solutions with initial conditions z1 > 0, Ax1 > 0 are in the class M+, and M- # 0 as follows, e.g., from Theorem 1 in Ref. 4, or from Theorem 6.10.4 in Ref. 1, with minor changes. = {z:]} its quasi-difference For any solution z of (l),denote by

]z !

= anlAznIasgn

Axn.

In view of the sign assumptions on a , b , if z E M+, then z and are positive increasing for large k. If z E M-, then 3: is positive decreasing and z[l]is negative increasing for k >_ 1. Thus M+,M- can be, a-priori, divided into the following subclasses:

M; = {z E M+ : limz, = t , < oo}, n

~ 2= {z, E M+ ~ : limz, n ~ 2= {z,E M+~ : limz, n

=

= limzkl =

031,

n

n

M; = {z E M- : limz, = l,, 0 n

M<[ = {z E M- : limsc, n

~6~= {z E M-

= 0, n

< e, < oo},

limzF1 = -t,, n

: limz, = limz!] n

e,, o < e, < 031,

= 03, limz:]

o < t, < 03>,

= 01.

2. Main results In the literature, solutions in classes M+,M- have been described from different points of view. For instance, in Ref. 2, a nonlinear system including

161

(1) is considered and a classification of solutions is given, depending on the convergence or divergence of the series C,"==, ( 1 / a n ) ' l a .In Refs. 5, 6, the same problem has been treated by means of the following series

The asymptotic behavior of solutions in M- depends on series V1,V2, while the behavior of solutions in M+ depends on series Z 1 , Z 2 , as the following results show.

Theorem A. [Ref. 6, Corollary 3.31 For solutions of (1) in M-we have: i l ) If & = V2 = 00, then M- = Mco # 0;

iz) I f Vl < 00 and V2 = 00, then Mynh = 0, Mce # 0. i3) I f Vi = 00 and fi < 00, then MF # 0, M<e = 0. i4) If V1 < 00 and Vz < 00, then MF # 0, MCt # 0. I n addition, for a 5 p, we have Mco = 0 an claims i 2 ) - i 4 ) . Theorem B. [Ref. 5, Theorem 71 For solutions of (1) in M+ we have:

# 0. i 2 ) If Zi = 00, 2 2 < 00, then M t = 0,M;,, # 0. i3) If 21 < 00, then # 0, M;,e = 0. I n addition, f o r a 2 p, we have MI&,, = 0 in claims i l ) If

21 = 2 2

= 00, then M+ = M+ m , C C

iz),

i3).

To study the possible coexistence of solutions in classes M- and M+, we use summation inequalities involving the series N

/ n

\ P

N

T,(A,B)

=

lim N-CC

CA"C Bk n=l

(krn

l/a

)

.

Such a result has been proved in Ref. 9 and completes analogous one stated in Ref. 7 for the case a = p.

162

Lemma A. [Ref. 9, Lemma 21 Let A = { A n } , B = {Bn} be two real positive sequences for n 2 1. il) If fa > ,B > 0 , then Sp(A,B)= 00 ===+ T,(A,B) = 00. iz) If P > fa > 0 , then T,(A,B) = 00 + Sp(A,B)= 00. From here we get the following relations for series V,, Zi (i = 1 , 2 ) .

Lemma 2.1. If fa > p > 0 , then 21)Z2=00 v2=00, i2)Z1=00 I f O < a < p , then i3) v 2 = 00 ----r. 2 2 = 00, 24) v1=00

*

=j v1=00.

==+

z1= 00.

Proof. Putting A, = (an+l)-'la, we obtain T l / p ( b , A ) = 6 and Sll,(b,A) = lim N-KI

N+l n=2

c

(a, 1

n-l b k )

=21.

k=l

Thus, from Lemma A claims i2), 24) follow. Putting A, = (an)-'/,, we have T,(A, b) = V2 and

and so from Lemma A, claim i l ) follows. Since for n 2 2 it holds

we have

hence

Sp(A,b) = 00 + 2 2 = 00. Applying again Lemma A, claim

23)

follows.

Note that the opposite statements to Lemma 2.1 need not to hold, as it follows from Section 2 in Ref. 9, with minor changes.

163

Hence the possible cases concerning the convergence or divergence of series V,,Zi (i = 1 , 2 ) are the following:'

C,)

:

C2) : C3) : C4) : C5) :

v1 = 2 2 = vz = 00; 21 = v 1 = 00, 2 2 < 00, v2 < 00; 21 < 00, v 1 < 00, 22 = v2 = 00; 21 < 00, v1< 00, 2 2 < 00, If5 < 00; 21 = v 1 = 00, 2 2 < 00, v2 = 00;

21 =

Observe that cases C1)-C4) may occur without additional conditions on a , p. When a # p, the cases C g , C8) may occur only if a > p, and CG), C7) only if a < p. When a = p # 1, i.e. in the half-linear case, the situation is slightly different, because the cases Cs), cfj)may occur only if a = p > 1 and CT), Cg) only if a = p < 1 (see Corollary 2 in Ref. 7). Finally, in the linear case, i.e. when a = p = 1, the only possible cases are C1)-C4) (see Section 2 in Ref. 7).

As claimed above, we have

M+#0,

M-#0.

The following results describe the decomposition of these sets.

Theorem 2.1. For the equation (1) with a > p we have: i f Cl) holds, then Mf = M+ W , W ) M-

= M-0,o;

if C2) holds, then M+ = M+ oo,el MCe = 01 MF # 0;

if C3) holds, then M+ = M l , MCe # 0, Mg

= 0;

if C4) holds, then M+ = Mz, M i e # 0, M; # 0; if C5) holds, then M+

= M+ oo,e,

M- = Mco;

if C8) holds, then MI[+= M+ e l M- = M0,O'

164

Theorem 2.2. For the equation (1) with a < ,B we have:

if C1) holds, then M+

= Mk,,,,

if CZ)holds, then ML,, # 0 , M t

= 8,

M- = M;;

# 0, M- = Mce; holds, then M"n+,e= 0,Mt # 0 , M , o = S,M& # 8,M; # 0;

if C3) holds, then M;,e if C4)

M- = Mc0;

if C6)holds, then M+

= 0, Mz

,, = M+

M- = M-o,e ;

if C7) holds, then M+ = M+ o o , ~ M, = M;. The proofs of Theorems 2.1, 2.2 easily follow from the above classification and Theorems A, B. The above theorems show the discrepancies between the superlinear equations (a < 0)and sublinear equations (a > p). In addition to the fact that cases Cs) - Cs) may occur only for either superlinear or sublinear equations, we can see discrepancies in the decomposition of classes M- and M+ in cases Cz) - C4). Observe that Theorem 2.1 does not give an answer about the existence of solutions in Mco when any of cases Cz),C3),C4) holds. The same concerns Theorem 2.2 and the existence of solutions in M&,,,, when any of these cases holds. This fact will be compared with the continuous case in the next section. The following examples show that these solutions can exist.

Example 1. Consider the nonlinear difference equation ( n 2 1)

where

*,

= n(n

+

2 2)>'

Then the case Cz) occurs and the sequence x, = l / n is a. solution in the subclass MI&. Hence, by Theorem 2.1 the unbounded solutions coexist with solutions which belong to MI; and Mco.

Example 2. Consider the nonlinear difference equation ( n >_ 1)

165

Then the case C2) occurs and the sequence z, = n(n- 1)is a solution in the subclass M&,w. Hence, by Theorem 2.2 any positive decreasing solution has a nonzero limit and it coexists with solutions in ML,, and 3. Discrete versus continuous case

In this section we compare behavior of solutions for difference equation (1) and for the differential equation

(a(t )Iz' 1 "sgn z')' = b(t ) IzI Psgn z,

(5)

where the functions a(.), b(.) are positive continuous on [0, 00) and a > 0, p > 0. Theorems 2.1, 2.2 are discrete analogues of some results stated for (5) in Ref. 8, where the same subclasses of solutions have been investigated. Nevertheless, there are some discrepancies between the discrete and continuous cases. The basic discrepancy consists in the fact that both classes Mf ,M- are nonempty in the discrete case, while this property may fail for (5) when a # p, see e.g. Ref. 8. Comparing Theorems 2.1,2.2 with Theorems 2,3 in Ref. 8, when any of # 0 in the contincases Cz),C3),C4) holds, we have Mc0 # 0 or uous case, according to a > p or a < p, respectively. This property is essentially due to the presence of the so-called singular solutions. Since the singular solutions do not exist in the discrete case (see, e.g., Ref. 4), it is an open problem whether the same result holds for the difference equation (1). Namely, if a > p, is it true that Mc0 # 0 in cases C2) - C4)? if (2: < p, is it true that ML,, # 0 in cases C2) - C4)? Another discrepancy is due to the relations between the convergence of the series 2 1 and its continuous counterpart

even in the linear case, as the following example shows.

Example 3. Consider for n 2 2 the linear difference equation A(n2gn(Ax,)) = eXp(n2)zn+l, where g = {g,} is the sequence given by g,

=e

+ e4 + e9 + ... + e(n-lI2 for n 2 2.

166

Let E = E ( t ) be a positive continuous function on [2, co),such that E(n) = gn and

~ ( tI:)(t - 1) exp(t - I ) ~ .

(7)

Such a choice is possible because gn 5 (n - 1)exp(n - 1)2. Jointly with (6) consider the corresponding linear differential equation (t2E(t)z’)’ = exp(t2)z (t 2 2). (8) We will show that all solutions of (6) are bounded, while any solution z of (8) satisfying z ( t )> 0, z’(t) > 0 for large t , is unbounded. Clearly,

c 03

21 =

1

c

n-l

-c e x p ( k 2 ) I :

O01 < co n

n2gn k = 2 n=2 and hence, from Theorem B, any solution of (6) is bounded. Fixed 0 < E < 1, we have n=2

lim t-03

Choosing T so that

Jiexp(s2)ds = co. exp[(t - ~

) 2 ]

Jiexp(s2)ds 2 exp[(t

Lrnt2E(t)

-

E ) ~ for ]

t 2 T , we have

exp(s2)ds dt

J1=

and, in view of (7),

Hence, from Theorem 1in Ref. 8 the differential equation (8) has unbounded solutions. It is worth to note that if we consider instead of (6) the difference equation A(n2gn(Azn))= exp((n

+ 1)2)zn+1,

(9)

then we can see that (9) has all solutions in M+ unbounded. Indeed, we have for (9) 03

03

k=l

n=2

exp n2

= 03,

167

and the assertion follows from Theorem B. The difference in the boundedness of solutions of ( 6 ) and (8) can be viewed in the context of the unified theory of the times scales for the continuous and discrete cases (see, e.g., Ref. 3). Our example illustrates that the convergence of the improper integral

depends on the choice of times scales T: for T = M the integral is convergent, while for T = R it is divergent.

References 1. R. P. Agarwal, Difference Equations and Inequalities, 2nd Edition, Pure Appl. Math. 228, Marcel Dekker, New York, 2000. 2. R. P. Agarwal, W.T. Li, P.Y.H. Pang, Asymptotic behavior of nonlinear difference system, Appl. Math. Computat. 140 (2003), 307-316. 3. M. Bohner, A. Peterson (Eds.), Advances in Dynamic Equations on Time Scale, Birkhauser, Boston, 2003. 4. M. Cecchi, Z. DoS16, M. Marini, Positive decreasing solutions of quasi-linear difference equations, Computer Math. Appl. 42 (2001), 1401-1410. 5. M. Cecchi, Z. DoS16, M. Marini, Unbounded solutions of quasi-linear difference equations, Computer Math. Appl. 45 (2003), 1113-1123. 6. M. Cecchi, Z. DoS16, M. Marini, Limit behavior for quasilinear difference equations, Proc. ICDEA 6, B. Aulbach, S. Elyadi, G. Ladas Eds., CRC Press, London, 2004, 383-390. 7. M. Cecchi, Z. DoS16, M. Marini, I. VrkoE, Summation inequalities and halflinear dzfference equations, J. Math. Anal. Appl. 302 (2005), 1-13. 8. M. Cecchi, Z. DoS16, M. Marini, I. VrkoE, Integral conditions for nonoscillation of second order quasilinear differential equations, to appear on Nonlinear

Analysis, T.M.A. 9. M. Cecchi, Z. DoS16, M. Marini, I. VrkoE, Asymptotic properties f o r half-linear difference equations, submitted to Mathematica Bohemica. 10. M. Tan,E. Yang, Oscillation and nonoscillation theorems f o r second order nonlinear dzfference equations, J. Math. Anal. Appl. 276 (2002), 239-247. 11. E. Thandapani, S. L. Marian, The asymptotic behavior of solutions of nonlinear second-order difference equation, Applied Math. Letters 14 (2001), 611-616. 12. P. J. Y. Wong, R. P. Agarwal, Oscillation theorems and existence of positive monotone solutions f o r second order nonlinear difference equations, Math. Comput. Modelling 21 (1995), 63-84.

168

THE COMPARATIVE INDEX FOR CONJOINED BASES OF SYMPLECTIC DIFFERENCE SYSTEMS

J. V. ELYSEEVA Moscow State University of Technology, Department of Applied Mathematics, Vadkovskii per. 3a, 101472 Moscow, Russia E-mail: [email protected] We present the new concept applied for the comparison of conjoined bases of symplectic difference systems. This concept is closely related to the concept of multiplicities of focal points introduced by W. Kratz. Thus, the comparative index p(i) = pl(i) p z ( i ) gives us possibility to describe the case when the image condition and the Riccati inequality for conjoined bases Yi,Yi do not hold. In this work we prove the main properties of p ( i ) and investigate connections between p ( i ) and the number of focal points m(i).

+

1. Introduction

We consider the symplectic difference system

associated with the Riccati difference operator

Rw[Q]= Ci - Qi+lAi + DiQi - Qi+lBiQi, where Wi, yi, J are real partitioned matrices with n x n blocks Wi =

[: 21

,yZ =

[21 , J

= [!I

i],

and I , 0 are the identity and zero

matrices. Note that the symplecity of any matrix W with n x n blocks A, B , C , D means that

WTJW = J

(1.2)

or ATD - CTB = ADT - BCT = DTA - BTC = DAT - CBT = I , and the matrices ABT, CDT, CTA, DTB, BAT, DCT, ATC, BTD are

169

symmetric. In particular, the matrix

[;I

A[W] = diag (CTBT)W = W T [;;]w-

(1.3)

is symmetric (here diag (C B ) denotes a diagonal block matrix with n x n blocks). It is well known that the Riccati matrix difference equation

R w [Q] = 0, i = 0 , . . . , N

(1.4)

has a symmetric solution QT = Qi iff there exists a conjoined basis of system (1.1)

YT J y i = 0, rank% = n such that the condition detXi

# 0, i = 0,. . . , N + 1

(1.5)

holds. If (1.5) does not hold, we have to consider generalized solutions of equation (1.4). According to the definition 2 , a conjoined basis of (1.1) is said to have a focal point in (i,i 11 if the conditions

+

KerXi+l

c KerXi,

XiX,f+,Bi 2 0

(1.6) (1.7)

do not hold (here, t denotes the Moore-Penrose inverse of the matrix A, KerA denotes the kernel of A, for a symmetric matrix A we write A 2 0 if A is positive semidefinite). For the case when conditions (1.6), (1.7) hold for i = 0 , . . . N the next separation theorem is proven '. Theorem 1.1. Let yi, $ be conjoined bases of the Hamiltonian difference system with the conditions

ImXi

c ImXi,

XT(Qi - Qi)Xi 3 0, i = 0 ,

(1.8) (1.9)

where XTQiXi = UTXi, ,XTQiXi = UTXi and does not have focal points in (0, N 11 then neither does $ and conditions (1.8),(1.9) hold for i = 0 , . . . N + 1.

+

Note that if condition in (1.6) does not hold, the matrix in (1.7) may loose the symmetry. This problem is solved in Ref. 5, where the following concept of the multiplicities of a focal point in (i, i 11 is offered.

+

170

Definition 1.1. A conjoined basis Yj has a focal point of multiplicity m l (i) in the point i 1 if ml(i) = rankMi, where

+

and this basis has afocal point of multiplicity mz(i)in the interval (i, i + l ) if m2(i)= ind(T?XiX!+,BiTi), Ti = I-MJMi, where indA is the number of negative eigenvalues of a symmetric matrix A. The number of focal points in (i, i -t13 is defined by m (i) = ml (i) m2 (i).

+

The main purpose of the present investigation is to derive the generalization of Theoreml.1 for the case when Yj has focal points in (0, N + 11. For the continuous case we have the formula (see Theorem 5.2.1 in Ref. 6) ind(0 - Q)(t+) - ind(Q - Q)(t-) = m(t)- k ( t ) ,

(1.10)

which connects the numbers of focal points m(t) = defX(t), k(t! = defk(t) (here defA is the dimension of KerA) of conjoined bases Y(t), Y(t) of the Hamiltonian differential system and ind(Q - Q), where Q, Q are the solutions of the differential Riccati equation . Note that the similar result for the discrete case remains in general an open problem. In this work we present the first results in this direction. So we introduce the comparative index p ( i ) for conjoined bases of (1.1) which gives us possibility to describe the case when (1.8),(1.9) do not hold. In the next section we prove the main properties of the comparative index (Theorem 2.1) and investigate the relations between the comparative index and the number of focal points (Lemmas 2.2 and 2.3). In Sec. 3 we develop our approach based on symplectic factorizations of a conjoined We show that for conjoined bases Y,,g factorized in form basis Y , Y, = LiWi[MT 0IT, = ,!,i!%i[&T 0IT with symplectic lower triangular block matrices Li, J?i we have that p ( i ) = indAy[W~(L~lJ?i)!fli], where Ay[W] is an extension of A[W] given by (1.3). In consequence of our index results we present (without the proof) the discrete version of (1.10) 738.

Ap(i)= p ( i

+ 1) - p ( i ) = m(i)- k ( i )

which we derive under assumption ImXi+l C ImXi+l, KerXi+l

KerXi.

2. Main results

Definition 2.1. For 2n x n-matrices Y, Y which obey the conditions

yTJ Y = 0, PTJP = 0,

(2.1)

rankY = rankY = n,

(2.2)

171

and w = w(Y,P) = Y T J Y define the comparative index p(Y,P): P(Y,Y) = P1 + P2, M = ( I - X t X ) w,

(2.4)

= rankM,

(2.5)

p2 = indP,

P

(2.6)

= 7~ (,~xtX)T , T = I - M ~ M .

(2.7)

The main properties of p(Y, p) are formulated in the next theorem. Theorem 2.1.

i) The matrix M in Definition 2.1 can be replaced b y

M = (I -XXt) x,

(2.8)

the matrixP in (2.6) is symmetric and

P = P ( X T [ Q- Q ] X ) 7

(2.9)

for any symmetric Q , Q such that X T Q X = XTU, X T Q X = X T U ,

ii) P(Y,?) = o i 8 1 mX c ImX, X T ( Q - Q)X 2 0, iii) p(Y, P) 5 min(rankw, rankx). The proof of Theorem 2.1 is based on the next lemma. Lemma 2.1. If conditions (2.1),(2.2) hold for Y, then

X X W ( I - X t X ) = 0,

(2.10)

rank(I - X X t ) U = rank(1- X X t ) U ( I - X t X ) = rank(1- X t X ) = n - rankX, I xtx), det C # 0, C = X - ( I - X X ~ ) U ( -

(2.11) (2.12)

det 93 # 0, 93 = x - ( I - X x t ) ~ .

(2.13)

Proof. Condition (2.10) follows from (2.1), while (2.11) is derived from (2.2) because of det(XTX U T U ) # 0. We have rank(XTX U T U ) ( IX t X ) = rank(1- X t X ) = rankUTU(I - X t X ) = rankUT(I - X X t ) U ( I -

+

+

172

<

<

X t X ) r a nk( 1- X Xt)U(I-XtX) 6 rank(1-XXt)U rank(1-XtX), where we use that rankAB rankA,rankB and (2.10). Then (2.11) holds. The matrix C = X - ( I - X X t ) U ( I - X t X ) is the orthogonal sum of X and R = - ( I - X X t ) U ( I - X t X ) , then rankC = rankX r a n M = n, C-' = Ct = X t + R t and 332 = C - ( I - X X t ) U X t X = C(I--RtUXtX) is nonsingular because of nonsingularity of C and I - R t U X t X . 0

<

+

Proof of Theorem 2.l(i) We have M = ( I - XtX)w(Y,?) = ( I XtX)( X T U - U T X ) = - ( I - x t x ) U T X = - ( I - x t x ) U y I - x x t ) X = = C T ( I - X X t ) X = C T M , then p1 = rankM = rankM because C defined by (2.12) is nonsingular. Moreover, M t M = M t M , then M can be replaced by M in (2.5),(2.6),(2.7). We have I m X T g ImX, then XT = X X t X T and P = (wTXtX) 7 = T(UT X - X-TXtT(XTU)XtX)T =

7 (X'[Q

- QlX) 7 and we complete the proof of (i). Note that (ii) follows from (i) and the proof of (iii) is similar to the proof of Lemma l(iv) in Ref. 5. The proof of Theorem 2.1 is completed. Consider p ( y i , c ) = p ( i ) for conjoined bases Y,, % of (1.1). Then we have that the Wronskian identity w ( Y , , e ) = wi = const holds. Using this identity it is possible to prove the next lemma.

Lemma 2.2. Let = [0 IIT , then

be a principal solution of (1.1) in the point k , i.e. P(Yk+l, %+l) = m ( k )

+

+

and pl(k 1) = m l ( k ) , p 2 ( k 1) = m 2 ( k ) , where ml(i),m2(i) are the according to Definition 1.1. multiplicities of focal points for Proof. Consider the Wronskian identity W k + l = Ycl??k+l = wk for the case ?k = [O 1IT. For this case we have that Wk = xr, Yk+1 = [BkT DkTIT, then the proof follows from Definition 1.1 and Theorem 2.1 (i). 0 Lemma 2.3. Let g, Y , be normalized conjoined bases of ( l . l ) , i.e. the matrix Zi = [RY,]is symplectic. Define Si = [Xixi]*, where Xi is the upper block of E. Then p(Si+l,Si) = m(i), and pi(i) = mi(i), p2(i) = m2(i),where p(Si+l,Si) is the comparative index and m(i) is the number of focal points for Yi.

Proof. Using the assumptions of this lemma we have Zi+l = WiZi or Zi+lZt:l = Wi, in particular Bi = SzlJSi = w(Si+l,Si). Then the proof follows from Definitions 1.1 and 2.1 (we also use that P = P T ) . 0

173

3. Symplectic factorizations and index results

In this section we present index results for the comparative index p ( i ) . Our approach is based on the special symplectic factorization considered for Introduce the following set of matrices.

x.

Definition 3.1. % E

if

F

G

%= [-GF]

with n x n blocks F, G which obey the conditions:

F+G=I, F 2 = F, F - F. Conditions (3.1),(3.2),(3.3) mean that % are symplectic orthogonal matrices and F,G = I - F are the orthogonal projections '. For example, if F, G are diagonal matrices, conditions (3.1),(3.2),(3.3) describe the set of permutation matrices for the symplectic case 8 . Introduce the symplectic and orthogonal matrices T x t x and % x x t which obey (3.1) with the blocks

F x t x = XtX, G x t x = I - XtX, F x x t = X X t , G x x t = I - XXt

(3.4) (3.5)

respectively. For any n x n matrix X matrices (3.4),(3.5) satisfy (3.2),(3.3). The main result of this section is the next theorem.

+

Theorem 3.1. The matrix Yi, i = 0,. . . N 1 is a conjoined basis of (1.1) iff there exist symplectic lower triangular matrices Li, %i E 0%and nonsingular matrices r)37i such that

hold with

Proof. The necessity is proved in Ref. 8, where we derive Li in terms of solutions of a transformed Riccati equation. Here we present two other factorizations which are connected with the case when %i = % x t x or

174

'Xi = '32xxt and X = X i is the upper block of 5 . So by Lemma 2.1 we have that any conjoined basis of (1.1) obeys (3.6) with Li = L x x t =

[ii:], Qi = QT, QiXi

'32i = 'Xxxt,

= XiXfUiXfXi,

= l)nxxt = X i - G x x t Ui, X = X i ,

(3.9)

W x x t has the blocks (3.5), where X = X i and the condition det #0 holds by Lemma 2.1. We also have that any conjoined basis of (1.1) obeys (3.6) with Li = L x t x , '32i = %XtX, mi = I , x = xi,

(3.10)

where L x t x satisfies the equation

~ x t x n x t x= [ ~ L x~x t]g ~ X x t [m$xtolT, x = X i with mxxt defined by (3.9). Note that for this case we have 9Jli z I , then Hi in (3.7) is the unit upper triangular block matrix by (3.8). Any matrix Y, factorized according to (3.6) obeys (2.1), (2.2) because of r a n k 5 = rank[!%?ZT0IT = n,X T J Y , = [!@ O]J[%QT0IT = 0 (we use that the symplectic matrices Li,'32i are nonsingular and satisfy (1.2)). Moreover, if (3.6) holds for yZ+l, Y,,then 5 + 1 = Wiyi iff [mL1olT = oIT or [ ( i ~ ~ i + l m ;oIT l ) ~= H ~ [ 0I1 ~ . Then Y, obeys (1.1) iff the symplectic matrix Hi in (3.7) is upper triangular 0 and (3.8) holds.

'32zl~21~i~i~i[mT

Corollary 3.1. If (3.6) holds for a conjoined basis of (l.l),then the symplectic matrix (3.11) in (3.6) defines Qi = ZiK,T1 = QT :

175

Then (3.12) is proved. To prove (3.13) rewrite the identity L2,WiLi = Xi in (3.7) in the form i z l W i i i = fii+lXifiil,

-

where Li =

[iiy] ,

Qi = ZiKF', Di = diag(KiK;lT).

In partic-

ular, [OI]LzlWiLi[IO]T = Rw[Qi] = KGITIOI]Xi[IO]TKF1. Then I]Xi[IOITFilMi and (3.13) holds iff X,T+,Rw[Qi]Xi = 1Mc+1Fi+l[O Fi+1[OI]Xi[IOITFi= 0. But the last condition is provided by (3.2), (3.3) and the structure of Hi in (3.7) which is upper triangular by (3.8). 0 Remark 3.1. According to (3.14) we have rankxi = rankFi, and I XiXi = T(I - FiF/)KtT1 = TGiKzT1, detT # 0, where we have Fi = F/ for the orthogonal projection Fi. Here we use the assertion (see Remark 2.5 in Ref. 7) that for X = d X B , det A # 0, det B # 0, we have I - X X t = T ( I - X X t ) d - l , I - X t X = B-l(I - X t X ) P , where matrices T, P are nonsingular. Corollary 3.2. Let conjoined bases fi, yi be factorized according to (3.6) with the matrices Li, &, ?Jki and Li, Wi, respectively. If the matrix Ci = LL1,!,i is separated into the blocks L i p , Ic,p = 1 , 2 , then p l ( i ) = rankM1, Ma = GiCCf,Fi,

P: =

(3.15)

p 2 (i) = indP:,

(3.16)

r.oiv,.oi = F i ~ c f T ~ ; , F iy, = I - M"M;,

(3.17)

where p(K,%)= p ( i ) = p l ( i )

+ p~a(i)is the comparative index for K , E .

Proof. According to Remark 3.1 and Theorem 2.1 we have hi = ( I - XiX!)Xi = TGiKF'k$i?Jki = TMZ?Jki, d e t T # 0, det?Jki # 0, where we use that Cil = KzTIKi and Xi = l?i&?Jki. Then p l ( i ) = rankMi = rankMa. Applying Remark 3.1 once again we have 7;: = I - M!Mi = ?JkjZi'(I - MPtMS)P = ?JkL1yP,d e t P # 0. Then pi = - Qi)Xi7;: = p*rfii&T(2iR;1 - ziK;1)k.fl.'7-*P 2 2 2 , = P T y F i C f T C i l F i y P , det P # 0, where we use Corollary 3.1 and Lal = KTZi - ZTKi. Then p 2 (i) = indPl and the proof is completed. 0

xXT(Qi

For any 2n x 2n symplectic matrix W and any n x n matrix Z define

A*,[W] = d i a g ( 2 I)AIW]diug(ZT I), A*,*[W]= diug(I Z)A[W]diug(I Z T ) , where h[W]is given by (1.3).

(3.18) (3.19)

176

Theorem 3.2. Let conjoined bases Then we have

e,Y , be factorized according t o (3.6).

p ( y Z , e ) = indA*,*[%TLi@i],Li = LT1,?,i, Z = fii,

(3.20)

m(i)= indR;[%i+1Hi%T] = indA;[LzlWiLi], 2 = Fi,

(3.21)

where p ( Y , , l $ ) = p ( i ) is the comparative index and m(i) is the number of focal points. Proof. The proof of (3.20),(3.21)is similar. Consider the proof of (3.20). Under the notations of Corollary 3.2 we have that

where the notation B-C means that B is congruent to C , i.e. B = A

ATCA, det A # 0. Then we apply Lemma 2.7 in Ref. 7. So we have that O M ind MT = ind f T ] = rankM + ind(1- M t M ) T D ( I - M t M ) ,

[

]

[

where DT = D and M, D are n x n matrices. Applying this result for the case D = Vi,M = - M t and using Corollary 3.2 we complete the proof of (3.20). 0

Remark 3.2. Consider Theorem 3.2 for the particular cases. Assume that 8, are factorized according to (3.6) with the factors defined by (3.9). Then, for this particular case we have p(Y,, = indhF[%TLi@i]for Li =

[ii ;] - Qi

e)

, Qi

= XiXjUiX!,

Qi

= X i X j f i i X j , 2 = X i X j and

%i,

have the blocks given by (3.5) for X = X i , X = X i respectively. For the case when l$, yi are factorized according to (3.10) we have that m(i) = indA5 [%i+1Hi%T] for the symplectic unit upper triangular matrix Hi and 2 = XjXi. Matrices %i have the blocks defined by (3.4) for X = Xi. Note also that Hi may be presented in terms of normalized with Y , by Lemma 2.3.

Theorem 3.3. Let conjoined bases C, Y, be factorized according t o (3.6). If the conditions ImXi+l ImXi+l, KerXi+l KerXi hold, then indA;[LZ,Wiii] = m(i)+p(i)= p ( i + l ) + k ( i ) , where 2 = &, m ( i ) , k ( i ) are the numbers of focal points in (i, i 11 for y i , fi and p(yZ,$) = p ( i ) .

+

Proof. The proof is presented in Ref. 10.

0

177

Corollary 3.3. If the conditions ImXi+l C ImXi+l, KerXi+l & KerXi hold, we have A p ( i ) = p(i 1) - p ( i ) = m(i)- rh(i).

+

References 1. L. Erbe and P. Yan, J . Math. Anal. Appl. 167,355 (1992). 2. M. Bohner and 0. DoSlf, Rocky M o u n t a i n Journal of Mathematics 27, 707 (1997). 3. F. Gantmacher, The Theory of Matrices, v.1, Chelsea Publishing Company, New York, 1959. 4. M. Bohner, Math. Inequal. Appl. 1(3), 375 (1998). 5. W. Kratz, J . Diflerence Equations and Appl. 9, 135 (2003). 6. W.Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin, 1995. 7. J. Elyseeva, Proceedings of I C D E A 2003, 127 (2005). 8. J. Elyseeva, Comp.Math. Appl. 47(1),123 (2004). 9. G. Golub and C. Van Loan, Matrix Computations, The Johns Hopkins University Press, 1996. 10. J. Elyseeva, Symplectic factorizations and Sturmian Theory, Linear Alg. and Appl., submitted.

178

A RENAISSANCE FOR A Q-UMBRAL CALCULUS

THOMAS ERNST Department of Mathematics Uppsala University P.O. Box 480, SE-751 06 Uppsala, Sweden E-mail: thomasQmath.uu.se

In this survey I will discuss a q-umbra1 method reinvented by the author, which also involves the Nalli-Ward-Alsalam q-addition and the Jackson-Hahn-Cigler qaddition. This kind of q-calculus has a close connection to the Jackson q-gamma function and the Heine R function. In this way we can find q-analogues of the results of Norlundfor generalized q-Bernoulli, q-Euler and q-Lucas numbers and polynomials, which were anticipated by Ward. This calculus has previously been used by P. Appell 1879, Ashton & Lindemann 1909, Smith & Pringsheim 1911 and Daum 1941. The Jacobi-Neville elliptic functions can easily be expressed in terms of the Heine R function and its tilde version. The advantages of this umbra1 method have been summarized in [ l o , p. 4951

We start with some definitions.

Definition 0.1. The power function is defined byqa ealog(q).We always use the principal branch of thelogarithm. The variables a , b , c , a l , a z,..., b l , b 2 ,... E C denote certain parameters. The variables i, j , k,1, m, n , p , r will denote natural numbers except for certain cases where it will be clear from the context that i will denote the imaginary unit. The q-analogues of a complex number a andof the factorial functionare defined by:

179

Definition 0.2. Let the q-shifted factorial be defined by

n = 0;

Let the Gauss q-binomial coefficient be defined by

for k = 0 , 1 , . . . ,n, and by

for complex a and3!, when 0 < IqI < 1. The q-multinomial coefficientis defined by

for {ki}f=l = 0,1,. . . , n and 1 ki = n. If the number of ki is unspecified, we denote the q-multinomial coefficient by

(3 ,

9

00

xki=n.

(7)

i=l

If 0 < IqI < 1 and IzI < 11 - ql-l,the q-exponential function E g ( z )was defined by Jackson 1904, and by Exton.

k=O

The Nalli-Ward-AlSalam q-addition (NWA), compare [l, p. 2401, [22, p. 3451, [29, p. 2561 is given by

c(r) n

( a cB9 b)"

5

k=O

a k l F k, n

= 0,1,2,. . . .

9

Furthermorelwe put

c(3 n

( a egb)"

E

k=O

9

~ ~ ( - b ) " - ~ n, = 0,1,2,. . . .

(9)

180

There is a q-addition dual to the NWA, which will be presented here for reasons to be given shortly. The following polynomial in 3 variables z, y, q originates from Gauss. Definition 0.3. The Jackson-Hahn-Ciglerq-addition (JHC), compare [3, p. 911, [17, p. 3621, [19, p. 781 is the function

Y z"(--;q)" 2

E P,,g(x,y), n = 0 , 1 , 2 , .. . .

(z El, y)" E P,,,(z, -y),

n = 0 , 1 , 2 , .. . .

(12)

For symbolic purposes, we will define a general q-addition. Definition 0.4. Let f ( k , n ) be a given function. addition is defined by

Then the general q-

and similar for (u e,,, b). The q-difference operator is crucial, we define it in a slightly different way.

ifx=O The umbra1 calculus has been used since the eighteenth century. Some of its important figures were Blissard (1803-1875), Glaisher (1848-1928) and Lucas (1842-1891). F.H. Jackson (1870-1960) followed this path in the early twentieth century, and fully understood the symbolic nature of the subject in his first investigations of q-functions. His investigations were followed by E-T. Bell (1883-1960), Rota (1932-1999) and Cigler. In [4] a special case of the following q-umbra1 calculus was used, the case q = 1 was treated in [5]. Definition 0.5. A q-analogue of [28, p. 6961. A q-umbra1 calculus contains aset A, called the alphabet, with elements called letters or umbrae. Assume that a , p, are distinct umbrae, then a new umbra is obtained by a * p, where * is @, EE, e,,El,, or any general q-addition.

181

There is a certain linear functional evaZ, @[[XI] x A 4 C , called the evaluation, such that eval(1,a) = 1,a E A. In the following, an arbitrary f E C[[z]]will be used. If a, /3, . . . , y are distinct umbrae, and i, j , . . . , k positive integers, evaZ(f, aipj. . . yk) = evaZ(f, ai)evaZ( f , pj). . . evaZ(f, 7‘).

-

(15)

/3 if Two umbrae a and p are said to be equivalent, denoted a eval (f,a ) = evaZ( f , p). The set of equivalent umbrae forms an equivalence class. There is a distinguished element E of the alphabet called the zero, such that and z El,z

e v a l ( f ,en) = 6,,0

-

(16)

E.

-

Elements a and ,B E A are said to be inverse to each other if a 83, ,8 E . We will use the following notation for the Ward number A, [29, p. 2581 A,

-

. @q 1,

1@q 1@,..

(17)

-

where the number of 1 in the RHS is n. If a1,. . . ,a, E A , a ai, i = 1,.. . ,n then a1 @, . . . $, a, Aqa. The last condition is a q-analogue of [23, p. 125, (13)],[24, p. 132, (49)]. N

where BNWA,,(Y) are given by (42). Here B can be changed to any qpolynomial sequence. Threeexamples of eval are the NWA, JHC and the general q-addition. Theorem 0.1. [22, p. 5’451. The q-addition (9) has the following properties, a , b E A; c E C:

-

( a @ , b) @, c

a@,b a@,O

ca@,cb

b@q

a

O@, a

-

N

a @ , (be,c)

C ( U @,

N

a

b).

The first three conditions mean that the umbrae of NalZi- Ward-Alsalam qaddition form a commutative monoid.

182

The following theorem shows how Ward numbers usually appear in applications. Compare with [l, p. 244, 3.161, where the notation Pk(n) was used.

Theorem 0.2.

where each partition of k is multiplied with its number of permutations. W e have the following special cases:

The formal power seriesform a vector space with respect to termwise addition and multiplication by complex scalars. In the rest of this paper, as in [29, p. 2581, unless otherwise stated, we assume that functionsf(z),g(z),F ( z ) ,G(z) E (C[[z]].

Definition 0.6. If 00

k=O

[29,p.258]F(z@, y)

[19,p.78]F(zH, y)

00

n

n=O

k=O

= C a,

= F [ z + y],

zkyn-'.

n

an

n=O

(23)

Q

c c(i) 00

f

(3

ykzn-k.

(24)

k=O

Definition 0.7. The Ward-Alsalam q-shift operator [l, p. 242, 3.11, a q-analogue of [24, p. 31 is given by

E(@,)"(zL.") = (z $, w)"

(25)

We denote the corresponding operator for the JHC by

E(H,).i.e.

=

E(H,)W(zn) (xH, w)"

When w = 1, we denote these operatorsE($,) and

(26)

E(Hq).

Definition 0.8. The linear difference operator for the NWA, a q-analogue of [24, p. 31, is defined by nNWA,q

where I is the identity operator.

= E ( @ q )- 1,

(27)

183

The following dual q-analogues of [5, p. 281, see [l,p. 242, 3.3, p. 243, 3.91, [29, p. 2641 hold.

E(@,)" = E,(wD,).

(28)

E(H,)" = E$ ( W D , ) .

(29)

In the following definition, theq-additions are written first in additive, then in multiplicative form. In the first case, we assume that the function argument operate from left to right when using the two q-additions. In the second case, we assume that the function argument operate from right to left in accordance with (28) and (29). So don't forget that the following two equations are not associative. Definition 0.9. If

k=O

We will now give a number of theorems for arbitrary letters which illustrate certain symmetry properties of this umbra1 calculus. Theorem 0.3. The NWA and the JHC are dual operators.

f ( x @, a 8, a )

f ( E ( H q ) - a E ( @ q ) a x= ) f(x).

Proof. Use (28) and (29).

(33) 0

By Goulden & D.M. Jackson I161 we obtain two furtherformulas of this type. Theorem 0.4. [16, p . 2281 (a4 P ) @, (7 4 6) a,P,7,6 E A

-

(a4 4 @q (74 P I ,

(34)

184

Theorem 0.5. [16, p. 2.281

The Leibniz theorem goes as follows. Notice the binomial coefficient on the right.

Theorem 0.6. A q-analogue of [5, p . 27, 2.131, [20, p. 97, lo], [21, p. 35, 21. Let f (x) and g(x) be formal power series. Then

Definition 0.10. If f(x) is the formal power series CEO akxk, the (Ward) q-sum is defined by m

moc)

k=n

k = n 1=0

where for each k the function value for the corresponding Ward number is computed. If n > rn, the sum = 0. The q-integral is defined by

la

f (t, Q) d,(t)

cf 00

= 4 1 - 4)

( a f , d q n , 0 < 141 < 1, a E

(38)

n=O

We now generalize this definition toa E C,, where the q-extended complex numbers C, is the set generated by C together with the operations B,, @, or any general q-addition. Similarly, we define

(-1)flq

= (-1)n.

(39)

We will now descrive the q-Appell polynomials, which already have been characterizedby Al-Salam [2], who described its algebraic structure. We will freely use the results from [2]. In the spirit of Milne-Thomson [21, p. 125-1471, which we will follow closely, we will call these q-polynomials @,polynomials, and express them by the NWA q-addition.

Definition 0.11. A q-analogue of [2l, p. 1241. The@, polynomials of degree u and order n have the following generating function

185

A special case of q-Appell polynomials are the generalized q-Bernoulli polynomials BF&A,u,q(x) of degree v and order n, which were defined for q = 1 in [21, p. 1271, [23] and for complex order in [l,p. 254, 10.31. Definition 0.12. The generating function for B$&A,u,q(~) is a q-analogue of [28, p. 7041.

The Ward q-Bernoulli polynomials [29] are given by BNWA,n,q(x)

(1)

(42)

BNWA,n,q(x).

Another example of a q-Appell polynomial is the q-Euler polynomialEg&A,u,q(x), which will be described in a paper in preparation. We will now follow Cigler [5] and give a few q-analogues of equations for Bernoulli polynomials. The first two of these equations are well known in the literature (q = 1). Definition 0.13. A q-analogue of [5, p. 131. n-1

SNWA,m,g(n)

x ( L q I m , sNWA,O,g(1) k=O

1.

(43)

Theorem 0.7. [l, p . 248, 5.131, 129, p . 265, 16.51, a q-analogue of [5, p . 13, p . 17: l , l l , p . 361, [25, p . 2371.

(G)g)- B N W A , m + l , g {m + 1),

BNWA,m+l,g( S N W A , ~ , ~= (~)

Theorem 0.8. A q-analogue of [5, p . 451, [23, 127, (17)].

186

This can be rewritten as a q-analogue of the well-known identity [15, p. 496, 8.21.

Cigler has given some examples of translation invariant operators. One of them is the Bernoulli operator.

Definition 0.14. A q-analogue of [5, p. 44-45], [6, p. 1541, [27, p. 591, [28, p. 701, 7031. The q-Bernoulli operator has the following form.

Theorem 0.9. A q-analogue of [5, p. 44-45]. We can expand a given formal power series in terms of the BNWA,k,q(x) as follows.

In another paper we will publish several Taylor-like q-analogues of Norlund's formulas. One example is

Theorem 0.10. A q-analogue of [23, (21) p. 16.31. The corresponding formula for n = 1 occurred in [l, p. 254 10.81.

Formula (47) implies the following theorem, which will be of considerable use in q-calculus.

Theorem 0.11. The q-Euler-Maclaurin summation theorem for formal power series. A q-analogue of [5, p. 541, [ l d , p. 303], [26, p. 251, [28, p . 7061.

187

References 1. Al-Salam W. A.,q-Bernoulli numbers and polynomials. Math. Nachr. 17 (1959),239-260. 2. Al-Salam W. A.,q-Appell polynomials. Ann. Mat. Pura Appl. (4) 77 1967 31-45. 3. Cigler J.,Operatormethoden fur q-1dentitaten.Monatshefte fur Mathematik 88,(1979), 87-105. 4. Cigler J.,Elementare q-1dentitaten.Publication de l’institute de recherche Mathkmatiqueavancke (1982) 23-57. 5. Cigler J.,Differenzenrechnung. Wien 2001. 6. Curry H.B.,Abstract differential operators and interpolation formulas. Port. Math. 10, 135-162 (1951). 7. Ernst T,The history of q-calculus and a new method, Uppsala , 2000. 8. Ernst T,q-Generating functions for one and two uariables.Simon Stevin, 13, no. 1, 2006. 9. Ernst T , A new method for q-calculus, Uppsala dissertations 2002. 10. Ernst T,A method for q-calculus. J. nonlinear Math. Physics 10 No.4 (2003), 487-525. 11. Ernst T,Some results for q-functionsof many variables.Rendiconti di Padova,ll2 (2004), 199-235. 12. Ernst T,q-Analogues ofsome operational formulas.Preprint 2003. 13. Gasper G & Rahman M.,Basic hypergeometric series,Cambridge, 1990. 14. Glaisher, J. W. L.Genera1 summation-formulae in finite differences. Quart. J . math. 29, 303-328. (1898). 15. Gould H.W.,Evaluation of sums of convolved powers using Stirling and Eu1eriannumbers.The Fibonacci quarterly 16, 6 (1978), 488-497. 16. Goulden, I. P. & Jackson, D. M.An inversion model for q-identities. European J. Combin. 4 (1983), no. 3, 225-230. 17. Hahn W.Beitrage zur Theorie der Heineschen Reihen.Mathematische Nachrichten 2 (1949), 340-379. 18. Heine E. ,Uber die Reihe ...J . reine angew. Math. 32, (1846) 210-212. 19. Jackson F.H., On basic double hypergeometric functions. Quart. J. Math. 13 (1942) 69-82. 20. Jordan, Ch.Calculus of finite differences. Third Edition. Chelsea Publishing Co., New York 1950. 21. Milne-Thomson L. M., The Calculus of Finite Differences. Macmillan andCo., Ltd., London, 1951. 22. Nalli P. ,Sopra un procedimento di calcolo analog0 alla integrazione.Rend. Palermo 47, (1923), 337-374. 23. NorlundN.E.,Mkmoire sur les polynomes de Bernoulli.Acta Math. 43 (1920), 121-196. 24. NorlundN.E.,Differenzenrechnung.Berlin 1924. 25. Lucas E.,Theorie des nombres, vol.1. Paris 1891. 26. Pearson.Calculus of finite differences. Cambridge 1850. 27. Finite operator calculus. Edited by Gian-Carlo Rota. With the collabora-

188 tion of P. Doubilet, C. Greene, D. Kahaner, A. Odlyzko and R. Stanley. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1975. 28. Rota, G.-C.; Taylor, B. D., The classical umbral calculus. SIAM J. Math. Anal. 25 (1994), no. 2, 694-711. 29. Ward M.,A calculus of sequences. Amer. J. Math. 58 (1936),255-266.

189

FOURTH-ORDER BESSEL-TYPE SPECIAL FUNCTIONS: A SURVEY

W.N. EVERITT, SCHOOL OF MATHEMATICS AND STATISTICS, UNIVERSITY OF BIRMINGHAM, EDGBASTON, BIRMINGHAM B15 2TT, ENGLAND, UK EMAIL: [email protected]

This paper is dedicated to the memory and achievements of George Neville Watson (1886 to 1965) This survey paper reports on the properties of the fourth-order Bessel-type linear ordinary differential equation, on the generated self-adjoint differential operators in two associated Hilbert function spaces, and on the generalisation of the classical Hankel integral transform. These results are based upon the properties of the classical Bessel and Laguerre second-order differential equations, and on the fourthorder Laguerre-type differential equation. From these differential equations and their solutions, limit processes yield the fourth-order Bessel-type functions and the associated differential equation.

1. Introduction

This survey paper is based on joint work with the following named colleagues: Jyoti Das, University of Calcutta, India D.B. Hinton, University of Tennessee, USA H. Kalf, University of Munich, Germany L.L. Littlejohn, Utah State University, USA C. Markett, Technical University of Aachen, Germany M. Plum, University of Karlsruhe, Germany 2000 Mathematics Subject Classification. Primary: 33C10, 34B05, 34L05. Secondary: 33C45, 34B30, 34A25. Key words and phrases. Bessel functions, Bessel-type functions, linear ordinary and partial differential equations, self-adjoint ordinary differential operators, Hankel transforms.

190

M. van Hoeij, Florida State University, USA

2. History

We see below that the structured definition of the general-even order Besseltype special functions is dependent upon the Jacobi and Laguerre classical orthogonal polynomials, and the Jacobi-type and Laguerre-type orthogonal polynomials. These latter orthogonal polynomials were first defined by H.L. Krall in 1940, see [13] and [14], and later studied in detail by A.M. Krall in 1981, see [12], and by Koornwinder in 1984, see [15]. In this respect see the two survey papers, [7] of 1990 and [6] of 1999. The Bessel-type special functions of general even-order were introduced by Everitt and Markett in 1994, see [8]. The properties of the fourth-order Bessel-type functions have been studied by the present author and the seven colleagues named in Section 1above, in the papers [2], [4] and [5].

3. The fourth-order differential equation The fourth-order Bessel-type differential equation takes the form

(zy”(z))”

-

((9z-1

+ 8M-’z)y’(z))’

= Azy(z) for all z E (0,co)

(1)

where M E (0, co) is a positive parameter and A E C, the complex field, is a spectral parameter. The differential equation (1) is derived in the paper [8, Section 1, (l.lOa)], by Everitt and Markett. This linear, ordinary differential equation on the interval (0,co) c JR, the real field, is written in Lagrange symmetric (formally self-adjoint) form, or equivalently Naimark form, see [18, Chapter V]. The structured Bessel-type functions of all even orders, and their associated linear differential equations, were introduced in the paper [8, Section 11 through linear combinations of, and limit processes applied to, the Laguerre and Laguerre-type orthogonal polynomials, and to the classical Bessel functions. This process is best illustrated through the following diagram, see [8, Section 1, Page 3281 (for the first two lines of this table see the earlier

191

work of Koornwinder [15] and Markett [17]):

I

Jacobi polynomials Ic(a,P)(l-a:p(l +z)P

i

-

Jacobi-type polynomials

k(a,P)(l - Z ) * ( l + z)P +MS(a: 1) N S ( s - 1)

+ +

1

1

Laguerre polynomials k(a)z" exp(-s)

Laguerre-type polynomials Ic(a)z" exp(-s) NS(a:)

1 Bessel functions K(a) 2*+1

--f

-

+

(2)

1 Bessel-type functions

+ MS(z)

tS(a)s2"+1

The symbol entry (here Ic and K are positive numbers depending only on the parameters a and P ) under each special function indicates a non-negative (generalised) "weight", on the interval (-1,l) or (0, oo),involved in:

( a ) the orthogonality property of the special functions ( b ) the weight coefficient in the associated differential equations.

-

It is important to note in this diagram that: ( i ) a horizontal arrow

indicates a definition process either by a linear combination of special functions of the same type but of different orders, or by a linear-differential combination of special functions of the same type and order (alternatively by an application of the Darboux transform, see [lo]) ( i i ) a vertical arrow J. indicates a confluent limit process of one special function to give another special function ( i i i ) the use of the symbol M6(.) is a notational device to indicate that the monotonic function on the real line R defining the weight has a jump at an end-point of the interval concerned, of magnitude

M>O ( i v ) the combination of any vertical arrow 1 with a horizontal arrow must give a consistent single entry.

-

Information about the Jacobi-type and Laguerre-type orthogonal polynomials, and their associated differential equations, is given in the Everitt and Littlejohn survey paper [7]; see in particular the references in this paper to the introduction of the fourth-order Laguerre-type differential equation by H.L. and A.M. Krall, Koornwinder and by Littlejohn. The general Laguerre-type differential equation is introduced in the paper [11]

192

by Koekoek and Koekoek; the order of this linear differential equation is determined by 4 2a with a E No = {0,1,2,..}. It is significant that the general order Bessel-type functions also satisfy a linear differential equation of order 4 2a (with a E NO), being an inheritance from the order of the general Laguerre-type equation. The purpose of this survey paper is to discuss the properties of the Bessel-type linear differential equation in the special case when a = 0, as given in the bottom right-hand corner of the diagram; this is the fourthorder differential equation (1) and involves the weight coefficient ~ ( 0 ) zits ; solutions should, in some sense, have orthogonality properties with respect to the generalised weight function ~ ( 0 ) z M6(0), where M > 0 is the parameter appearing in the differential equation (1);see [8, Section 41. Our knowledge of the special function solutions of the Bessel-type differential equation (1) is now more complete than at the time the paper [8] was written. However, the results in [8, Section 1, (1.8a)], with a = 0 , show that the function defined by

+

+

+

~,">"(z):= [ ~ + M ( x / ~ ) ~ I J ~ ( x ~ ) - ~ ~J I~(AX) ( x /for ~ all ) ~z( Ex (0, z )oo), -~

(3) is a solution of the differential equation (l),for all X E C , and hence for all A E C, and all M > 0. Here:

(i) the parameter M > 0 (ii) the parameter X E C (iii) the spectral parameter A and the parameter M , in the equation (l), and the parameters M and A, in the definition (3), are connected by the relationship

A

= A(X, M ) = X2(X2 + 8M-l)

for all X E C and all M > 0 (4)

( i w ) JO and 51 are the classical Bessel functions (of the first kind), see [24,Chapter 1111. Similar arguments to the methods given in [8] show that the function defined by

Y,">"(z):= [1+M(X/2)2]Yo(Xz)-2M(X/2)2(Xz)-'Y1(Az) for all z E (0, oo), (5) is also a solution of the differential equation (l),for all X E C, and hence for all A E C and all M > 0; here, again, YOand Y1 are classical Bessel functions (of the second kind), see [24,Chapter 1111.

193

The earlier studies of the fourth-order differential equation (1)failed to find any explicit form of two linearly independent solutions, additional to the solutions J;lM and However, results of van Hoeij, see [22] and [23],using the computer algebra program Maple have yielded the required two additional solutions, here given the notation I;1M and K i . M , with explicit representation in terms of the classical modified Bessel functions 10,KO and 11, K1. These two additional solutions are defined as follows, where as far as possible we have followed the notation used for the solutions J:>M and Y:lM, Y;lM.

E C , with arg(X) E [0,2n), M E (0,oo) and using the principal value of J , define

(i) given X

c = c(X,M ) :=

Jm and d

(ii) define the solution I:>M,for all z

E

E d(X, M

) := 1

+ M(X/2)2

(0, oo),

+

I,JyT) := -dIo(cz) ;cMz-1Il(Cz) := -[1+ M(X/2)2]Io(zJX2 8M-l) + J I M (2 M(X/2)2)z-111(zJX2 + 8M-l)

+

+

(iii) define the solution K:'M, for all z

+

E

(0, oo),

I , > M ( z ) := dKo(cz) ;cMz-lKl(cz) := [1+ M(X/2)2]Ko(zJX2 8M-l) +JM (2 M(X/2)2)~-1K1 (zJA2

+

+

+ 8M-l)

Remark 3.1. We have (1) The four linearly independent solutions J:.M, Y f . M , provide a basis for all solutions of the original differential equation (l),subject to the (A, A) connection given in (4). (2) These four solutions are real-valued on their domain (0, oo) for all X E R. (3) The domain (0, oo) of the solutions J:IM and I:'M can be extended to the closed half-line [0, 00) with the properties

J;IM(O) = I:lM(0) = 1 for all X E R and all M E (0, oo).

The classical Bessel differential equation, with order a = 0, written in a form comparable to the fourth-order equation (l),is best taken from the left-hand bottom corner of the diagram (2); from [8, Section 1, (1.2)]with a = 0 we obtain -(zy'(x))'

= ~ ~ z y ( for z ) all z E

(0, oo);

(9)

194

here X E C is the spectral parameter. It is to be observed that, formally, if the fourth-order Bessel-type equation (1) is multiplied by the parameter M > 0 and then M tends to zero, we obtain essentially the classical Bessel equation of order zero (9), on using the spectral relationship (4) between the parameters X and A. This Bessel differential equation (9) has solutions Jl(Xx) and Yl(Xx) for all x E (0, co) and all X E C. For the need to apply the F’robenius series method of solution we also consider the differential equation (1) on the complex plane C:

+

+

w ( ~ ) ( z )~ z - ~ w ( ~ ) ( z( ) 9 ~ - 8M-’)w”(z) ~ +(9zP3 - ~ M - ’ z - ~ ) w ’ ( z )A w ( z )= 0

(10)

for all z E C. In this form the equation has a regular singularity at the origin 0, and an irregular singularity at the point at infinity 00 of the complex plane C; all other points of the plane are regular or ordinay points for the differential equation. It should be noted that the classical Bessel differential equation (9) has the same classification when considered in the complex plane C. A calculation shows that the Frobenius indicial roots for the regular singularity of the differential equation (10) at the origin 0, are {4,2,0,-2}. The application of the Frobenius series method, using the computer programs [l]and Maple (see [23]), yield four linearly independent series solutions of (lo), each with infinite radius of convergence in the complex plane C. If these solutions are labelled to hold for the Bessel-type differential equation (1) then we have four solutions { y r ( . , A , M ) : r = 4,2,0,-2}, to accord with the indicial roots, to give the theorem, see [2, Section 31: Theorem 3.1. For all A E C and all M > 0 , the diflerential equation (1) has four linearly independent solutions { y r ( . ,A, M ) : r = 4,2,0, -2}, defined o n (0,co) x C, with the following series properties as x -+ O+,

where the 0 - t e r m s depend upon the complex spectral parameter A and the parameter M ,

+ +

+

0(x8) y4(2, A, M ) = x4 iA4-l~~ y2(x, A, M ) = k x 2 O ( x 4 11n(x)1) Y o ( Z , A, M ) = 1 + 0 ( x 4 Iln(.)I) y-~(x,A, M ) = m x P 2 O(lln(x)I).

(11)

+

Here the fixed numbers Ic, 1, m E R and are independent of the parameters A and M ; these numbers are produced by the Frobenius computer program (11 and have the explicit values:

Ic

=

-(27720)-l,

1 = (174636000)-l, m = -(9779616000)-1

195 4. Higher-order differential equations

As mentioned in Section 3 above there exist Bessel-type linear differential equations of all even-orders 4 2 a , where a! E No is any non-negative integer. The definition and some properties of these differential equations, and the associated Bessel-type functions, are considered in detail in [8, Sections 2 and 31. Here we give the form of the sixth-order and eighth-order differential equations, as given in [8, Section 1, (1.10b) and (l.lOc).]. (Note that there is a printing error in the display (1.lOb); the numerical factor 255 is to be replaced by 225. Also printing errors in the display (1.10~)which are now to be corrected using the form of the differential equation (13) below.)

+

(i) The sixth-order equation derived from the corrected differential expression for [8, Section 1, (1.9) and (1.10b)l is

-(~~y(~)(z))(~)+ (33zy”(z))” - ((2252-1 =

(x6 + ~

- 9G111-1z3)y’(z))’

4 - (93 )~x2)z3y(z) for all z E (0, m),

(12) where, as before, the parameters M E (0, m) and X E @. When this equation is considered in the complex plane @ the Frobenius indicial roots for the regular singularity at the origin 0 are {6,4,2,0,-2, -4}, using the methods provided by [23]. (ii) The eighth-order equation derived from the corrected differential expression for [8, Section 1, (1.9) and (1.10c)l is

+

( ~ ~ y ( ~ ) ( z-) )(78~~y(~)(z))(~) (~) (1809ay”(z))” - (( 110252-1 - 26(4!)M-’z5)y’(x))’

= (As

+ M-126(4!)X2)z5y(z)for all z E (0, oo),

(13)

where, as before, the parameters M E (0, oo) and X E @. When this equation is considered in the complex plane @ the Frobenius indicial roots for the regular singularity at the origin 0 are {8,6,4,2,0, -2, -4, -6}, using the methods provided by [23]. We note that, formally, if the equations in (12) and (13) are multiplied by M > 0, and then letting M tend to zero we obtain, respectively, the two Sturm-Liouville differential equations, see [8, Section 1, (1.2)], -(z3y’(z))’ = X2z3y(z) and

-

(z5y’(z))’ = X2z5y(z) for all z E (0, m).

(14) For the solutions of these equations in classical Bessel functions see [8, Section 1, (1.4)].

196

5. The fourth-order differential expression L M We define the differential expression LM with domain D ( L M )as follows:

D ( L M ):= {f : (0,m)

--t

C : f“) E AC1,,(O,oo) for T = 0,1,2,3},

(15)

and for all f E D ( L M ) L M [ ~ ] ( z:= ) (zf”(2))”- ((9z-l

+ 81M-’z)f’(z))’(z E ( 0 , ~ ) ) ; (16)

it follows that

LM : D(LAf) --t L:,c(O, W).

(17)

The Green’s formula for LAf on any compact interval [a, p] C (0, +a) is given by

where the symplectic form defined by

[.,.I(.)

: D ( L k f ) x D ( L M )x (O,+m) + C is

If,gI(z) := 3(.)(.f”(.>>’ - (d’(.))’f(.) -

z (3’(.”’’(.)

- (92-1

- G”(.)f’(.))

+ 8 M - l ~ (g(z)f’(z) ) - g’(z)f(z)).

(19)

The Dirichlet formula for L M on any compact interval [a, p] c (0,+m) is given by

: D ( L M )x D o ( L M x ) (0, +m) + C is defined where the Dirichlet form [., by, for f E D ( L M )and g E D o ( L M ) with ,

D o ( L M ):= {g : (0, +m) -+ C : g(‘) E AClOc(0,+m) for T = 0 , l ) and

(21)

197 6. Hilbert function spaces

The spectral properties of the fourth-order Bessel differential equation

(zy”(x))” - ((9.-1

+ 8M-’x)y’(x))’ = R x y ( x ) for all x E (0, co),

(23)

with A E C as the spectral parameter, are considered in two Hilbert function spaces: (1) The Lebesgue weighted space

L2((0,co);x) := f : (0, +co) 4 c :

{

I”

z

lf(x)I2 dx < +co

(24) with inner-product and norm defined by, for all f,g E L2((0,co); x),

This space takes into account the weight function x on the righthand side of (23). (2) The Lebesgue-Stieltjes jump space L2([0,00); m k ) , as suggested by the results in [8, Section 41. Let the monotonic non-decreasing function riZk : [0,co) -+ [ O , o o ) be defined by, where k > 0 is a real parameter, i&(z) = -k for x = 0 = x2/2 for all z E (0, +m).

Then riZk generates a Baire measure m k on the u-algebra B of Borel sets on the interval [ O , c o ) ; in turn this measure generates a Lebesgue-Stieltjes integral for Borel measurable functions. The Hilbert function space L2([0, 00); m k ) is defined on all functions with the properties:

(i) f : [0, co) -+C and is Borel measurable on [0, co) (ii)

Imx

lf(x)I2 ds < +co.

The norm and inner-product in L 2 ( [ 0 00); ,

llflli := and

/

[OF)

dmrc(x) =

m k )

lf(o)12 +

/

0

are defined by 00

5

If(x)I2da: (26)

198

Note that the first integrals in both these definitions are LebesgueStieltjes integrals taken over the set [0, m), whilst the second integrals can be taken as Lebesgue integrals.

7. Differential operators generated by L M The Lagrange symmetric differential expression L M generates self-adjoint operators in both the Hilbert function spaces L 2 ( ( 0 , m ) ; x ) ,and in L2( [ 0,m);m k ) for all k E (0, m). In the space L2((0,00); z ) the expression L M generates a continuum { T } of self-adjoint operators, including the significant F'riedrichs operator F ; these properties are developed and considered in Sections 8 to 13 below. For each k E (0, m) the expression L M generates a unique self-adjoint operator SI, in the space L2([0,m);m k ) ; the properties of this operator are considered in Sections 14 and 15. 8. Differential operators in L 2 ( ( 0 ,00); z)

The maximal and the minimal differential operators, denoted respectively 7'1 and TO,as generated by the differential expression L M in the Hilbert function space L 2 ( ( 0 ,00); z ) , are defined as follows, see [18, Chapter V,

199

From standard results we have the operator properties, see [18, Chapter

VI 7

To C Tl,T,* = TI and T; =To,

(33)

thereby noting that both TO and TI are closed linear operators in

L 2 ( ( 0 00); , 9. Self-adjoint operators in L 2 ( ( 0 ,00); z)

In the weighted space L 2 ( ( 0 00); , x) the Lagrange symmetric (formally selfadjoint) differential expression has the following endpoint classifications at the singular endpoints 0 and +m (for additional details see [2, Section 61):

(i) At O+ the singular endpoint is limit-3 in L 2 ( ( 0 m); , x) (ii) At +00 the singular endpoint is Dirichlet and strong limit-2 in L 2 ( ( 0 ,00); X I . Based on this information the self-adjoint extensions of the closed symmetric operator TOare determined by the GKN-theorem on singular boundary conditions as given in [18, Chapter V] and [9]. In particular, for the operators TOand T I ,any self-adjoint operator T = T* generated by L M in L2((0,m);x) is a one-dimensional extension of TOor, equivalently, a onedimensonal restriction of T I . Let the domain D ( T ) as a restriction of the domain D(T1) be determined by

D ( T ) := {f E W

l ):

[f,cp1(0)= 01,

(34)

where the function cp E D(T1) is a non-null element of the quotient space D(Tl)/D(TO)which satisfies the GKN symmetry condition [cp, cpI(0) = 0.

(35)

Then the differential operator T defined by

T f := x P 1 L ~ [for f ] all f E D ( T )

(36)

satisfies T* = T , and is self-adjoint in the Hilbert space L2((0,m);x). All such self-adjoint operators are determined in this way on making an appropriate choice of the boundary condition function cp. 10. Boundary properties at O+

The results of the following theorem are essential to obtaining the explicit forms of the boundary conditions at O+ to determine all self-adjoint extensions of TO.

200

Theorem 10.1. Let f E D ( T 1 ) ; then the values o f f , f’, f” can be defined at the point 0 so that the following results hold:

(9 f E A C P , 11 (ii) f’ E AC[O,11 and f’(0) = 0 (iii) f ” E ACl,,(O, 11 and f” E C[O,11 ( i w ) f (3) E ACl,,(O, 11 and lim,,o+ (xf (3)(x))= 0. For the proof of this theorem see [2, Section 81. We consider the functions 1,x, x2 on the interval [0,1] but “patched”, see the Naimark patching lemma [18, Chapter V, Section 17.3, Lemma 21, to zero on [2,00) in such a manner that the patched functions belong to the domain D(T1);we continue to use the symbols 1,x, x2 for the patched functions. A calculation shows that the results given in the next lemma are satisfied:

Lemma 10.1. The patched functions 1,x, x2 have the following limit properties in respect of the symplectic form [.,.] and the maximal domain D ( T 1 ) :

( i ) 1,x2 E D(T1) but x 4 D ( T 1 ) (ii) [l,1](0+)= [x,X](O+) = [x2, x2](0+) = 0 (iii) [x,x2](O+)= 0 and [1,x2](O+)= 16 ( i w ) [l,x](O+) does not exist. The lemmas and corollaries now given below are taken from [2, Section 91, where proofs are given in detail. The results of Theorem 10.1 and Lemma 10.1 now provide a basis for the two-dimensional quotient space D(T1)/D(TO);

D(Tl)/D(To)= span{l, x2} = { a

+ bx2 : a, b E C}.

(37)

The linear independence of the functions (1,~’)within the the quotient space follows from the property [l,x2](O+)= 16 # 0. A calculation now gives, recall Theorem 10.1,

Lemma 10.2. Let f E D ( T 1 ) ; then the following identities hold:

(4 [f,W+)= -8f”(O) (ii) [f,x2](0+)= lSf(0). Similarly we have

Lemma 10.3. Let f , g E D(T1); then

20 1

( i ) [f,91(0+) = 8 [f(o)iw)- f ” ( 0 ) I m l (4 [f,910 (O+) = 8 f ” ( O ) m . We have the corollaries:

Corollary 10.1. The domain of the minimal operator TO is determined explicitly by

D(T0) = {f E D(T1) : f(0) = 0 and f”(0) = 0). Corollary 10.2. For all

Srn

{X

f

(38)

E D(T1)

+

lf”(x)I2 (9x-l

+ 8M-’x)

lf’(x)12)dx < 00.

0

(39)

Corollary 10.3. For all f,g E D(T1) the Dirichlet formula takes the form ( T l f l 9 )= Sf”(O)i7(0>

+

00

+

{ x f ” ( x ) g ” ( x ) (9x-1

+ 8 M - l ~ f)’ ( x ) # ( x ) }dx.

(40)

11. Explicit boundary condition functions at O+ We can now determine all forms of the boundary condition function cp satisfying the symmetry condition (35) to determine the domain of all selfadjoint extensions T of the minimal operator TO.

Lemma 11.1. All self-adjoint extensions T of TOgenerated by the differential expression L M in L 2 ( ( 0 00; , x ) are determined by, using the patched functions 1,x 2 ,

D ( T ) := {f E D(T1) : [f,cp](O+)

and

=0

+

where ( i ) cp(x) = a Px2 (ii)a , P E IR and a2 + P 2 # 0). (41)

( T f )( x ) := x - l L ~ ( f ) ( xfor ) all x E ( 0 , ~ and ) all f E D ( T ) .

(42)

There is an equivalent form of this last result, using the results of Lemma 10.2:

Lemma 11.2. All self-adjoint extensions T of To generated by the differential expression LM in L 2 ( ( 0 ,00; x ) are determined by

+

D ( T ) := {f E D(T1) : ( i ) -af”(O) 2Pf(O) = 0 (ii)a,P E IR and a2 p2 # 0)

+

(43)

202

and ( T f )( x ) := z - l L M ( f ) ( x )f o r all x E ( 0 , ~ )and all f E D ( T ) .

(44)

Remark 11.1. We note the two special cases:

( i ) When a = 0 the boundary condition is f(0) = 0; this boundary condition plays a special role, and gives an explicit form of the domain of the F’riedrichs extension F of To; see Section 13 below. (ii) When p = 0 the boundary condition is f ” ( 0 ) = 0. 12. Spectral properties of the fourth-order Bessel-type operators Theorem 12.1. The minimal operator TO,defined in (31) and (32), is , x ) by the null operator 0 , i.e. bounded below in the space L 2 ( ( 0 00); (TOfl

f ) 2 0 for all f

E

D(T0).

(45)

Proof. Since TOis a restriction of the maximal operator TI the result of Corollary 10.3 can be applied to give, using also Corollary 10.1,

+

+

+

{ X I f”(x)I2 (9x-l 8 M - l ~ )If ’ ( x ) I 2 }d x (Tof,f ) = Sf”(O)f(O) = { x lf”(x)I2+ (92-1 + 8 M - l ~ )If’(x)I2}d x 2 0 for all f E D(T0).

0

Theorem 12.2. (1) Let T be a self-adjoint extension ofT0; then:

( i ) The essential spectrum aess(T)is given by oess(T) = acont(T) =

[O,.).

(46)

(ii) There are no embedded eigenvalues of T in the essential spectrum. (iii) T has at most one eigenvalue; i f this eigenvalue is present then it is simple and lies in the interval (-00~0). (2) Every point p E (-00~0) is the eigenvalue of some unique selfadjoint extension T of TO. Proof. The proof of this theorem is given in detail in [2, Section 131.

203

13. The F’riedrichs extension F The closed symmetric operator TOis bounded below in L 2 ( ( 0 co); , z),see Theorem 12.1, and the general theory of such operators implies the existence of a distinguished self-adjoint extension F, called the F’riedrichs extension of TO. This F’riedrichs operator has the properties:

(i) To c F = F* c Ti (ii) D ( F ) = {f E D(T1) : f(0) = 0) (iii) The essential spectrum oess(F)is given by oess ( F ) = ocont ( F ) =

[o,~)

(47)

(iv) F has no eigenvalues. For a discussion of the definition and properties of this F’riedrichs extension see [2, Section 151.

14. Self-adjoint operator s k in L 2 ( [ 000); ,

mk)

In this section, given any k E (0, co),we define the operator s k generated by the differential expression L M in the Hilbert function space L2([0,03); m k ) , where this space is defined in Section 6 above.

Definition 14.1. Let k E (0,co) be given; then the operator s k :D(sk)

c L2([01m);mk)

--f

sk

L2([0100); m k )

(48)

is defined by (see (28) and (29), and Theorem 10.1 for the definition and properties of the domain D(T1) c L2((0,co); z)) (2) D ( s k ) := D(T1) ( i i ) for all f E D ( s k )

{

( S k f ) (Z):= -8k-’f”(O)

for Z = 0 := Z - l L M [ f ] ( Z ) for all z E (0, co).

Theorem 14.1. For all k E ( 0 , ~ ) :

(i) (ii) (iii) (iv)

The The The The

linear manifold D ( S k ) is dense in L2([0,c o ) ; m k ) . operator S k is hemitian in L 2 ( [ 0co); , mk). operator s k as symmetric in L2([0,00); m k ) . operator s k is bounded below in L 2 ( [ 0 00); , mk) ( S k f ,f ) k

2 0 for all

f E D(Sk).

(49)

204

For the proof of this theorem see [4, Theorem 5.21. Theorem 14.2. Let k E (0,oo) be given; then the symmetric operator s k o n the domain D ( s k ) is self-adjoint an the Halbert function space L2([0,00); 4.

For the proof of this theorem see 14, Theorem 5.41. Theorem 14.3. Let k E ( 0 , ~ be ) given; then the operator 5’1, o n the domain D ( S k ) is the unique self-adjoint operator generated by the diflerential expression L M in the Hilbert function space L 2 ( [ 0 00); , mk).

For the proof of this theorem see [4, Theorem 5.51. 15. Spectral properties of the self-adjoint operator S k

The spectral properties of the self-adjoint operator SI,in L2( [ O , o o ) ; m k ) are given by Theorem 15.1. For any k E ( 0 , ~ let ) the self-adjoint operator SI, in L 2 ( [ 0 , m ) ; m kbe ) defined as in Definition 14.1 above; then the spectrum a(Sk) of s k has the following properties:

(i) s k has no eigenvahes (ii) the essential spectrum o f s k is given by vess(Sk) = ccont(Sk) = [o, W ) . For the proof of this theorem see [4, Theorem 6.11. 16. Distributional orthogonality relationships

Recall that from the properties of the classical Bessel function JO we have the result that Jo(.) $ L2((0,m);x). However from IS, Section 1, (1.7)] we have the following distributional (Schwartzian) orthogonal relationship for the classical Bessel function Jo, in the space D’of distributions, h ~ m r J ~ ( X r ) J ~ (d px = r )S(X - p) for all X,p E ( 0 , ~ ) ;

(52)

here 6 E 23’ is the Dirac delta distribution. This is the generalised orthogonality property for the solutions JO of the classical Bessel differential equation, of order 0, given by (9); this result mirrors the spectral properties of this equation, when considered on the half-line (0, oo),in the space

205

L 2 ( ( 0 ,00); x); in particular the result that every self-adjoint extension T of the corresponding minimal operator TOhas the property ueSs(T) = [0, m). The distributional proof of ( 5 2 ) is discussed in the forthcoming paper [ 5 ] ,where the result is also related to the properties of infinite integrals of Bessel functions as originated by Hankel, see [24, Chapter XIII]. As above for the Bessel function JO we have, from the explicit representation ( 3 ) ,the fourth-order Bessel-type function J:lM $! L2([0,00); m k ) for all k , M E (0,m). To obtain a distributional orthogonality for J:l‘, given any M > 0, it is necessary to choose a special value of the parameter k , i.e. k = M / 2 . Then it is shown in [8, Section 4,Corollary 4.31 that we have the following distributional (Schwartzian) orthogonal relationship for the fourth-order Bessel-type function J:IM, in the space D‘ of distributions,

X [1+M ( X / 2 ) 2 ] - 2{ s , ” x J : ’ M ( ~ ) J , ” ~ Md(xx+) ~ M J : ’ M ( 0 ) J , ” ~ M ( O ) } = S(X - p ) for all A, p E (0, m). (53) The distributional proof of (53) is discussed in the forthcoming paper [ 5 ] . As a formal representation it follows that (53) may be written as, using the inner-product for the space L2 ([0, 00); m M p ) ,

x [I + M

( x / ~ ) ~( JI: -? ~ ( .J) :, * ~ ( . ) ) = b ( -~ p ) for all ~ , Ep(0, m). M/2

(54) As another connection between the classical Bessel (9) and the fourthorder Bessel-type (1) differential equations it is to be noted that, formally, the orthogonality result (53) tends to the orthogonality result ( 5 2 ) , as the parameter M tends to zero.

17. The generalised Hankel transform From the general theory of symmetric integrable-square transforms given in [20, Chapter VIII] one form of the classical Hankel transform, for the Bessel function JO and working in the Hilbert function space L 2 ( ( 0 m; , x), is:

(4 Let

f E L 2 ( ( 0 00); , z) then the Hankel transform g E L 2 ( ( 0 00); , s) is given by, for s E (0, m), g(s) =

Lrn

EJo(sE)f(E)dE

with convergence of the integral in L 2 ( ( 0 co); , s)

(55)

206

(ii) With g E L 2 ( ( 0 00); , s ) the inverse transform, to recover f , is given by, for z E (0,00), roo

with convergence of the integral in L 2 ( ( 0 0, 0 ) ; z) (iii) The Parseval relation holds between g and f

There is also a direct convergence form of the Hankel transform which is best written as, starting with f E L1((O,o0); z),

f(z) =

Lrn

sJo(zs) ds

Loo

EJo(sE)f(E)

dJ

(58)

with z E (0,m). Here the integrals are Lebesgue or limits of Lebesgue integrals as discussed in [20, Chapter VIII]. There is a n equivalent generalised Hankel transform involving the fourth-order Bessel-type function J,’M(.) and working now in the Hilbert function space L2 ([0,00); m ~ / 2; )note again these results require the unique choice of k = M / 2 . The complete discussion of the following results for the generalised Hankel transform are to be found in the forthcoming paper [5]. To state these results the Lebesgue-Stieltjes Hilbert function space L 2 ( ( 0 ,00); n) is required. Let the function fi : [0, 00) 4 [0,00) be defined by

+ M ( x / ~ ) ~ I for - ~ all x E [o,

f i ( ~:= )

[I

f i ’ ( ~= ) x [I

+ M ( x / ~ -2> ~2Io for all x E [o, cm)

00);

(59)

then

so that f i is monotonic increasing on [0, 00) and generates a Baire measure on the a-algebra B of Borel sets on the interval [0,00). The Hilbert space L2((0,oo);n) is then defined as the set of all Borel measurable complexvalued functions f on [0,00) such that

207

Remark 17.1. This norm ll.lln and inner-product (.,.)n for the space L2((O,0o);n)are not to be confused with the norm l l . l l k and inner-product (., .)k, introduced in Section 6, for the space L 2 ( [ 000); , mk).

-

We note that the weight function X X [1+ M(X/2)2]-2 in the integral in (60) is the factor in the distributional orthogonal relationships (53) and (54). (1) The L2-theory of the generalised Hankel transform is given by the following results:

Theorem 17.1. Let f E L2 ( [ O , o o ) ; m ~. pT )h e n there exists ex, n) with the property that actly one function g E L 2 ( ( 0 00);

here g is defined by, for almost all X E (O,m),

(-TMf)

J y( x ) f ( x ) d m M / Z

(A) := g(X) =

7

(62)

thereby defining also the generalised Hankel operator

FA4 : L2 ( [ O , 00); m

M p )

+

L 2 ( ( 0 ,00);

(63)

In addition g satisfies

Remark 17.2. Note that the result (62) has to be interpreted as follows, in ( i )and (ii): J x ’ M ( x ) f ( x ) d m M / 2 ( x E) L2((0,co);n) for all X E [O, m) (ii) lim

x-m

/m 0

Ig(X) -

/

JxtM(x)f(x)dm~/:!(x)

IOJl

(iii) Note that the Cauchy-Schwarz inequality shows that if g E L2((0,co);n) then g E L’((0,m);n).

208

Theorem 17.2. Let g E L 2 ( ( 0 ,00); n). Then there is exactly one function f E L2 ( [ 0 ,00); m ~ p with ) the property that (61) is satisfied; here f is defined by

(GA4g) (x):= f(x) :=

g(A)dn(A)

1;:

ifx=O

J i ' M ( x ) g ( X ) d n ( A for ) z E (0, co), (65)

thereby defining also the inverse generalised Hankel operator GA4 : L 2 ( ( 0 , m ) ; n+ ) L2 ([O,oa);mA4,2).

(66)

Remark 17.3. Note that the result (65) has to be interpreted as follows:

(i) i A J : I A 4 ( x ) g ( A ) d n ( X )E L2 ( [ O , c o ) ;

m~4/2) for

all A E (0,co)

(ii)

(2) The direct convergence of the generalised Hankel transform is given by the following results:

-

Theorem 17.3. Let y E (0, co). I f f : (0, co) + R has the property 2

&f (x)E L1(0, co)

(67)

and is of bounded variation in a neighbourhood of y,then

Let p E (0, co). If g : (0, oa)+ R has the property A-

+

1 M(X/2)2

E L l ( 0 ,00)

and is of bounded variation in a nezghbourhood of p , then

(69)

209

R e m a r k 17.4. The integrals in Theorem 17.3 are either Lebesgue integrals or limits of such integrals over compact intervals of ( 0 , ~ ) . Corollary 17.1.

( i ) If y E ( 0 , ~ is) a point of continuity of the function f then (62), (65) and (68) imply ( G M ( F M f ) ) ( Y )=

/

00

0

JX”,(r)( F h f f ) ( X ) W X )

since

(ii) Ify

=0

then

since

from (72), the use of the Fubini integral theorem and noting that J;lM(0) = 1. (iii) I f p E ( 0 , ~ is) a point of continuity o f g then (62), (65) and (70) imply ( . T M ( G M S ) ) ( P ) = g(1-1).

18. The P l u m partial differential equation The Plum equation is a fourth-order linear partial differential in the Euclidean space R2 of two dimensions, derived from a linear partial differential expression which is connected with the fourth-order Bessel-type ordinary differential equation. If the Laplacian V2 partial differential expression is written in polar co-ordinates

210

then the Plum equation has the form, with u = u ( r ,e), 47 = Ru. V 4 u - yV2u - -u T2

(74)

Here y > 0 is determined by y = 8M-1 where M > 0 is the parameter in the fourth-order Bessel equation (l),and A E C is a spectral parameter. Written out the equation (74) becomes, see [16, Section 1, (2)],

+ -~

- --

= Ru.

(75)

From the results given in [19] and [16] assume that a solution for (74) is of the separated form

u(r,8) = v(r)w(O)

(76)

where w is required to be a solution of the second-order Sturm-Liouville differential equation

-.Itye)

= 4w(e).

(77)

Note that w is then of the general form

w(8)

= Acos(28)

+ B sin(28)

(78)

for, say, 8 E [O,T] and scalars A, B. Also note that the factor 4 in the equation (77) is critical, and has to be fixed, for the separation method to be effective. Substitution of (76) into (75) yields, see [16, Section 11 and [19],

= Av(r)w(O)for

all r E ( 0 , ~ and ) all 8 E [O,T].

(79)

For (79) to hold requires that the function v(.), on gathering up terms, has to satisfy the ordinary differential equation, see [16, Section 1, (4)], for all T E (O,W),

+r

+ (--r:

v ( 4 ) ( r ) -2d 3 ) ( r )

-

)

y v”(r)+

:( :) -

v’(r)-Aw(r) = 0. (80)

211 This last equation may be written in the Lagrange symmetric form ( T ~ ” ( T ) ) ”-

+

( ( 9 ~ - 1 y ~ ) v ’ ( ~ )=) ’Arv(r) for all

T

E (0, co),

(81)

which is t h e Bessel fourth-order differential equation (1) when y = 8M-’. Thus separated solutions of the partial differential equation (74) can be written in the form U(T,

0) = v(r)w(0) for all T E (0, co) a n d 0 E [0,27r],

(82)

where w(.) is any trigonometrical solution (78) of (77), a n d v(.) is any solution of t h e fourth-order Bessel equation (81) for any choice of the spectral parameter A. Defining t h e partial differential expression PY[.], for y E (0, co),by

(83) it is shown in [16] that PY is a formally symmetric linear partial differential using polar co-ordinates ( T , 0). expression in L2(E2), Some early studies indicate t h a t there may be problems in applied mathematics, for which t h e partial differential equation PY[u]= Au is involved in one or more of t h e associated mathematical models.

References 1. B.M. Brown, W.N. Everitt, L.L. Littlejohn and D.K.R. McCormack. The F’robenius Program: A User’s Guide. USU Publication: Department of Mathematics and Statistics, Utah State University, Logan, Utah, USA: 1998. 2. Jyoti Das, W.N. Everitt, D.B. Hinton, L.L. Littlejohn and C. Markett. The fourth-order Bessel-type differential equation. Applicable Analysis. 83 (2004), 325-362. 3. W.N. Everitt. The Plum and Bessel fourth-order differential equations. (Report; University of Birmingham, England: 04 August 2005: available in LaTeX file report3.tex). 4. W.N. Everitt, H. Kalf, L.L. Littlejohn and C. Markett. Additional properties of the fourth-order Bessel-type differential equation. (To appear in Math. Nachr.) 5. W.N. Everitt, H. Kalf, L.L. Littlejohn and C. Markett. The fourth-order Bessel equation; eigenpackets and a generalised Hankel transform. (In preparation.) 6. W.N Everitt, K.H. Kwon, L.L. Littlejohn and R. Wellman. Orthogonal polynomial solutions of linear ordinary differential equations. Jour. Computational Appl. Math. 133 (2001)’ 85-109. (The Proceedings of the Fifth International Symposium on Orthogonal Polynomials and Applications, held at the University of Patras, Greece in September 1999.)

212 7. W.N. Everitt and L.L. Littlejohn. Orthogonal polynomials and spectral theory: a survey. Proceedings of the I11 International Symposium on Orthogonal Polynomials and Applications, Erice, Italy, 1990. I M A C S Annals o n Computing and Applied Mathematics 9 (1991), 21-55; J. C. Balitzer, A. G. Basel. 8. W.N. Everitt and C. Markett. On a generalization of Bessel functions satisfying higher-order differential equations. Jour. Computational Appl. Math. 54 (1994), 325-349. 9. W.N. Everitt and L. Markus. The Glazman-Krein-Naimark theorem for ordinary differential operators. Operator Theory: Advances and Applications 98 (1997), 118-130. 10. F.A. Griinbaum and L. Haine. Orthogonal polynomials satisfying differential equations: the role of the Darboux transformation. C R M Proc. Lecture Notes 9 (1996), 143-154; Amer. Math. SOC.,Providence, RI, USA. 11. J. Koekoek and R. Koekoek. On a differential equation for Koornwinder’s generalized Laguerre polynomials. Proc. Amer. Math. SOC.112 (1991), 10451054. 12. A.M. Krall. Orthogonal polynomials satisfying fourth order differential equations. Proc. Roy. SOC.Edinburgh (A) 87 (1981), 271-288. 13. H.L. Krall. Certain differential equations for Tchebycheff polynomials. Duke Math. J. 4 (1938) 705-718. 14. H.L. Krall. On orthogonal polynomials satisfying a certain fourth order differential equation. The Pennsylvania State College Studies: No. 6. (The Pennsylvania State College, State College, PA.: 1940.) 15. T.H. Koornwinder. Orthogonal polynomials with weight function ( 1 - ~ ) ~ ( 1 + z ) ~M 6 ( z 1) N ~ ( -z 1). Canad. Math. Bull. 27 (2) (1984), 205-214. 16. L.L. Littlejohn. Remarks on the Plum partial differential equation. (Report; Utah State University, USA; May 2004: available in LaTeX file PlumPDE.tex). 17. C. Markett. Product formulas and convolution structure for Fourier-Bessel series. Constr. Approz. 5 (1989), 383-404. 18. M.A. Naimark. Linear dzflerential operators: 11. (Ungar, New York: 1968. Translated from the second Russian edition.) 19. M. Plum. Manuscript on a fourth-order partial differential equation connected with the Bessel-type special function. (University of Birmingham, England; Summer 2003.) 20. E.C. Titchmarsh. The theory of Fourier integrals. (Oxford University Press: 1948.) 21. E.C. Titchmarsh. Eigenfunction ezpansions: I. (Oxford University Press; second edition: 1962.) 22. M. van Hoeij. Formal solutions and factorization of differential operators with power series coefficients. J. Symbolic Comput. 24 (1997), 1-30. 23. M. van Hoeij. Personal contribution. (International Conference on Difference Equations, Special Functions and Applications; Technical University Munich, Germany: 25 to 30 July 2005.) 24. G.N. Watson. A treatise o n the theory of Bessel functions. (Cambridge University Press; second edition: 1950.)

+

+ +

213

LAPLACIANS AND THE CHEEGER CONSTANTS FOR DISCRETE DYNAMICAL SYSTEMS

SARA FERNANDES~,CLARA GRACIO~AND J. SOUSA R A M O S ~ We consider discrete laplacians for iterated maps on the interval and examine their eigenvalues. We have introduced a notion of conductance (Cheeger constant) for a discrete dynamical system, now we study their relations with the spectrum. We compute the systoles and the first eigenvalue of some families of discrete dynamical systems.

2000 MCS Primary 37B10,05C35; Secondary 37305, 37315 Keywords: Discrete Laplacian, Cheeger constant, discrete dynamical system, systole Dedicated to the Memory of Bernd Aulbach

1. Introduction

Our subject is the study of discrete dynamical systems arising from the iterates of a map in the interval and our main concern is to search for invariants which could differentiate systems with the same topological entropy. In this paper is made a strong connection between a geometric point of view and a discrete dynamical system point of view. We explore the connections Cheeger constant/conductance, Laplace-Beltrami operator/discrete laplacian and systoles and isoperimetric inequalities in both contexts. We begin with the geometric motivation, see Grbcio-Sousa Ramos5. The idea of a Riemann surface is a central one in mathematics, and appears in such seemingly diverse areas as low dimensional topology, algebraic, differential and hyperbolic geometry, complex analysis, group theory and even number theory. This paper concerns compact Riemann surfaces endowed with a hyperbolic metric. It is possible to describe it in some ways, these include its representation as an algebraic curve; as a period matrix; as a Schottky group; as a Fuchsian group and as a hyperbolic manifold, in particular, using Fenchel-Nielsen (F-N) coordinates. It is exactly this last way that we will use, Riemann surface theory based on closed geodesics using Fenchel-Nielsen (F-N) coordinates. This geometry has always played

214

an important part in the study of Riemann surfaces. There are directly related to very active subjects such as spectral theory, Teichmuller spaces and number theory. One of the most natural objects in this context are the systoles, the shortest closed geodesics of a Riemann surface. They provide perfect analogues to classical problems of lattice sphere packings. In this paper we will introduce and evaluate explicitly the systoles in discrete dynamical systems. Given a compact Riemannian manifold M , the Laplace-Beltrami operator A on functions on M is an elliptic operator with discrete spectrum 0 = 110

<

< p2 < .. < pk... f

00.

Whereas in the case of surfaces of genus 0 or 1 the explicit computation is possible, this is not the case for surfaces of higher genus. For surfaces of genus g = 2 we can compute numerically the geodesic length spectrum of M (lengths of closed geodesics) 0 =lo

< (1 < c 2 < ... < !?k...

-+ 00,

and using this spectrum to computing the pi,i E N. These surfaces can be obtained by gluing together pairs of pants with no twists on the boundary components. This corresponds to Riemann surfaces in Teichmuller space for which the Fenchel-Nielsen (F-N) coordinates are of the form (e(yl),l(y2),t(y3),0, O , O ) , where l(ri)are the lengths of the three geodesics on the surface.

..------

Figure 1. Bitorus: closed Riemann surface of genus 2.

One of the possible ways to describe a closed Riemann surface M of genus g are the F-N coordinates, that consists in the lengths and twists of 39 - 3 disjoint simple closed geodesics. This space of coordinates is homeomorphic

215

to the Teichmuller space and the Teichmuller modular group acts on any such space of F-N coordinates as a group of algebraic diffeomorphisms. In this context we can state the definition of the Cheeger constant, see Brooks4.

Definition 1.1. Given a Riemannian manifold M (compact, no boundary), define the Cheeger constant by area(X)

h ( M ) = inf x min(woZ(A),wol(B))’ where the infimum is over all compact separating hypersurfaces in M (A and B are the resulting disconnected sets).

A basic model for the discrete context is a graph G = (V, E ) consisting of a set V of vertices and a prescribed set E of pairs of vertices. For a subset X of vertices in G, the volume vol(X) is defined by vol(X) =

c

d,,

VEX

where d, is the degree of w. We denote the volume of G by vol(G) =

c

d,

21

and the edge boundary is a(X) = {{x,y} E E : x E X, y E V\X}

.

Definition 1.2. The Cheeger constant of a graph G with vertex set V is defined to be h(G) = inf

lW)l

x min(wol(X), wol(V\X))’

where the infimum is over all subsets X of the vertex set and la(X)l is the number of edges joining the vertices of X with vertices of V\X, (See Bollobbs3 for details). The smallest non-trivial eigenvalue p1 of the Laplacian of a graph is closely related to the isoperimetric Cheeger constant. The Cheeger inequality for a graph G states that

Related with this kind of constants, the isoperimetric constants, is another geometric quantity, the systole, which has also a correspondent in graph theory : In 1972 Marcel Berger defined metric invariants that capture the “size” of k-dimensional homology of a manifold. He asked if these invariants can

216

be constrained by the volume. The invariant that captures the notion of the "size" is the systole, see Berger2.

Definition 1.3. A systole of a Riemann surface M of genus g is the length of the shortest closed geodesic of M . As usual there are the graph-theoretical analogue of this notion. One of our interest is to pass back and forth between the geometric and graphtheoretic pictures. Comparing various results may be an important source of insight. After the geometric motivation we present now the dynamical settings. Consider a discrete dynamical system described by the iterates of a map f in the interval. Suppose that f is a Markov map, that is, a piecewise monotonic transformation that maps each interval of the partition onto a union of intervals of the partition. Let A, be the transition matrix associated to this system and let P be the corresponding weighted stochastic matrix (see Parry''). A row vector T E R" is a stationary distribution of the system with weighted transition matrix P if and only if a) ~ ( i2)0 , for all i E V ; b) CiGV.rr(i)= 1; c) 7r = TP. A Markov chain with transition matrix P is said to be ergodic if it has a stationary distribution. It is known that any finite, irreducible Markov chain is ergodic. To each such dynamical system, there is an underlying graph G, whose vertex set V is the set of intervals of the Markov partition and whose edges i j correspond to non-zero entries aij of the transition matrix A f , also denoted in the graph context as A (G) . Denote by { 1 , 2 , . . . ,n } the set of vertices and by e = (ij)the edge that joins the vertex i with the vertex j. We say that i is the starting point of e and that j is the end point of e. Let w(ij) = 7riPij the weight of the edge and let (Gf, w) be the weighted graph associated to f. The volume of (Gf, w) denoted by vol(Gf, w) is the sum of the weight of its edges vol(Gf, w) =

w(e). eEE

The systole of (Gf, w) is defined by sys,(Gf) = inf { p , ( c ) : c is a non-loop cycle of Gf}

217

where pw ( c ) is the product of the weights of its edges, compare to Balacheffl. The geometric quantities we've seen in the geometry's motivation can be viewed as quantities in graph theory, and so, as quantities of a discrete dynamical system. 2. Discrete Laplacian and conductance

Let D(G) = diag(di : i is in V ) be the diagonal matrix indexed by V and with vertex degrees in the diagonal, that is, dii is the number of edges beginning or ending in the vertex i. Define the Laplace matrix of G by

A ( G ) = D(G) - A( G ) . It is easy to see that A ( G ) satisfy the relation

A(G) = Q

Q~,

where Q is the incidence matrix with columns indexed by the edges and rows indexed by the vertices, defined as follows qie =

{

-1 if i is the starting point of e +1 if i is the end point of e 0 if i is not incident on e

,

see Moharg. Note that in the construction of the incidence matrix Q we had to chose arbitrarily an orientation and ignore the loops. In fact in the case of an unoriented graph it will make no difference. The case we are concerned with in this paper corresponds to an oriented and weighted graph, i.e., we have no longer a symmetric transition matrix. Anyway we will define the Laplacian in a such way that the relation (1) will be satisfied and the Laplacian will be still a symmetric matrix (see Fernandes-Sousa Ramos')

+

A ( G ) = D(G) - ( A A T ) where D(G) = diag(dii : i is in V ) ,with n

n

j=1

j=1

is the matrix degree, or, for a weighted version

+

A * ( G )= D*(G)- ( P P T ) ,

218

where D*(G)= diag(d;i : i is in V ) ,with n

n

j=1

j=1

Now we define the conductance of a discrete dynamical system as follow.

Definition 2.1. The conductance of a discrete dynamical system with stationary distribution r and stochastic transition matrix P is defined by

@=

min

iEU,j€V\U

Era

0#UCV 0 # IUI 5 ;

(ELI

The conductance may be viewed as a weighted version of edge expansion of the graph underlying the Markov chain. For a fixed U , the quotient in the definition is just the conditional probability that the chain in equilibrium escapes from the subset U of the state space in one step, given that it is initially in U . Thus @ measures the ability of the system t o escape from any small region of the state space, and hence to make rapid progress to the stationary distribution. It is used to bound the mixing rate of a discrete dynamical system, see Bollob&s3. Let’s observe the behavior of certain families of maps. These maps are piecewise linear models, semiconjugated to quadratic and cubic maps. We begin with the definition of the tent map. Let I = [0,1], the unit interval, we consider a family of unimodal maps fs : I + I defined by

o < x < ~ ;
if fs(z)={ysz+sif

These are Markov maps for certain s E (1,2). Now we define a family of bimodal maps. Consider f s , , : I + I defined by

f s , J x )=

{

if O < z < c l

-sx + e if sz + 1 - s if sx

c1 cg

< z < cg ,
where c1 = e/2s and cg = (e+ s - 1)/2s. These are Markov maps for certain s E (1,3). With this structure we are able to codify the trajectories using the symbolic dynamics in both cases. As usual for the unimodal case we use

219

the alphabet { L ,C, R } and the rule

L if x < 112

R if x > 1/2 to codify the addresses of the trajectory of the critical point c = 1/2, attributing the correspondent symbol to the iterates of c. For the bimodal case, denote by L (left), M (middle) and R (right) the three laps of the interval I and by A and B the two critical points according to a similar rule:

For this case we need the sequences of both critical points {fi(cl)} and {fZ(cz)}, with i E N,the pair of kneading sequences, to have the topological classification of the maps by their trajectories (in true topological semicongugacy), see Milnor-Thurston'O. Our results are strongly supported by the observation of the tree with the trajectories of the unimodal family, ordered according to the topological entropy, see Lampreia et d7for details about the tree of unimodal maps.

LL

LLL

Figure 2. tropy.

LR

LLR

LRR

RR

LRL

RRL

RL

RRR

RLR

R LL

Tree I of the trajectories of unirnodal maps, ordered a s the topological en-

220

So, we observed a certain monotonicity of both quantities, conductance and first non-zero eigenvalue of the laplacian, when we follow a "path" in the tree, from the first branch (corresponding to the trajectory of the fixed point, i.e., with period 1) until a final (obviously never attempted) branch (nonperiodic trajectory), observing these two values for each period, k, in the path. In Figure 3 and 4 is the graphic result of this observation for a family of unimodal maps and a family of bimodal maps.

0.1 8

10

12

14

d i m (A)

Figure 3. T h e higher curve is 29, the middle is pL,andthe lower is a 2 / 2 , for the family (CRLL(RLk)m), k = 1,2, ....

dim ( A ) Figure 4. The higher curve is 2 9 , the middle is pland the lower is a 2 / 2 , for the family ( ( A R M k ) " , (BLMk)" ), k = 1,2, ....

221

The observation of these quantities for a large number of such families of maps leads us to formulate the following result, which is known and prove for the simple random walk on the graph, see Mohar'.

Theorem 2.1. Let f be a map of the interval and Af the transition matrix associated to the Markov partition. Then the first non-zero eigenvalue pLl(Af)of the laplacian and the conductance @f of the Markov graph verify the following inequality

This upper bound is a discrete version of the Cheeger inequality. As it was pointed out in Moharg it has several applications concerning the convergence rates of Markov chains to the equilibrium. 3. Systoles in discrete dynamical systems

We've talked about systoles in geometry and graphs. To bring this concept to the context of discrete dynamical systems we will use again the tools given by graph theory. We can formulated the following result that establish a relation between the systole and the Betti number of the Markov graph G f given by bf = aij - n 1, where Af = ( a i , j ) is the adjacency matrix of G f , see i,j=l Balacheff' .

+

Theorem 3.1. Let f be a map of the interval, (Gf, w)the weighted Markov graph and htop(f) the topological entropy of the map f. Then

We applied this inequality to the families we've treated before. First for unimodal family and secondly for the bimodal family.

222

t

6

8

10

12

14

16

k

Figure 5 . The higher curve is the topological entropy times the systole and the lower is the logarithm of the Betti number, for the unimodal family (CRLL(RL)k)", k = 1 , 2 , 3,....

1 6

* 5 0'

8

10

12

14

16

k

Figure 6. The higher curve is the topological entropy times the systole and the lower is the logarithm of the Betti number, for the bimodal family (BLMk)OO), k = 1 , 2 , 3,....

Acknowledgments

First two authors acknowledge CIMA-UE for financial support. Third author partially supported by FCT/POCTI/FEDER. References 1. F. Balacheff,

Volume entropy,

systole and stable norm on graphs.

math.MG/0411578.

2. M. Berger, D u cdte' de chez Pu, Ann. Scient. Ec. Norm. Sup. 5 (1972) pp.1-44 3. B. Bollobb. Volume estimates and rapid mixing, Flavours of Geometry.

223 MSRI Publications, Vol. 31,(1997). 4. R. Brooks, P. Perry and P. Petersen. On Cheeger’s inequality. Comm. Math. Helv. 68, pp. 599-621, (1993). 5. C. Gracio and J. Sousa Ramos, Spectrum of the laplacian on hyperbolic surfaces, Submitted to Grazer Mathematische Berichte. 6. S. Fernandes and J. Sousa Ramos, Conductance: from electrical networks through graphs to dynamical systems, Proceedings of NDES04 (2004). 7. J. P. Lampreia, A. Rica d a Silva, J. Sousa Ramos, Subtrees of the unimodal maps tree. Boll. Un. Mat. Ital. C (6) 5 (1986), no. 1 (1987), 159-167. 8. B. Mohar, Isoperimetric inequalities, growth, and the spectrum of graphs, Linear Algebra Appl. 103, pp.119-131 (1988). 9. B. Mohar, The Laplacian Spectrum of Graphs, Graph Theory, Combinatorics and Applications, V01.2, (1991), pp.871-898. 10. J. Milnor and W. Thurston, On iterated maps of the interval, Dynamical Systems: Proc. Univ of Maryland 1986-1987, Lecture Notes in Mathematics vol. 1342 (Springer, Berlin, New York), pp.465-563. 11. W. Parry, Intrinsic Markow chain, Trans. Amer. Math. SOC. 112 (1964), 55-65.

224

ALGORITHMS AROUND LINEAR DIFFERENTIAL EQUATIONS

A. FREDET Iut de Saint-Denis Place du 8 Mai 1945 93206 Saint-Denis cedex, f i a n c e E-mail: fredetQstix.polytechnique.fr, fredetQiutsd.univ-paris13.fr

When one considers a linear differential equation any(,)

+ a,-ly(n-') + . . . + a l y ' + aoy = 0,

one is usually interested in solving this equation. But what does solve mean ? Is it to compute one solution or a basis of solutions ? And more precisely, is it t o compute a numerical approximation of the solutions or t o find a global form ? In this paper, we focus on closed form the solutions i.e. solutions that can be fully expressed using the usual functions. For example, one considers rational solutions, i.e., solutions in the field that defines the coefficients of the equation, since this problem is the starting point of many others algorithms like the factorization of linear differential operators or the computation of liouvillian solutions of linear differential equations (solutions which can be built up from the rational functions by algebraic operations, by taking exponentials and by integration). If many works were done on some special linear differential equations, the study of the general case is uneasy. It usually leads to heavy computations. So, the use of computer algebra is natural for solving such problems. But if the theoretical approach is well known, the algorithmic methods are recent, even for rational solutions of linear differential equations with coefficients in C(z) or in exponential extensions, and furthermore for more general solutions like solutions involving special functions. In this paper, we summarize actual algorithms to compute closed form solutions of linear differential equations. Generalizations to linear differential systems and linear difference equations are briefly discussed.

1. Introduction The study of linear differential equations is not recent. During the 17th century, many people were already interested in the integration in finite terms and the linear differential equations with order one. Since these problems appear in physical phenomenons, Newton (1642-1727), Euler (17071783), Bernouilli brothers (Jacques: 1654-1705 ; Jean : 1667-1748), Rccati

225

(1676-1754), Lagrange (1736-1813) worked on such subject. Many articles were published on special cases of linear differential equations, mainly with order one, sometimes with order two. The works of Liouville (1809-1882) are usually considered as the starting point of the study of linear differential equations with a general form:

L(y)

= any(,)

+ . . . + a1y’ + aoy = b,

(1)

The first problem consists to determine the form of the considered solutions. For example, many works were done on local solutions23. But some physical properties of the solutions can not be deduced from the local approach. So, one should also be interested in global solutions. If one considers the following linear differential equation II

y +y=o

&+ 2n

Z2n+l

one finds that a local solution has the form c1 c2(2n+l)! where c1 and c2 are constants. One can also see that the functions cos(x) and sin(x) are solutions, and so are the functions exp(ix) and exp(-ix) where i is the complex number such that i2 1 = 0. Each form of the solution gives some indications on the physical phenomenons described by the equation. In this paper, we are interested in closed f o r m solutions, i.e. solutions that be completely expressed using usual functions. This generalize the problem of integration in finite terms, which consider a linear differential equation with order one 28 29 30. We now consider a differential field ( K ,D ) , i.e. a field K together with its derivation D . We denote by C the constant field C = { a E KlDa = 0). Furthermore, we assume that C is algebraically closed. The first closed form solutions one is interested in are rational and exponential solutions:

+

Definition 1.1. Let (K, D ) be a differential field and L = C:=,aiDi be a linear ordinary differential operator with coefficients in K .

A rational solution of L is any element y E K such that L(y) = 0. An exponential solution of L is any element y such that L(y) = 0 and EK.

9

Even if these solutions were studied since a long time, it is in 1991 that M.F. Singer presented the first method to find rational, and exponential solutions of linear differential equations with coefficients in almost all liouvillian extensions of C(x) 32. This method although effective was not

226

very efficient. One will see that even in C(z), and even with a powerful computer, the rational solutions are not easy to compute. 2. Rational and exponential solutions of linear differential

equations Consider a linear differential equation L(y) = y(”)+. . .+aly’+aoy = 0 with coefficients in C(z). Assume that one is interested in exponential solutions i.e. solutions with the form exp(J f(z)d z ) for some function f(z)in C(z). Let p be an irreducible polynomial appearing in the denominators of the coefficients ai. Assume that, using padic expansion, one has:

+ . . . where deg(u,) < deg(p) and a 2 1 y = exp(/$ + . ) where deg(u,) < deg(p) and a 2 1

f

u, Pa

=-

+

and if one denotes

+

+

then, since L(y) = 0, there exists i and j such that ai ia = aj j a so a is in the set 5 i < j 5 n}. Furthermore, one finds the indicia1 equations:

{WIl

c

ai,aiui = 0 mod p or

il ai + i a = m a x ( a j +ja) i- 1

This gives us a bound for the exponants and a finite set of choices for the the coefficients u, since deg(u,) < deg(p). Changes of variable lead us to I

+

consider exponential function with the form exp(J Cj s) where s is a polynomial and pj are irreducible polynomials that do not occur in the denominators of the coefficients ai. Using expansions at infinity, one can also find possibilities for the polynomial s. So, the problem is reduced to compute rational solutions of associated linear differential equations. Remark that the possibility for u, is usually not unique, so this gives us

227

several possibilities for the changes of variable and one will consider several associated differential equations. The number of possibilitites can be decreased using modular information 17. This reduction from exponential to rational solutions can always be perEK formed in extension with the form K ( t )where t is such Dt E K or or t is algebraic over K 32. So, we now focus on rational solution.

?

2.1. LDE with coemcients in C ( x ) Let

L(y)

+ ~ , _ l y ( ~ - ' )+ . . . + aly' + aoy with ai E C(z)

= y(%)

The first algorithm concerning linear differential equations implemented in computer algebra systems compute y E C(z) such that L(y) = 0. There are mainly three steps to proceed: (1) Step 1: find P E C [ x ]such that P y E C [ x ] . Change the variable and focus on polynomial solutions. (2) Step 2: find M such that L(y) = 0 with y E C[x]implies deg(y) 5

M. (3) Step 3: findyi'sin Csuch that y = zM+y~-~zM-'+...+y1z+yo

Let us consider each step in details: (1) Step 1: Denominator Assume Y = p~l.,.p;m

with N E C[z] and p l , . . . ,p, irreducible polynomials in C[x].For all k, 1 5 k 5 m 21 -++. . . and by iteration y(i) = arc(arc +

y = Pik

2).

. . (arc + 2 - 1). +. . .

p;k+i

If one denotes vi ai=-+ P?

...,

then

This indicia1 equation gives us a finite set of choices for the integer a and then a bound for the exponent. An alternative method to the padic expansions exists avoiding the complete factorization of the denominators of the coefficients. A change of variable lead us focus on polynomial solutions

228

(2) Step 2: Bound on the degree Usually, one changes the derivation: instead of one uses D = XD such that Dx = x. So

L = D" + f-J,-lD("-l)

D such that Dx = 1,

+ . . . + UlD + a0

becomes

L

=

D" + bn-1D("-1) + . . . + blB + bo

+.

Consider a polynomial solution y = xm+ym-lxm-l . . +ylx+yo and plug it into the linear differential equation. This gives us the indicia1 equation

c

m(m-l)...(m-i+l)wz

=o

zlai=max(aj)

So, finite set of possibilities for the degree and then a bound can be found. (3) Step 3: coefficients We plug y = xm+ym-lxm-l+. .-+ylx+yo in the linear differential equation. This gives us a linear system and we find the yi's. This method, although algorithmic, might take time when the order of the linear differential equation is high because one the wants the integer roots of a polynomial with a high degree. Furthermore, to consider a bound on this root might not be the best way since we compute all the integer roots. Variations exist6, proposing to use all the integer roots and making the computation faster when the polynomial solution has a big degree and few non-zero coefficients. Furthermore, when the bound on the degree is big, the linear system found to compute the coefficients is also big. But this system has a special form and a reccurrence can be used instead Sometimes, the computation is heavy to realize at the end that there is no solution (for example if the linear system is unconsistent). It might be interesting to detect the non-existence of rational solutions first, and this is done recently '. So, even with a powerful computer, the efficiency of the algorithms to compute integer roots of polynomial, to find solutions of a linear system or detect their non-existence is crucial. Algorithms t o compute rational and exponential solutions of linear differential equations rely on classical problems, which are also studied in computer algebra.

'.

229

2.2. LDE with coeficients in exponential extensions

When one consider linear differential equation appearing in physical problems, it involves others variables than x. For example, the exponential variable appear in many biological phenomens so it is natural to be interested in linear differential equation with coefficients involving exponential variable:

Definition 2.1. 8 is exponential over K if it is transcendental over K such is in K . that Furthermore, we assume that the constant field is not extended, i.e., if Const(K) := {a E K J D a= 0) then Const(K(8)) = Const(K).

9

Let L(y) = y(,)

+ a,-ly(n-l) +. . . + al{ + aoy with ai E K(8)

Assume that one wants to compute y E K(8) such that L(y) = 0. The first algorithm presented in 1991 32 has three steps, as in C ( x ) : (1) Step 1: find P E K[8] such that P y E K[8-’,8] (2) Step 2: find M and m such that L(y) = 0 with y E K[8-l,8] deg,(y) 5 M and vals(y) >_ m. (3) Step 3: find yi’s in K such that y = yMOM . . . ym8”

+

+ +

Consider each step in details: (1) Step 1: Denominator Using padic expansions, one remarks that the polynomials appearing in the denominator of rational solutions are factors of the coefficients ai. Furthermore, one finds indicial equations that give us bounds on the power with which these polynomials appear in the denominator. (2) Step 2: Bounds on the degree and the valuation Let M and m be such that if L(y) = 0 for y E K[O-l,8] then dege(y) i M and v.le(y) L m. We have

+ an-1Dn--l + . . . + a l D + a0 = 8”L” + . . . + fPL, with Lk E K[D] Y = yMeM + . .. + ymem L

= a,Dn

Then L(y) = 0 + L,(ym8m) = 0 and L,(yM8’)

=0

230

So, we now focus on exponential solutions of linear differential equations with coefficients in K . We compute all the exponential solutions of L, and L, and we use Itisch's integration to find the ones with the form f Or with f E K and y E Z. An improvement l4 is proposed for the field C(z, e.f dz)d z ) : since one is interested in solutions with the form f exp(J yg(z) dz), and more precisely in the integer y,the authors use expansions (padic or at the infinity) to find a finite set of possibilities for such y. This avoid to compute the unnecessary solutions and do not rely on heavy integration results. This improvements is generalized to a larger class of exponential extension 19. (3) Step 3: coefficients We plug Y = y M O M + . . .+ymOm into the linear differential equation. This gives us a linear differential system with coefficients in K . We use non-commutative algebra to triangularize this system and we solve it by iteration. An improvement is also proposed l4 for the field C(z, e . f g ( z ) d z ) : the -+ authors remark that the the linear differential system M Y = 0 has a special form:

1 0

* *

..

,..

0

...

0

..

I * . . .

Y6

* 0

**-.. *

M=

Y6+1 4

**... o r ...

*

0 0

.

*

0 0

...

0

* *

and Y =

Yr- 1 %Y

*

So, one uses this special form to avoid to use heavy noncommutative algebra to triangularize this system and one considers a recurrence to compute the coefficients. This improvements is also generalized to a larger class of exponential extension 19.

Let us follow the original algorithm and its improvements on one example:

23 1

Example 2.1. Consider a linear differential equation with coefficients in C ( x ,ex):

L

+ 2s)e3z + (2 + + +((7 - 4z2)e3"+ (-4z2 + z ) ~ - ~ " )+D (6 + 12z2 18s)e3" + (-82 - 4 - 8 ~ ~ ) e - ~ " = ((7 4z2)D+ 6 + 12z2 - 18z + (-3 + 2 ~ ) D ~ ) +((2 + 2z)D2 - 8z - 4 - 8z2 + ( - 4 ~+~2)D)e-2X

=

((-3

-

-

e ~ ~

One then has

L3 = ((7 - 4 x 2 ) 0 + 6 + 122' - 18z

+ (-3 + 2z)D2)

One wants to find y1 = es f(.) with f E C(z) such that L3(y) = 0. Using the algorithm to compute exponential solutions of linear differential equations in C(z), one finds that f(z)= c1 exp(z2) c2 exp(3z) where c1 and c2 are constants. One now searches g E C(z), y E Z and c1,cz constants such that es f = gerx. This gives us that g = 1, y = 3, c1 = 0 and c2 = 1. But one can directly compute the y:

+

y = f(z)eYz

y(') = (+f(.)

+ . . . e(Y--l)l+ . . . + . . . f'(z)+ . . .f(i)),YZ

And, since at infinity f = cxa

+ ...

+ . . . , we have

y(') = (y'm'~+ . . . )eYx + . . .

+ 2z)e3x + (2 + + +((7-4x2)e3" + (-4z2 + 2 ) e - 2 z ) ~+ (6 + 12x2 - 18z)e3" + (-8s - 4 - 8 ~ ~ ) e - ~ " so L(y) = 0 =+ -47 + 12 = 0 * y E {0,3} L = ((-3

This method avoid the computation of unnecessary exponential solutions (ex' in the example) and the use of the integration process. So this natural idea makes the algorithm faster and this algorithm is now available in computer algebra systems. 3. Liouvillian solutions of linear differential equations

One is usually interested in a larger class of closed form solutions:

Definition 3.1. Let ( k , D ) be a differential field. We say that K = k ( t 1 , . . . ,tn) is a liouwillian extension of K if for all i:

232

9

(1) E k ( t l , . . . ,ti-1) (exponential case) (2) Dti E k ( t 1 , . . . , t i - 1 ) (logarithm case) (3) ti is algebraic over k(t1,. . , t i - 1 )

-

Let L = Cy=oaidi be a linear ordinary differential operator with coefficients in k. A liouvillian solution of L is any element y in a liouvillian extension and such that L ( y ) = 0. This class of solution is very large and it would be hard t o compute such solutions without additional informations. But, as for a classical polynomial, you can associate a linear group to each linear differential equation. The properties of this group give you informations about the solution. This theory is known as the differential Galois theory 2 7 . The Galois group of a linear differential equation with order n is a subgroup of GL,. Properties of these subgroups give us informations on the form of the solutions. For example, one reduces the possibilities for the form of liouvillian solutions:

Proposition 3.1. 31 A linear differential equation L has liouvillian solution if and only if it has rational solution, or exponential solution or solutions with the form y = e s u where u is algebraic with degree bounded by a function depending only of the order n of L. The bound on the algebraic degree can be explicitely computed and is usually high. Improvements on this bound are done for linear differential equations with order two:

Proposition 3.2.

25

Let k be a differential field. Let L ( y ) = y" - ay with

E Ic

This linear differential equation has liouvillian solution i f and only i f it has solutions with the form y = es 11 where u is algebraic with degree 1,2,4,6 or 1zl solution of the associated Riccati equation

R(u)= u2 + 21 - a Same result holds for order three:

Proposition 3.3.

33 34

Let k be a differential field. Let 111

L(y)=y

- al;

+ aoy with a l ,

a0

E k

This linear differential equation has liouvillian solution if and only if it has solutions with the form y = e s u where u is algebraic with degree 5',4,6,7,9,12,21 or 36.

233

To compute the solutions with the form es where u is algebraic with degree m, there are two possibilities: the first one is to find the minimal polynomial P ( u ) = urn bm-lurn--l . . . blu bo: one computes an associated linear differential equation LBm(y) where the solutions are products of m solutions of L. Then, b,-l = u1 . .. u, is the logarithmic derivative of one solution of L@"(y). So, one computes exponential solutions of LBm(y) and proceed by iteration to find the coefficients of the minimal polynomial. Note that the order of L@"(y) is greater than the order of L , so these computations will take time. The second possibility consists in remarking that this solution u 1 + . . u, is a semi-invariant of the Galois group of LBm(y). A semi-invariant factoring into a product of linear form will be paired with a liouvillian solutions

+

+ +

+

+ +

+

35.

Proposition 3.4. A linear differential equation L(y) = 0 has a liouvillian solution whose logarithmic derivative is algebraic of degree m i f and only if there is a semi-invariant of degree m which factors into linear forms. One then needs algorithm to compute semi-invariants of Galois group, to factorize multivariate polynomials, and more precisely to compute linear factors of multivariate polynomials.

4. Solutions of linear differential equations in term of special functions The study of non-liouvillian closed form solutions is new in computer algebra. An algorithm l5 for computing special function solutions of the form y(z) = m(z)F(e(z)) of second order linear differential equations, where m ( z )is an arbitrary Liouvillian function, ((z) is an arbitrary rational function and F satisfies a given second order linear ordinary equation (in particular Airy, Bessel, Kummer and Whittaker functions). Another algorithm l3 is given for considering rational solutions of linear differential equations with doubly-periodic coefficients. Such equations appear historically in many instances, for example in the Lam6 equation y"(z) - (n(n l)P(z) B)y(z) = 0 where n is a positive integer, B is an arbitrary constant and P = P ( z ;92,93) is the Weierstrass function such that P'(z)= J ~ P ( z-) gZP(z) ~ - g3.

+

+

234

5. Linear differential systems A linear differential system Y' = AY where A is a square matrix can be reduced to a linear differential equation using a cyclic vector. But the linear differential equation has then a large order and big coefficients so one usually try to consider directly the system. The approach to compute rational solutions of linear differential systems is more or less the same than the approach to compute rational solutions of linear differential equations: one first compute the denominator of solutions, then a change of variable allows us to focus on polynomial solutions. We then try t o bound the degree of such polynomial solutions. But this step is harder than in the case of linear differential equations, since the indicia1 equation sometimes do not give possibilities for the exponants:

Example 5.1. Consider the following system: dY dx

- = M Y with

M=

Assume one is interested in polynomial solutions. The naave method would be to assume that such a solution exists, with the form

One then plugs this solution in the system, collects with respect to x and vanishes the coefficients. In this example, one obtains coefficient of x'+~

:

coefficient of xXfl : coefficient of 'x

:

coefficient of xx-l :

(o0

(; (; (; =

0 -23 ) Y * = o

$)YA4+(i

gO)y*=O

;9Yx-2+(;

gO)y*-l=o

;$)Y'4+(;

gO)YA-2+(;

-

1)YX-1

;)y*=hy*

235

One solves these systems and finds

YX=(";)

(

YX-1 = )l;;

YX-2

=

YX-3 =

When one computes

(; One finds that y~ =

y ~ - 4 , one

0")

3Cl(X -

(";) # (:).

YX--4

(;J

(

c43X 3c3 - T C 1

finds condition on A:

+ (;:;)

(

= (A - 1) )l;;

2 ) = 0 and then X

Vanishing

the constants parameters: ci : system (2) have the form c1

)

y ~ - 3=

(x2'>

=

2 because c1

# 0

since

y-1, one finds also conditions on

c4 = c2 =

0, c3

= c1.

So, solutions of the

where c1 is constant.

One can prove that a condition on the degree X will always appear but there is not bound on the number of iterations that will be needed. Anyway, if one changes the system (but not the problem), the condition may appear at the first step:

Example 5.2. Consider the same system with the form x z = M Y where

If one considers the matrix

then the previous system is equivalent to Ax% solutions, one obtains

(:

-2

1-X

)

yx = 0.

=

AMY. If one plugs a

236

),

The coefficient yx is in the kernel of

which must be non-

)

(:

-$ vanishes. The 1-X parameter X appear in the indicia1 equation but the condition does not give us a finite set of possibilities for A. Remark that if one changes the variable, considering Y = T Z where

trivial. This implies that the determinant of

one considers the following system dZ

x-

dx

= M Z where M = T-lMT

-

=(

2 x 20 - 1

If one notes A

=

(s0 1’), and considers A X E

a solution with the form z

(X 0 X 2;) ),

= zxxx

T-~xT’

(3)

-””> 2

=

A M 2 then, plugging

+ . . . , one finds that the determinant of

must vanish. This gives us a finite set of possibilities for

(here = 2 and then a bound on the degree. The coefficients can be computed using- the indeterminate coefficients method. One then finds that Z = c1

(x”,a’)

is a solution of (3), and then Y = TZ

= c1

(“2

I ) is

a solution of the first linear differential system.

On this example, one sees that when the determinant is equal to zero, there is no condition on the degree appearing at the first step. In the case of linear differential system with coefficients in C ( x ) , several works are made to find a suitable equivalent system such that the bound appear a t the first step. The first method use the notion of superirreducibility 21. Another one changes the system into another one which is not really equivalent, since one needs also additional conditions, but where the conditions on the degree can be easily computed. The first method is generalized l 8 to exponential extensions, but many coefficients fields can not be considered until now.

237

6. Linear difference equations Linear difference equations are equations with the form any(z

+ n) + . . . + aly(z + 1)+ aoy(z) = O

The study of these equations is not new and the link between the linear differential equations and the linear difference equations is now clear. An algorithm' to compute the rational solutions of this two types of equations with coefficients in C(z) is given and is extended to linear q-difference equations (equations of the form aiy(q2z) = 0 ) . M. Bronstein" adapted the algorithm given in to monomial extensions, and used lo the methods to find the solutions of linear difference equations in their coefficient field. As for the linear differential differential equations, one focuses on some special closed form solutions: we consider the map cr such that d u o , a1, a2,. . = (a17 a2,. * . ).

Edo

Definition 6.1. A sequence a is hypergeometric if (TU = f a for some f E K . There is an evident link between the exponential solutions of linear differential equations and the hypergeometric solutions of linear difference equations.

Definition 6.2. Let m > 0. The mth spread of a is the sequence a' given by (am>n =

{

=

anlm if n 0 mod m 0 if n $ O m o d m

Definition 6.3. Let a and b be sequences and f be a function. The ring L of the liouvillian sequences is the smallest difference subring containing C(z) such that (1) ab = f b =+ b E L (2) i f a E L , c b = b + a = + b E L (3) a E L Vi,m such that 0 5 i 5 m, u - ~ ( u ' ) E L

*

Definition 6.4. The interlacing of m sequences a('),... ,a(m)is the sequence

238

There is also a Galois group associated to the linear difference equations and properties of these Galois groups give us properties of the liouvillian solutions: Proposition 6.1. 2o Let L be a linear difference operator with coeficients in C ( x ) where C is algebraically closed and u is the automorphism of C ( x ) over C mapping x to x 1. Let G be the difference Galois group of L over

+

C(X). (1) All the solutions of L ( y ) = 0 are liouvillian i f and only if G is solvable group (2) L ( y ) = 0 has a liouvillian solution i f and only i f it has a solution y such that y is the interlacing of m hypergeometric sequences where

1Lrnsn. This gives us an algorithm to compute the liouvillian solutions of linear difference equations. More or less, algorithms to consider linear differential equations are adpated or equivalent to algorithms to consider linear difference equations.

7. Conclusion Many algorithms appear in the computation of closed form solutions of linear differential equations: integer roots of a polynomial, linear differential systems, factorization,. .. Since general linear differential and difference equations usually lead to heavy calculus, any improvements in these algorithms improves the computation and help to propose efficients methods to users. Computer algebra is very helpful but even with powerful computers, one needs efficient algorithms implemented in computer algebra systems. So it is crucial to improve the naive approaches. The first results on complexity recently appeared. This should help to choose between one approach or another for a given linear differential equations. The computer algebra applied to linear differential and difference equations is a field of research full of applications and discoveries. Many works are in progress, and many new algorithms are in the pipe. References 1. S. Abramov, Rational solutions of linear differential and difference equations with polynomial coeficients, Journal of Computational Mathematics and Mathematical Physics vol. 29 (1989), p. 1611-1620

239 2. S. A. Abramov, Rational solutions of linear difference and q-difference equations with polynomial coeficients, Proceedings of ISSAC’95 (1995), p. 290-296 3. S. A. Abramov and M. Bronstein, O n solutions of linear functional systems, Proceedings of ISSAC’04 (2004) 4. S. A. Abramov and M. Bronstein and M. PetkovSek, O n polynomial solutions of linear operator equations, Proceedings of ISSAC’95 (1995), p. 290-296 5. S. A. Abramov and M. Bronstein and M. PetkovSek, O n polynomial solutions of linear operator equations, Proceedings of ISSAC’95 (1995), p. 290-296 6. M. A. Barkatou, A fast algorithm t o compute the rational solutions of systems of linear differential equations, rapprot de recherche LMC-Imag (1997) 7. M.A. Barkatou, On rational solutions of system of linear differential equations, Journal of Symbolic Computation vol. 28 (1999), p. 547-567 8. M. Barkatou and E. Pfl’ugel, A n algorithm computing the regular formal solutions of a system of linear differential equations, Journal of Symbolic Computation vo1.28 (1999), p. 569-587 9. A. Bostan and T. Cluzeau and B. Salvy, Fast algorithms f o r polynomial solutions of linear differential equations, Proceedings of ISSAC’05 (2005) 10. M. Bronstein, O n Solutions of Linear Ordinary Diflerence Equations in their Coeficient Field, Journal of Symbolic Computation vo1.29 (2000), p.841-877 11. M. Bronstein, O n Solutions of Linear Ordinary Differential Equations i n Their Coeficient Field, Journal of Symbolic Computation vol. 13 (1992), P. 413-440 12. M. Bronstein, Computer Algebra Algorithms for Linear Ordinary Differential and Diflerence Equations, Proceedings of the third European Congress of Mathematics, vol. 2, Progress in Mathematics 202, Birkhauser, 105-119 13. R. Burger and G. Labahn and M. van Hoeij, Closed f o r m solutions of linear odes having doubly periodic coeficients, Proceedings of ISSAC’O3 (2003) 14. M. Bronstein and A. F’redet, Solving linear ordinary differential equations over C ( x ,e s f(z)dz),Proceedings of ISSAC’99 (1999), p.173-179 15. M. Bronstein and S. Lafaille, Solutions of linear ordinary differential equations in term of special functional, Proceedings of ISSAC’O2 (2002) 16. T. Cluzeau and M. van Hoeij, Computing hypergeometric solutions of linear difference equations, to appear in Applicable Algebra in Engineering, Communication and Computing 17. T. Cluzeau and M. van Hoeij, A modular algorithm f o r computing the exponential solutions of a linear differential operator, Journal of Symbolic Computation vol. 38 (2004), p. 1043-1076 18. A. Fredet, Re‘solution SOUS forme finie d’e‘quations diffLrentielles line‘aires et extensions exponentielles, PhD thesis, Laboratoire Gage - Ecole polytechnique (2001) 19. A. n e d e t , Linear diflerential equations in exponential extensions, Journal of Symbolic Computation vo1.38 (2004), p.975-1002 20. P. A. Hendriks and M. F. Singer, Solving difference equations in finite terms, Journal of Symbolic Computation vo1.27(3) (1999), p. 239-260 21. A. Hilali and A. Wazner, Formes super-irre‘ductibles des systbmes difft?rentiels line‘aires, Numerische Mathematik vol. 50 (1987), p. 429-449

240 22. M. van Hoeij and J.-F. Ragot and F. Ulmer and J.-A. Weil, Liouvillian solutions of Linear Differential Equations of Order Three and Higher, Journal of Symbolic Computation vol.11 (1998), p. 1-17 23. E.L. Ince, Ordinary differential equations, Dover publications, INC (1956) 24. M. Karr, Summation in finite terms, jacm vol. 28 (1981), p. 305-350 25. J. J. Kovacic, A n Algorithm f o r Solving Second Order Linear Homogeneous Differential Equations, Journal of Symbolic Computation v01.2 (1986), p. 3-43 26. M. PetkovSek, Hypergeometric solutions of linear recurrences with polynomial coeficients, Journal of Symbolic Computation vol. 14 (1992), p. 243-264 27. M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren Der Mathematischen Wissenschaften, 328, Springer Verlag (2003) 28. R.H. Risch, The Problem of Integration in Finite Terms, Trans A.M.S. vol. 38 (1969), p. 1676-189 29. R.H. Risch, The Solution of the Problem of Integration in Finite Terms, Bulletin A.M.S. vol. 76 (1970), p. 605-608 30. M. Rosenlicht, On Liouville’s theory of Elementary Functions, Pacific Journal of Mathematics vo1.65 (1976), p. 485 - 492 31. M. F. Singer, Liouvillian solutions of n t h order homogeneous linear differential equations, American Journal of mathematics vol. 103 (1981), p. 661-681 32. M. F. Singer, Liouvillian Solutions of Linear Differential Equations with Liouvillian Coeficients, Journal of Symbolic Computation vol. 11 (1991), p. 251-273 33. M. .F. Singer and F. Ulmer, Galois Groups of Second and Third Order Linear Differential Equations, Journal of Symbolic Computation vo1.16 (1993), p. 936 34. M. .F. Singer and F. Ulmer, Liouvillian and Algebraic Solutions of Second and Third Order Linear Differential Equations, Journal of Symbolic computation V O ~16 . (1993), p. 37-73 35. M. F. Singer and F. Ulmer, Linear differential equations and products of linear forms, Journal of pure and applied algebra vol. 1178~118(1997), p. 549-563

241

STABILITY OF DIFFERENTIAL-DIFFERENCE SYSTEMS WITH QUADRATIC RIGHT-HAND SIDES D E W S KHYSAINOV Faculty of Cybernetics, Taras Shevchenko National University of Kyiv, VladirnirskayaStr., 64 Kyiv, 01 033,Ukraine IRYNA GRYTSAY Faculty of Cybernetics, VladirnirskayaStr., 64 Kyiv, 01033, Ukraine

In this paper the differential-difference systems with quadratic right-hand sides are considered. Conditions for stability, estimation of domain of stability and estimation of rate of convergence of solutions were given.

1. Introduction

A solution of a system with delay is investigated by Lyapunoff second method to have double modifications [3,4]. The first modification is the Lyapunoff-Krasovsky method. In this case, a trajectory segment is identified with a point in the Banach space. The main ideas of the functions method are transferred to functionals. The theorems about stability of this method have necessary and sufficiently conditions. The second modification is using finitedimensional Lyapunoff functions with an estimate of the derivative, when the trajectory of the system is located within level surface of the Lyapunoff function. This condition was named the Razymihin condition. 2. Preliminary

In this paper we will consider the zero solution of the differential-difference system with quadratic right-hand side x(t) =Ax(t-r)+XT(t)Bx(t-z), (1) where t 2 0, z is a positive constant, x(t) E R" , A is a square constant matrix. Matrices XT(t) and B are n x nz and nzx n rectangular matrices

242

x T ( t )= {x:(t),xT(t),...,xf(t)}, B~ = {B,,B,,...,B"}.

-

We suppose that square matrices Bi , i = 1,n are constant and symmetric, and

-

all elements of square matrices X'(t), i = 1,n are zeros except of i -th line which equals x(t) = (x,(t),x,(t),..., xn(t)) [1,2]. Stability investigation of the zero solution x(t) = 0 of the system (1) and estimation of domain of stability will be realized by finite-dimensional Lyapunoff functions with the Razymihin condition, inasmuch as the linear part of the system (1) has not any phase coordinates component in the present moment time, and the functionals method has some problems [ 5 ] . The Lyapunoff function is V(x, t) = e"xTHx with a positive-define matrix H is the solution of the Lyapunoff matrix equation [6,7] A'H+HA = -c. (2) The exponential factor e", y > O does not guarantee an existence infinitesimal higher limit for function V(x, t) [7,8]. It allows to obtain an upper bound of solutions decrease in domain of stability of the zero solution. If the matrix A is asymptotically stable, then the matrix equation (2) has an unique solution, the positive-define matrix H , for any positive-define matrix C. We will use the vector, and matrix norms

Where Amin(*), A,,,m(*) are smallest, and largest eigenvalues of corresponding symmetrical positive defined matrices. aV; is a level surface of the Lyapunoff function, and V; is a domain in the space R" x R , i.e.

av; = {(x, t): v(x, t) = a }, v; = {(x, t): v(x, t) < a } . 3. Main results 3.1. Linear case

Consider the linear system with delay k(t) = Ax(t - T) . Lemmal. Let a solution x(t) satisfies (x(t), t)E VL ,where t > -T . Then

(3 )

243

(4)

Pr o o f . The Lyapunoff function satisfies conditions [9] e*hm,.(H)Ix(t)I25 v ( w , t ) 5 e”h,,(H)Ix(t)12

(5)

That is, h,,.(H)IX(t)12 < a . So inequality (4) holds. Lemma2. Let a > 0 , y > 0 exist, and a solution x(t) of the system (3) in the moment t = T holds (x(T),T) E aV,l, in the moment T - 2z < t < T holds , . Then (x(t), t) E VY

Pr o o f . The system (3) is written an integral form t

x(t) = x(t - z) + IAx(s - z)ds . t-7

In the moment t = T ,inequality is got Ix(T) - x(T - z)l I JlAIIx(s - z)lds . T-T

From the Lemma2 condition and the inequality ( 5 ) , where T - T I s < T we get e + T ) ~ m , ( ~-tj2 )l~5 ( ~V(X(S

-t),~-T)<

v(~(T),T)
Therefore The obtained inequalities are substituted in the subintegral expression, we have

Lemma3. For a solution x(t) of the system (3) on interval 0 < t I z , the inequality is true

IWl (1 + ~A~.]lx(O)ll,. Proof . The system (3) is written an integral form

(8)

244

The asymptotic stability conditions of the system with pure delay (3) is obtained, by using this results. Theoreml. Let a matrix A is asymptotically stable. Then for T < T, ,where

the system with pure delay (3) is asymptotically stable too. And its solutions satisfy the exponential valuation of decrease

where 0 < y < y' , y' is a solution of the equation y'(hmin(C)-y*hmX(H))= 4 & @ $ l A ~ ~ A ~ e ~ ' " -1). ( e ~ ' " (11)

P r o o f . Let x(t) is any solution of the system (3). Then, it follows from Lemma3 that IxWl 5 (1 + IAI.C)IX(O)IIT in the case 0 5 t I T . On this interval, x(t) satisfies (x(t), t) E VJ ,where y > 0 is some value, and 9

Let us show that it is true on t > T . Suppose that it is not true, a moment T > T exists, where (x(T),T)E aVJ . Let us obtain the total derivative for the system (3) d -V(x(t))= e"yx'(t)Hx(t)+ dt +e"{xT(t)(ATH+HA)X(t)+2xT(t)HA[x(t-T) -x(t)]}. If the matrix A is asymptotically stable, then it executes for any positivedefine matrix C and H d -V(x(t))l e" {yh,,(H) - h,, (C))lx(t)l' + 2e"IHAIIx(t)llx(t) - x(t - z)l . dt As it follows from the Theoreml condition an inequality (7), in the moment t = T ,the inequality holds

If the inequality

245

is true, then the total derivative of the Lyapunoff fitnction will be negative. This implies, that an velocity vector of a motion x(t) is direct to inside, in the moment t = T , and (x(t), t) E VJ for any t > 0 . It follows from the inequalities (4),(8), for t > z

that is the inequality (10) is true. Let us find conditions when inequality (12) are executed. If y -++O then the inequality (1 1) has the form

kmin (c>- 2IHA(IHl&Er

>0

and if z < z, ,then r,

=

‘mi,(‘)

2 ( H A ( ( A I r n. That is, the maximum allowed delay z, has the form (9). Let z < r,, . Then “some stock remains” for an exponential solution decrease, the solution of the equation (1 1) defines it. Remarkl. The condition (9) is raw, but it is easy calculate for systems. For the scalar equation x(t) = -ax(t - z) , a > 0 the stability condition is r < n/2a. From the inequality (1 1) we get T < l/a for the Lyapunoff function V(x, t) = e’x’ . 3.2. Nonlinear case

Let us consider the system with delay of the general form (1). Lemmad Let a > 0 , y > 0 exist, and a solution x(t) of the system (1) in the moment t = T holds (x(T), T) E aV,l, in the moment T - 2r I t < T holds (x(t), t) E VJ. Then

1 %

-t-ez

cp(H)lBl(er - l]x(T)lz.

Y Proof. The system (1) is written an integral form

246 I

x(t) = x(t - T) + ~[Ax(s - T) + X'(s)Bx(s -

.

I-T

In the moment t = T ,the inequality is got T

Ix(T) - x(T - T)l I ICAllx(s - T)l +IX(S>IIBIIX(S- .)lkls . 7-T

From the Lemma4 condition and inequalities (7), where T - T I s I T we get Ix(s - r)1 < e f l ( T - " " ~ l x ( T ) ,I Ix(s)l < eiy(T-s),&@lx(T)I . The obtained inequalities are substituted in the subintegral expression, we have

or 2 1, Ix(T) - x(T - r)( < -e2 ,&@]A\( e,'

1 1, + -ez cp(H)IBl(e,

- l]x(T)I'. Y Lemma5. For a solution x(t) of the system (1) on the interval 0 5 t I T ,the inequality is true

Y

- l)lx(T)l

~~Il~(O)lll,*

Ix(t>JI (1+ IAl+(O)ll, P r o o f . The system (1) is written an integral form

(14)

I

x(t) = x(0) + ~[Ax(s - T) + XT(s)Bx(s - T

.

) ~ S

0

Then Ix(t)l< )X(O)I+ jlAIIX(S - 71 + IX(S)IBIIX(S - +s

I

0

Theoremt Let a matrix A is asymptotically stable. Then for T < T, ,where T, defines in (9), the zero solution of the system with delay (1) will be asymptotically stable. And the domain of stability contains a sphere U, with a radius

(H) 21BI&@e%, and for a solutions x(t) from this sphere an upper estimate is true

where

P r o o f . Let us obtain the total derivative for the system (1) d -V(x(t)) = e"'(t)Hx(t)+ dt er fiAx(t - r) + X'(t)Bx(t -.r)bx(t)

+ x'(t)H[Ax(t

- r)+ X'(t)Bx(t - r)l,

or d

-V(x(t))= e*x'(t)(yH + A T H +HA)x(t)+ 2erxT(t)HA[x(t-.r)-x(t)]+ dt + 2evxT(t)HXT(t)Bx(t - T) . Using the matrix norm and the Lyapunoff function, then we get -v(x(t))< d dt

yv -

U V + A e + m ( e+ - 1)w.V L_W) Y

+ 21Bl~cpoe+Vf 7

or 3

V<-aV+bVr, where

The solution of the Bernuli equation is

From this equation the estimation of the domain of stability is

And the estimation of solutions of the system (1) is

The upper estimate (16) is got by (15), (17). The domain of asymptotic stability is R . That is, if an initial value of a function x(t) belong to sphere

248

U R , then the upper estimate (16) is true for a solution corresponds the initial value of the function x(t) . References 1. V.F.Davydov Majorizing Estimates of Solutions of Quadratic Differential Systems with Delay// Ukr. Math. Journ., v.47, N24, P.542-545. 2. D.Ya.Khysainov, V.F.Davydov System of Quadratic Form Stability/iRAS Ukraine, N27, 1994. -P.11-13. 3. L.G.Korenevskiy Stability of Dynamikal Systems with Chance Disturbance of Parameters. Algebraic Criterions - Kyiv, Science idea, 1989. - 208 pp. 4. N.N.Krasovskiy Some Problems of Stability Theory of Motion - M. Fizmetgiz, 1959. -212 pp. 5 . E.A.Barbashin Lyapunoff Functions. - M., Nayka, 1970. - 240 pp. 6. D.Ya.Khusainov, A.V.Shatirko Lyapunoff Functions Method in Stability of Differential-fbnctional Systems Investigation. -K., 1997. - 236 pp. 7. A.M.Lyapuniff General Problem about Dynamic Stability. - M.-L. 1950 472 pp. 8. B.P.Demidovich Lecture about Mathematical Stability Theory. - M., Nauka, 1967. - 472 pp. 9. L.E.Elsgolc, S.B.Norkin Introduction ib Differential Equations with Delay. - M., Nayka, 1971. - 296 pp. 10. F.R.Gantmaher Theor of Matrices. - M., Nayka, 1966. - P.576.

249

CHARACTERISTIC ALGEBRAS OF DISCRETE EQUATIONS

ISMAGIL HABIBULLIN Ufa Mathematical Institute Department of Mathematical Physics Chernyshevskaja Street 112 450077 Ufa,Russia E-mail: habibullin-iC8mail.rb.m Characteristic Lie algebra of a discrete equation is studied. An algorithm of finding the algebra for a given equation is discussed.

1. Introduction

Fully discrete equations i.e. equations for which all independent variables are integer while the dependent variables are continuous essentially differ from their differential and differential-difference analogs. For instance, even the standard algebraic problem like computing the Lie symmetries or higher symmetries are reduced to a rather hard problem of solving non-local functional equations. But the Lie algebraic approach is still the most powerful tool to investigate nonlinear models. It is well known that the characteristic Lie algebra introduced by A.B.Shabat in 1980, plays a crucial role in studying the hyperbolic type partial differential equations. For example, if the characteristic algebra of the equation is of finite dimension, then the equation is reduced to a pair of ordinary differential equations. Numerous examples approve the widely known hypothesis that if the algebra is of finite growth then the equation is integrated by the inverse scattering method. More details and references can be found in Recently it has been observed by A.V.Zhiber that the characteristic algebra provides an effective tool for classifying the nonlinear hyperbolic equations. Remind that years ago the characteristic Lie algebra has been used t o classify integrable systems of a special type ’. However, the characteristic algebras have not yet been used to study the discrete equations. In spite the discrete equations have become very

’.

250

popular the last decade (see, for instance, the survey 3 ) . In this paper we show that the characteristic algebra can be defined for any discrete equation of the hyperbolic type (or for a system of such equations) and it inherits most of the important properties of its continuous counterpart. However it has essentially more complicated structure. I am pleasant to acknowledge that this work was stimulated by the paper by R.S.Ward, where the discrete field theory is studied and the question is posed which kind of algebraic structures are associated with the finite field discrete 3D Toda chains. 2. Invariants and vector fields

Consider a discrete nonlinear equation of the form

t,, = f ( t , t,,

(1) where t = t(u,w) is an unknown function depending on the integers u, w and f is a smooth function of all three arguments. The following notations are used to shorten formulae: t , = t(u 1,w), t , = t(u,w l ) , and t,, = t(u 1 , w 1). So without shortening the equation (1) looks as follows t(u 1,w 1) = f ( t ( u ,w),t(u 1,w),t ( u ,w 1)).Actually, equation (1) is a discrete analog of a partial differential equation. Particularly, the class of equations (1) contains difference schemes for the hyperbolic type PDEs on a quadrilateral grid. The notations above are commonly accepted, but not very convenient to indicate the iterated shifts. Therefore we introduce the shift operators D and D acting as follows D f ( u ,w) = f(u 1,w) and Df(u, w) = f(u,w 1). For the iterated shifts we introduce the notations fj = D j ( f ) fj = Dj(f), so that t , = t l , t, = 51 and so on. Equation (1) is supposed to be hyperbolic. It means that it can be rewritten in any of the forms: t, = g(t,t,,t,,), t, = r(t,tU,,,tu),and t = ~ ( t ,t,,, , t,) with some smooth functions g , T , and s. A function F = F(w,t,t l , E l , ...), depending on w and a finite number of the dynamical variables is called w-invariant, if it is a stationary "point" of the shift with respect to w so that (see, also, ')

+ +

+ +

tv),

+

+

+

+

+

+

D F ( w t, , t i , E l , ...) = F ( w ,t , t l ,51, ...),

+

(2)

and really the function F is to solve the equation F(w 1,t,, f,f1, i ? ~ ,...) = F(w,t,tl,fl,...). Notice that in contrast to the continuous case the equation is completely nonlocal. Examining carefully the last equation one can find that:

251

Lemma 1 The w-invariant doesn’t depend on the variables in the set {fj>j”=l.

If any w-invariant is found, then each solution of equation (1) can be represented as a solution of the following ordinary discrete equation F ( t ,t,, ...,t j ) = c(u),where C ( U ) is a function on u. Due to Lemma 1, equation (2) can be rewritten as

F ( v + 1, t,,

f,f i ,...) = F ( v ,t ,t i , t z , ...).

(3)

The left hand side of the equation contains t , while the right hand side doesn’t. Hence the total derivative of D F with respect to t, vanishes. In other words the operator X1 = D-l&D annihilates the w-invariant F : X1F = 0. In a similar way one can check that any operator of the form X 3. = D - j d t ,D j , where j 2 1, satisfies the equation X j F = 0. Really, the right hand side of the equation D j F ( w , t , t l , f I ,...) = F ( w , t , t l , f l ,...) (which immediately follows from (2)) doesn’t depend on t , and it implies the equation X j F = 0. As a result, one gets an infinite set of equations for a function F . For each j the operator X j is a vector field of the form

“ d X=C..-. dtj j=O

(4)

Notice that the coefficients of the vector fields X j for j = 1 , 2 , 3 , ... depend on the variable t-l,tLz, ...t - j , but the function F does not. So actually, the invariant should solve additional set of the equations X - j F = 0, where X - j = and j 2 0. Consider now the Lie algebra L, of the vector fields generated by the operators { X j } ; z r , with the usual commutator of the vector fields [ X i , X j ] = X i X j - X j X i . We refer to this algebra as the characteristic algebra of equation (1). Theorem 1. Equation (1) admits a nontrivial w-invariant if and only if the following two conditions hold: 1) linear envelope of the vector fields { X j } g l is of finite dimension; 2) suppose the set of the vector fields { X i l , X i 2 ,..., X i p } , where il < iz < ... < i,, is a basis of the linear envelope from the previous part. Then the Lie algebra L generated by the operators {Xj}iz?ip+l is of finite dimension. Proof. Suppose that the equation (2) admits a non-constant invariant. Then the following system of equations

&

X j F ( v , t , t l , t z ,...t ~= )0 ,

-W

< j < 00

(5)

252

has a non-constant solution. It is possible only if the linear envelope of the vector-fields { X j } Z , is of finite dimension. Evidently, due to the classical Jacoby theorem the second condition of Theorem 1 is also satisfied. Thus, one part of the theorem is proved. It is worth mentioning an appropriate property of the vector fields above. If for a fixed j 2 2 the operator X j is linearly expressed through the operators X I , X2, ... Xj-1, then any operator X I , is a linear combination of these operators. Really, we are given the expression X j = alX1 ... aj-1Xj-1. Note that Xj+l = D - l X j D for j 2 1, hence Xj+l = D-1 (a2)Xz D-'(aj-l)xj. Suppose now that conditions 1) and 2) are satisfied, so the dimension of the algebra L is finite and equals say N , show that in this case equation (1) admits a v-invariant. Evidently there exists a function G ( t , t l ,...t ~ ) , which is not a constant and that X G = 0 for any X in L. Such a function is not unique, but any other solution is expressed as h ( G ) . Due to the construction the map X + D - l X D acts from L into L , hence G1 = DG is also a solution of the same system X G = 0 and therefore G I = h ( G ) . In other words one gets a discrete first order equation: DG = h ( G ) , write its general solution in the following form: C = F ( v , G ) where C doesn't depend on v. Evidently the function F found is just the v-invariant needed.

+ +

+ + 1..

3. Computation of the characteristic algebra

In this section the explicit forms of the operators { X j } are given. We show that the operators are vector fields and give a convenient way to compute the coefficients of the expansion (4). Start with the operator X I . Directly by definition one gets X 1 F ( t ,t l , ...) = D-l&F(t,, f , f 1 , ...). Computing the derivative by the chain rule one obtains -

X i F ( t , t l , ...) = D-'(-

a + -af a afl a + ...) F ( t , , f , f l , . . . ) , at, at, a f + -at, afl

and finally

a

x1= -+D

at

af a (at,>

(2)&+...+

z+D-'

D-1

wj a

(at,) at,+l+"''

(6) So the operator X1 = CEO xi& where xj = D-l af at, j-1 for j > 0 and xo = 1. The coefficients can be computed by the following very convenient formula

(

2j+l

= 21D(21)D2(x1) ...D j ( Z l ) ,

)

(7)

253

or

D

~ j + l=

a

zjDj(z1).

Really, D x ~ +=~ at, q g )... = D(Dj(Z1)...D(Z1)zl). To find X2 use the following formula

D(g)g

-

a

a

X2F = D-lXIDF(t,tl, ...) = D-l(-++xl-+zz-+ at at,

= Bfjafj-l...E afj-1 afj-2 at, =

a at2

...) F ( t , , f , f l , ...).

After opening the parentheses and some transformation the right hand side of the last formula gets the form

XZF(t,tl, ...) = = D-l

(

d + X l ( f l ) - - - + Xl(fZ)-- a + ...) F ( t , , f , f l , ...). af df afz

a

Xl(f)--

1

So the operator can be written as

Prolonging this way one gets

The following statement turns out to be very useful for studying the characteristic algebra. Lemma 2. Suppose that the vector field

" j=O

a atj

satisfies the equation

then X 0. Proof. It follows from (11)that 00

j=O C D ( Z j ( f l , f - l , t,tl,tZ, " ' 7

a

tkj))at,+l

00

a

- ~ X j ( t ; , t - l , t , t l , t 2..., , trc,)-. atj - j=O

Comparison of the coefficients before the operators equation yields: xo = 0, XI = 0 ... and so on.

& in both sides of this

254

4. Characteristic algebra for the discrete Liouville equation

The well known Liouville equation the form 1

& = e" admits a discrete analog of

t,, = -t( t u - l)(tw - 11,

(12)

which can evidently be rewritten as

t,+

1 =t-v + l . t - 1 t,

(13)

Specify the coefficients of the expansions (6)-(8) representing the vector fields XI and X2 for the discrete Liouville equation (12). Find the coefficient x1 = D-' remind that f (t,t,, t v ) = +(t,- l)(t, - 1) and

(g),

dt-w,

t l

tu)

=X

1 L

-

V

+

1 1

Similarly

n",=', 3

It can be proved easily by induction that xj = for j 2 1, here to := t. Remind also that xo = 1. So that the vector field is a tlt2t3 a +..., tlt2 (t - l)(tl - 1) at2 (t - l)(tl - l)(t2 - l)'at, (14) It is a more difficult problem to find the operator Xs, it can be proved that

.-+

XI =

(t - 1)t (Xl - x-11, (t-, - l)t-,

where the operator X-1 is defined as follows X-1 = D&D-l. For the coefficients x-,j of the expansion X-1 = Cj"=,x-,j one can deduce the formula x - j =

nii=', w .so that the operator is represented as

x-1= a + -.tl - 1 a + (tl - l)(t2 - 1) . - a +

(16) at1 ttl at2 (tl - l)(t2 - l)(t3 - 1) . - + .... ttl t2 at, Theorem 2. The dimension of the characteristic Lie algebra of the Liouville equation (12) equals four.

at

t

+

a

255

Proof. It is more easy to deal with the operators Y+ = (t - 1)Xl and Y- = tX-1 rather than the operators X I and X-l. In order to prove the theorem it is enough to check the formula

[Y+,Y-]- Y+ +Y- = 0.

(17)

Denote through X the left hand side of the equation (17) and compute D ( X ) to apply the Lemma 2. It is shown straightforwardly that

tl - 1 D(Y+) = T ( Y +

-(t-

a a tl 1 ) ~ )and D(Y-) = tl - 1 (Y- - t-). at

Compute now the shifted operator D([Y+,Y-J) = [D(Y+),D(Y-)] which can be represented as follows

A1

= [Y+,Y-] -

A3 =

1 1 Y- - -Y+, tl - 1 tl

t,+t-ia t1-1

at’

Summarizing all computations above one gets the result 1

1

D[Y+,Y-]= [Y+,Y-]- -Y- - -Y+ tl - 1 tS

+

-1 t d + ( 1 + ttl + tl - 1 at -1-1

+

which implies D([Y+,Y-] - Y+ Y-) = [Y+,Y-] - Y+ Y-. Now apply Lemma 2 to the function X = [Y+, Y- J - Y+I- Y- to get X z 0. The next step is to prove that the operators X Z and X3 are proportional. It can be checked that X 3 = ! B { X 2 . The characteristic algebra is generated by the operators XI, X 2 , X O and X-1. It is of dimension four.

256

5. How to find the invariants?

In this section two examples of equations with their invariants are shown. Start with a simple one. Example 1. Consider a linear equation of the form tu,

+

= t,

+t, - t + 1

(18)

+

so that f ( t ,t,, t,) = t , t, - t 1 and g(t-,, t , t U )= t , - t is easy t o see that the algebra is of one dimension

d

x1

a d + - + - + ..., at at, at2

=-

x2

+ t-,

- 1.

It

0.

The first integral of the smallest order for the equation X1 F = 0 can be taken as GO= tl - t. Evidently it solves the equation DGo = GO 1. Now, the invariant is to be taken as I = tl - t - v,because D I = t,, - t , - v - 1 = t, - t - v = I . Any other v-invariant of the equation can be represented as F = F ( I , D I , D21,...D k I ) . Example 2. Turn back to the discrete Liouville equation discussed above. Find the intersection of the kernels of the operators X I and X-1 hence they constitute the basis of the characteristic algebra of the Liouville equation (12). To this end first solve the equation X1F = 0 which is reduced to the following infinite system of ordinary differential equations:

+

A,

So the invariants of the vector field X I are 10 = I1 = D ( l o ) , I2 = D ( I l ) , ... . Change the variables in the vector fields by setand ting f = t , f 1 = Io, & = I I , ... . Then one gets X I = t ; + l f 2 + 1 x-l = ----- - .... Now solve the equation at t(t - 1) at; t;t(t - 1 ) f2 -ddi1 _ f i (f22 1 +11 ) t o find the common solution FO = 1)(t l 1) of both equations X1F = 0 and X-1F = 0. The invariant is I ( t , t l , t 2 ) = Fo(t,t l , t 2 ) = (& l ) ( y1) (see, also 5 ) . Evidently the basis of common invariants can be chosen as follows Fo,D(Fo),O2(Fo),.... It can be easily verified by direct computation that Fo solves the equation DFo = Fo. Corollary. In the case of the Liouville equation each solution of the system of the equations X1F = 0, X-1F = 0 gives the v-invariant.

a

a

&

a at",

+

(A+

+

+

+

257

Proof. Each solution of this system can be represented as = F ( F o , D ( F o ) , D 2 ( F )...), , hence D F ( F o , D ( F o ) , D 2 ( F )...) , = F(D(Fo),D ( D ( F o ) )D , ( D 2 ( F ) )...I , = F(F0,D(Fo),D 2 ( F ) ,...).

F

6. Conclusion

The notion of the characteristic algebra for discrete equations is introduced (see, also7). It is proved that the equation is Darboux integrable if and only if its characteristic algebras in both directions are of finite dimension. The notion can evidently be generalized to the systems of discrete hyperbolic equations. It would be useful to compute the algebras for the periodically closed discrete Toda equation, or for the finite field discrete Toda equations found in * and ’, corresponding in the continuum limit to the simple Lie algebras of the classical series A and C.

Acknowledgments. The author thanks A.V.Zhiber for fruitful discussions. The work has been supported by the grants RFBR# 05-01-00775, RFBR#04-01-00190. References 1. A.N.Leznov, M.V.Savel’ev, Group methods of integration of nonlinear dynamical systems. M.:Nauka, 1985 (in Russian). 2. A.B.Shabat, R.I.Yamilov, Exponential systems of type I and the Cartan matrices, Preprint Ufa, 1981. 3. A.V.Zabrodin, Hirota’s equation and the Bethe anzats, Theoret. and Math. Physics, (in Russian) 116 (1998) 54-100. 4. R.S.Ward, Discrete Toda field equations, Phys. Letts A 199(1995) 45-48. 5. V.E.Adler and S.Ya.Startsev, Theor. Math. Fizika, 121, N2, (1999) P.271284. 6. R.Hirota, J.Phys. SOC.Jpn. 46 (1979) P.312-319. 7. I T Habibullin, Characteristic algebras of the discrete hyperbolic equations, arXiv:nlin.S1/0506027, 2005. 8. I.T.Habibullin, Discrete Toda field equations, arXiv:nlin.S1/0503055, 2005.

258

ON THE ASYMPTOTIC BEHAVIOUR OF SOLUTIONS OF NEURONIC DIFFERENCE EQUATIONS

YOSHIHIRO HAMAYA Department of Information Science, Okayama University of Science 1-1 Ridai-chyo, Okayama 700-0005, Japan E-mail: [email protected]. ac.jp Sufficient conditions for the global attractivity of the equilibrium point of the delay difference equations, which appears as models of the actions of neurons with dynamical threshold effects, are obtained by applying the method of Liapunov functionals and others.

AMS:

39A12.

KEY WORDS: tionals

1

neuron models, global attractivity, Liapunov func-

Introduction In this paper, we first consider the following difference equation z,+1

n = 0,1,. ..

= a tanh (z, - bz,-k),

(1) where Ic is a nonnegative integer and a, b E R. Equation (1) is a discrete analogue of the delay differential equation

dz(t) = -z(t) + a tanh [ z ( t )

-

dt

bz(t - r ) ] ,

t20

which has been proposed by Gopalsamy and Leung [3] as a model of the dynamics of a single isolated neuron (see also [1,2]). Here z(t) denotes the activation level of a neuron at time t and a is real constant which are related to the range of the continuous vraiable z(.), while b denotes a measure of the inhibitory influence of the past history, a > 0, b 2 0 and r is the time delay, r E [ O , o o ) (for details see [1,3]). By the condition of a , b, we see that the existence of solution {z,} is guaranteed for equation (1)whenever the initial conditions are bounded. We let yn E 2, - bz,-k,

nE

[-Ic,o~)

259

and we obtain from Eq.(l) that Yn+l = atanhy, - abtanhy,-k,

n

> 0.

(2)

It is easy t o see that equation (2) has the equilibrium point y*, which given by y* = a(1- b) tanh y*.

Global Attractivity of Eq.(l) and Others

2

We study the global attractivity of the equilibrium point x* of equation (1). First, we need the concept of the semicycle of a sequence (cf. [ S ] ) . Definition. A positive semicycle of a solution {x,} of equation (1) consists of a string of terms {xl,xl+1,.-. ,xm}, all greater than x*, with 1 2 -k and m 5 0;) and such that

either

1 = -k

1

OT

> -k

and

<

and xm+l

21-1

5 x*

and

either m = 0;)

OT

m

0;)

5 x*.

A negative semicycle of a solution {x,} of equation (I) consists of a string of terms { X L , X ~ + ~ , .,x,}, -. all less than x*,with 12 -k and m 5 0;) and such that either 1 = -k

1 > -k

and

21-1

m < 0;) and

Xm+1

OT

2 x*

and

either m = 0;)

OT

2 x*.

The main results in this paper are the following: Theorem. 1 Suppose that a ( l + b)

5

1 and a ( l - b)

5 1,

Then, the every solution {x,} of equation (1) satisfies lim x, = 0.

n+oo

(3)

260

Remark. This paper is motivated by the research in Goplasamy and Leung [3]. They have obtained a global stability result for the equilibrium z* of equation (E) when a > 0 b 2 0, and we also regard the following Theorem A as an answer in the affirmative for their research using a Liapunov functional. For the equation (E), [l]has established the global attractivity of the equilibrium point z* under the condition (3) when a, b E R. Otherwise, Liapunov-method in [3] can not apply directly the delay difference equation ( 1 ) when a , b E R. Theorem A [cf.3]. For a > 0, b

Then every solution

2 0, assume that a ( l - b ) < 1, a ( l + b ) < 1. (2,) of Eq.(l) goes to 0 as n + m.

Our results are the global attractive conditions for difference equation (1) of a,b E R and other models, which are independent of delay terms. Recently, [7] has shown the stability criteria for functional differential equations with delay. Moreover, as oscillation results for equation ( l ) ,see also

[4,5$1. We next consider the following model with several delays.

where b* = Czl bi,a > 0,bi argument for Eq.(l), we have

20

for an integer m

> 0.

bi tanh yn+,

n

By the same

m

yn+l = a tanh yn - a

> 0.

(5)

i=l

It is also easy to see that equation (5) has the equilibrium point y*, which given by y* = a ( l - b*) tanhy*.

We also have the following result to Eq.(4).

Theorem. 2 Suppose that a ( l + b*) < 1 and

a ( l - b*) < 1,

Then, the every solution { z n } of equation (4) satisfies lim zn = 0.

n+w

261

Now, we will start to prove the theorems. The idea of these proofs are based on [1,3,5]. Proof of Theorem 1. Assume that x, is a solution of Eq.(l). It follows from 0 < < 1 that xn is bounded. Therefore there exist L , M E R such that

L = lim inf x, 5 M = lim sup xn. n+w

n+w

Thus for any

E

> 0 there exists no 2 0 such that L-E<xn<M+E

n2no.

(7)

The case where sequence { x n } is eventually nonnegative or eventually nonpositive is easy to prove and will be omitted. Therefore, it remains to prove the case where { z n }is not eventually sequence. If it is necessary, we can choose two subsequences (semicycle-sequences) {nl}and { f i l } such that { n l } ,{ f i l } + 00 as 1 + 00, xnt+,

2 xn,

lim xnI = M

(8)

1403

and

5 ZA,

2Al+1

lim xAI = L,

(9)

1400

Assume, for the sake of contradiction that L < M. We consider first the case where ab 2 0. Using (7), (8) and Eq.(l), we obtain Zn,+,

5 atanh(xn, - b(L - E ) )

1 2 10

+ Ic,

1 2 10

+ Ic.

and also (7),(9) and Eq.(l) imply xAl+,

Let I

+ 00

2 a tanh(xA, - b(M + E ) )

in the previous inequalityes. Then, we have respectively that M

5 atanh(M - b(L - E ) ) , and

L 2 atanh(L - b(M Since 6

+

6)).

> 0 is arbitrary small, we have M

5 atanh(M - bL), and

L 2 atanh(L - b M ) .

262

Assume that M

5 0. Then L < 0 which, by ( l l ) , yields if

tanh(L - b M ) = From 0

<

a>O

< 1, we conclude that L-bM,

tanh(L - b M ) =

if if

a>O

a<0.

Thus (11) leads to L

> a(L - b M ) ,

then L ( l - a)

> -abM 2 0.

But 1 - a 2 0, then the above inequality implies that L > 0 which is a contradiction. If M > 0, using similar arguments as above, we find from (10) and 0 < < 1 that

M

< a(M - bL).

Rearranging, we have M ( l - a) < -abL, hence L

< 0 so that (12) holds which in turn implies that -abM

< L ( l - a).

From the previous two inequalities we get M ( l - a(l

+ b)) < L(l-

a ( l + b)),

or, equivalently ( M - L)(1 - a ( l + b ) ) < 0 which is impossible in view of (3) and the fact that M > L. This contradiction implies that L = M . On the other hand when ab < 0, the similar reasoning implies that

M 5 atanh(M(1 - b ) ) ,

(13)

and L 2 atanh(L(1 - b)). Assume that M

> 0. Then

(14)

(13) yields

tanh(M(1 - b ) ) =

if

a
263

From 0

<

< 1, we conclude that >M(l-b),

tanh(M(1-b)) =

<M(1-b), It follows from (13) and (*) that M

if a < O if a > O

< ~ ( 1 -b)M 5 M

which is impossible. Therefore, M 5 0 and hence L < 0. Using the negativity of L it is easy to see that (14) leads to the inequality

L > a(1- b)L 2 L which is impossible, too. Then L = M when ab < 0 or ab 2 0 provided that (3) holds. Since the trivial solution is the unique equilibrium, due to the second inequality of (3), we get L = M = 0 which is our desired conclusion. The proof is complete. Proof of Theorem 2. defined by

We consider a Liapunov-like functional V,

i=l

kn-k;

Since (15), it is easy to see that for all integer n,

v, > IYnl. Calculating the differences of V, along the solutions of Eq.(4), AV, = Vn+l - V,

5

-IYnl m

+ al tanhynl

+ x ( a b i l tanh9n-k; I + abilynl - abiIYn-k; I) i=l

5 -(1- a ( l + b*))ly,I,

n

> 0.

(16)

It follows from (16) that

c

n-1

v, + (1- a ( l + b*)) ly1l 5 vo, n > 0. By hypothesis in (6), a(1+ b*) < 1, and hence it follows from (16) that the 1=0

sequence {V,} is nonincreasing for all solutions {y,} is convergent. Letting n -+ 00 into (16) we obtain lim AV, = 0 5 -(1 - a(1

,--to3

of Eq.(5) and so {V,}

+ b')) ,lim lynl +a

264

which implies, lim lynl = y* = 0 ,

n--tm

and then, this completes the proof of Theorem 2. Moreover, we consider the another type of neural model. %+I

= ax,

+ a t a n h (2, -

m

bizn-+),

n = 0 ,1 ,. . .

(17)

i=l

where a E R : 0 5 a < 1,b, = Czl bi, a > 0 , bi 2 0 for an integer m By the same argument for Eq.(4), we have

> 0.

m

yn+i = ay,

+ a tanh yn - a C bi tanh yn+,

n > 0.

(18)

i=l

It is easy t o see that equation (18) has the equilibrium point y*, which given by 1 y* = -a(1 - b , ) tanhy*. l-a We obtain the following result to Eq.(17).

Theorem. 3 Suppose that

a

+ a(1 + b*) < 1

Then, the every solution {z,}

and

a

+ a(1 - b*) < 1,

of equation (17) satisfies lim

,--too

2,

= 0.

Proof. We consider a Liapunov-like functional V, defined by m

n-I

i=l

l=n--ki

Since (20), it is easy to see that for all integer n,

v, > IYnl. Calculating the differences of V, along the solutions of Eq.(17),

265

It follows from (21) that

Vn

+ (1-

n-1 ~r

- a ( l + b*))

C 191I 5 ~ 0 ,

n > 0.

l=O

By hypothesis in (19), a+a(l+b,) < 1, and hence it follows from (21) that the sequence {Vn} is nonincreasing for all solutions {yn} of Eq.(18) and so {Vn}is convergent. Letting n. -+ 00 into (21) we obtain lim AV, = 0 5 -(I- a - a(1

n+m

+ b,)) n+o3 lim lynl

which implies,

and then, this completes the proof of Theorem 3.

Acknowledgements The author wishes to thank Professor T. Kohno for his help to figure in the conference’s talk by using Maple 8.

References 1. H. A. El-Morshedy and K. Gopalsamy, O n the oscillation and asymptotic behaviour of solutions of a neuronic equation, Funkcialaj Ekvacioj, Vol. 44 (2001). 83-98. 2. E. R. Caianiello and A. De Luca, Decision equation for binary systems: A p plication t o neuronal behavior, Kybernetik, Vol. 3, (1966). 33-44. 3. K. Gopalsamy and I. K. C. Leung, Convergence under dynamical thresholds with delays, IEEE Transactions on Neural Networks. vo1.8 No.2 (1997). 341348. 4. Y. Hamaya, Global attractivity in Wazewska-Lasota difference model, Differential Equations and Dynamical Systems, vo1.5 (1997). 187-195. 5. Y. Hamaya and Shinsuke Sato, Global attractivity of some neural net models, Int. J. Pure and Appl. Math., in press. 6. V. L. Kocic and G. Ladas, ”Global Behavior of Nonlinear Difference Equations of Higher Order with Applications”, Kluwer Academic Publishers,l993. 7. M. Pituk, More o n scalar functional differential equations generating a monotone semifEow, Acta Sci. Math. (Szeged), Vol.69 (2003). 633-650.

266

PERTURBATION OF NONNEGATIVE TIME SCALE QUADRATIC FUNCTIONALS

R. HILSCHER* AND v. R~JZICKOVA~ Department of Mathematical Analysis, Faculty of Science, Masaryk University, Jandtkovo ndm. 2a, CZ-60200 Bmo, Czech Republic E-mail: [email protected], xruzzckoQmath. muni.cz

In this paper we consider a bounded time scale T = [a,b ] , a quadratic functional F ( z ,u ) defined over such time scale, and its perturbation G ( q u ) = F ( z ,u ) CY l l ~ ( a ) 1 1where ~, the endpoints of F are zero, while the initial endpoint %(a)of G can vary and z(b) is zero. It is known that there is no restriction on %(a) in 0 when studying the positivity of these functionals. We prove that, when studying the nonnegativity, the initial state .(a) in G must be restricted to a certain subspace, which is the kernel of a specific conjoined basis of the associated time scale symplectic system. This result generalizes a known discretetime special case, but it is new for the corresponding continuous-time case. We provide several examples which illustrate the theory.

+

1. Introduction and Motivation

In this paper we consider perturbations of certain continuous-time, discretetime, and time scale quadratic functionals. Consider the continuous-time quadratic functional

+

over admissible pairs (z,u), i.e. z'(t) = A ( t )z ( t ) B ( t )u ( t ) on [a,b ] , where A , B , C are n x n piecewise continuous real-valued matrix functions such that B(t) and C ( t ) are symmetric on [u,b],and where [a,$ is continuous-time (connected) interval. Such functionals arise as second variation in calculus of variations and control problems, see for example [l]. The 'Work is supported by grant 1K04001 of the Ministry of Education, Youth, and Sports of the Czech Republic. tWork supported by grant 201/04/0580 of the Czech Grant Agency.

267

functional Fc is nonnegative if F c ( x , u ) 2 0 for all admissible ( x , u ) with .(a) = 0 = x(b), and it is positive definite if F c ( x ,u)> 0 for all admissible ( x , ‘ ~with . ) .(a) = 0 = x(b) and x +?4 0. Sometimes we neglect the word “admissible” and simply say that FCis positive definite or nonnegative over certain endpoint constraints. The following result for positive definite quadratic functional Fc is widely known and can be derived e.g. from [2, Lemma 3.171, see also [3, Theorem 13.21, [4, Theorem 12.2.71, or [5, Lemma 21.

Result 1.1. The quadratic functional Fc is positive definite over zero endpoints .(a) = 0 = x(b), x $ 0, i f and only if there exists a > 0 such that the perturbed quadratic functional

Gc(x,u):= Fc(x,u )

+ a! IIx(a)I12

is positive definite over free initial state .(a)

and x(b) = 0, x $ 0.

Similar statement holds also in the discrete case, see e.g. a special case in the proof of [6, Theorem 41. Namely, let A k , B k , c k , and Dk be real n x n matrices such that the 2 n x 2n matrix s k := ( A k B k ) is symplectic for all k E [0, N ] , that is, s,‘J’sk = 3 with J’ := GkODlf ( - I o ) . Here [0, N] denotes the discrete interval { 0 , 1 , . . . ,N } . A pair ( z , u ) is admissible if B k u k on [O, N]. The discrete quadratic functional

xk+l = Akxk

+

N

3 d ( x ,U ) :=

{xrC,‘AkZk

+2

T

T

Xk c k Bkuk

+

T

Uk

T

Dk

BkUk}

k=O

is nonnegative if F d ( X , u)2 0 for all admissible ( x ,u)with xo = 0 = X N + 1 , and it is positive definite if F d ( x , u ) > 0 for all admissible ( x , u ) with xo = 0 = xN+1 and x $ 0 .

Result 1.2. The discrete quadratic functional 3 d is positive definite over zero endpoints xo = 0 = xN+l, x $ 0 , i f and only i f there exists a > 0 such that the perturbed quadratic functional

+ CX ((xO(12

Gd(x,u):= 3 d ( x ,u)

is positive definite over free initial state xo and

XN+1 = 0,

x $ 0.

We refer t o [7] for the elements of the time scales calculus and t o [8, 9, 101 for the basic notions about time scale symplectic systems. Let T = [a,b] be a bounded time scale with the forward and backward jump operators a ( t )and p ( t ) a t t and with the graininess p ( t ) := a ( t )- t. Let d,l?, C , and D be real-valued piecewice rd-continuous n x n matrix functions on [a,p ( b ) ]

268

such that the 2 n x 2 n matrix S ( t ) := identity

S T ( t )J-

($:; E::;)

satisfies on [a,p(b)] the

+ J-S(t)+ p ( t )S T ( t )J-S(t)= 0.

(1)

Consider the time scale quadratic functional

:=L b

3(x,u)

{~~C~(I+pd)x+2px~C~f?u+~~(I+pD~)f

+

over admissible pairs ( 5 ,u),i.e. x A ( t ) = d ( t )x ( t ) D ( t )u ( t )on [a,p(b)]. The functional 3 is nonnegative if F ( x , u ) 2 0 for all admissible ( x , u ) with .(a) = 0 = x ( b ) , and it is positive definite if 3 ( x , u ) > 0 for all admissible ( x , u ) with .(a) = 0 = x(b) and x $ 0. When the time scale [a,b] is continuous-time (connected) interval, then the time scale functionals 3 and Q reduce to the functionals 3cand Q,. While if the time scale [a,b] = {0,1,. . . ,N 1 ) is a discrete interval, then they reduce to the functionals 3 d and Q d . The following time scale generalization (and unification) of Results 1.1 and 1.2 is a special case of [ll,Theorem 10.51, see also the proof of [ l o , Theorem 71, but the latter one requires a certain normality assumption.

+

Result 1.3. The time scale quadratic functional 3 is positive definite over zero endpoints .(a) = 0 = x ( b ) , x $ 0 , i f and only i f there exists Q > 0 such that the perturbed quadratic functional

Q(., u):= 3 ( x ,u)+ Q 11x(a)1I2 is positive definite over free initial state .(a)

and x ( b ) = 0 , x $ 0.

We stress the fact that the initial endpoint of the perturbed quadratic functionals Q,, Q d , and 6 in Results 1.1-1.3 is fwe. Recently in [12, Theorem 31, the authors discovered that in the discrete case a corresponding statement regarding the nonnegativity of 3 d holds, but the initial state xo of the perturbed functional Qd may no longer be free but must be restricted to a subspace.

Result 1.4. The discrete quadratic functional 3 d is nonnegative over zero endpoints xo = 0 = x N + 1 i f and only i f there exists Q > 0 such that the perturbed quadratic functional Qd is nonnegative over x,+lxo = 0 and x ~ =+0, where ~ 0)is the solution of the associated discrete symplectic system

(x,

xk+l

=AkXk

+B k u k ,

uk+l

=ckxk

+ Dkuk,

E

[o, N ] ,

(Sd)

269

starting with XO= I and

I!&= 0 .

The proof therein is based on the characterization of the nonnegativity of 3 d with general endpoints. The continuous-time analog pertaining the nonnegatiuity of the functionals 3c and Gc is to our knowledge an open problem, although an incorrect statement (with the free initial endpoint) can be found in [2, Lemma 3.171, see also Example 4.1 in Section 4. In this paper we approach this problem via time scales. Thus, we generalize the discrete case in Result 1.4 to time scales and at the same time obtain as a special case the continuous-time statement. The advantage of the present proof is that it does not use a characterization of the nonnegativity of F with general endpoints, which itself is rather involved and is a content of a separate study, see [ll].However, the main results of this paper are independent of [ll].Hence, we provide at the same time a simpler proof of Result 1.4 than the one in [12]. We illustrate the theory in several examples. Finally, let us mention that the results of this paper remain valid if we consider complex-valued coefficients and replace the transpose of a matrix by the conjugate transpose. 2. Main Results - Nonnegativity

Let d(t),B ( t ) ,C ( t ) , and D ( t ) be the coefficients of the time scale quadratic functional 3 and let S ( t ) be the corresponding 2n x 2n coefficient matrix introduced in the previous section. The time scale symplectic system is

xA = d ( t )x

+B(t)

U,

uA = C ( t ) z -t- D ( t )U ,

t E [a,p(b)].

(S)

The defining property (1) implies that the matrix I + p ( t ) S(t)is symplectic, hence invertible. This means in the time scale terminology that the matrix S ( t ) is regressive and, by the time scale existence and uniqueness [7], the system (S)has unique solutions on the whole time scale [a,b] for any initial time to E [a,b] and arbitrary initial values. Hence, matrix solutions of (S), typically ( X ,V ) or (X, are denoted by the capital letters, while vector solutions of (S) and admissible pairs, typically (z,u)or (Z,ii), by small letters. It is known that if a solution ( X , U ) of (S) satisfies the property that X T ( t o )U ( t o ) is symmetric and rank ( X T ( t 0 )VT(tO))= n at some point t o E [a,b] (for example at the initial point t o = a ) , then it satisfies this property for all t E [a,b]. Any solution of (S)having this property is called a conjoined basis.

a),

270

For a function f (t) defined on [a,b] we adopt a common convention that the difference f (b) - f ( a ) is denoted by f (t) Next we present the main results of this paper. For the rest of this section denote by (X,o)the conjoined basis of (S) satisfying the initial conditions X ( a ) = I and o ( a ) = 0.

. :1

Theorem 2.1. The quadratic functional 3 is nonnegative over zero endpoints .(a) = 0 = x(b) if and only if there exists a > 0 such that the perturbed quadratic functional G is nonnegative over X(b)x(a) = 0 and x(b) = 0. Theorem 2.2. The following statements are equivalent. (i) 3 is nonnegative over zero endpoints .(a) = 0 = z(b). (ii) 3 is nonnegative over X(b) .(a) = 0 = x(b). (iii) There exists a > 0 such that the perturbed quadratic functional Q is nonnegative over T(b)~ ( a=) z(b). Observe that the value of x(b) in condition (iii) is allowed t o be nonzero. While Theorem 2.1 follows easily from the equivalence of (i) and (iii) in Theorem 2.2, we stated it separately because of its direct connection to the motivation in Section 1. Before proving Theorem 2.2 we display the following example.

F]

Example 2.1. Let T = [a,b]:= [0,5]U [$, U {r} be the union of two connected intervals and a singleton. Then p(t) = 0 for t E [0, U [$, $) and p(t) = for t E {;, Let n = 1 and define the coefficients as:

5)

F}.

5

z ) u [$, %), for t E {E, F}, for t E [O,

d(t)= D(t) = 0, d(t)

= D(t)

-&,

B(t) = -C(t)

= 1,

B(t) = - C ( t )

E

g.

Then the principal solution of (S) is ( X ( t ) , ~ ( t ) ) = ( s i n t , cost) on T and the associate solution from Theorem 2.1 is (X(t), o(t)) = (cost, - sint) on T. By [13, Theorem 5.11 modified to this setting, the time scale functional 3 is nonnegative over x(0) = 0 = x(T). But there is no a > 0 such that the functional 9 is nonnegative over free ~ ( 0 and ) X ( T ) = 0, since for admissible pairs (xg, ua)defined by

i

( ( g - t ) 2 + p s i n t , 2t-:+Pcost),

(4% 4)) := ( p sint, p cost),

for t E

[$,

f o r t E [O,;],

F]u { T } ,

p E R, we have that XP(T) = 0 and G ( x ~ UP) , = (r2/18) [ a/2 - ,B - (r3/2160) + ( r / 9 ) ]

with a parameter

271

+

Thus, for example for p > ( a 1) we have g(zp,u p ) value of z p ( 0 ) = is not in the kernel of A(..) = -1.

$

< 0.

Of course, the

The remaining part of this section is devoted to the proof of Theorem 2.2. First we establish several auxiliary lemmas. For two admissible and ( 5 , U ) we define pairs (z,u)

A(z7 u,5,W t ) := {zTCT(I pd) 5

+

+ pzTCTBii + puTBTC5 + u T ( I + pD)TBii}(t).

Then A(z, u,5,U ) = A(Z,U, z, u) and A(z, u,z, u)is the integrand of the quadratic functional F.

Lemma 2.1. Let (z, u) be admissible on [a,b] and let (5, U ) be a solution of (S). Then

J," A(z, u,2 , U ) ( t ) At

. ;1

= z T ( t )U ( t )

Proof. It is a direct application of the time scale integration by parts formula Jf f " ( t ) g A ( t ) A= t f ( t ) g ( t ) - Jf fA(t)g(t)At.

:1

Lemma 2.2. Let (z,u) be admissible, ( X , V ) be a solution of (S), and c E Rn. Then the pair ( 5 ,U ) := (z, u) ( X ,U )c is also admissible and

+

F(Z,'11)

;1 + CTXT(t)V ( t )c lab.

+

= F(2,u) 2 CTUT(t)z ( t )

Proof. The admissibility of (?.,a) follows from the fact that ZA = zA X A c . Since F ( z , u ) is quadratic in z, u and F ( X c , V c ) = Jf A ( X c , Uc,X c , V c ) ( t )At, we have

+

F ( 5 ,U ) = F ( z ,u)+ 2

A(z, u,X c , U c ) ( t )At

+ F ( X c ,V c )

J," V ( t )c ;1 + CTXT(t)U ( t )c lab, + 2 zT(t)

(2)

= F ( z ,u)

where we used twice Lemma 2.1 to evaluate the last two terms in equation (2).

Lemma 2.3. Let (z,u) be admissible with z(b)= X ( b ) z ( a ) and define ( 2 , G ) := (z, u)- ( 2 , V ).(a).

Then

(Z,G)

(3)

is admissible, % ( a )= 0 = Z ( b ) , and F ( z , u )= F ( 3 ,a)

+ z T ( a )gT(b)V ( b ).(a).

(4)

272

o),

ProoJ We apply Lemma 2.2 with (Z,.ii) := ( 2 , E ) , ( X ,V ) := (X, and c := --x(u). Then ( 3 , E ) is admissible, %(a) = x(u) - X ( u ) x ( u ) = 0, T ( b ) = x(b) - X ( b ) x(u)= 0, and since X T ( t ) o(t)is symmetric, we get

;1 + x T ( u )X T ( t )O ( t )x ( u );1 = F ( x , u )- xT(u)OT(b)x(b ) + 2 ( u ) OT(b) [ X ( b )x ( u ) - x(b)]

F(T,a) = 3 ( z , u )- 2 x T ( u )O T ( t )z ( t )

= F ( x ,u ) - xT(u)X-T(b) O ( b )x ( u ) .

Hence, this lemma is now proven. Now we are ready t o prove Theorem 2.2.

Proof of Theorem 2.2. Each of the conditions (ii) and (iii) implies condition (i) trivially. Conversely, condition (i) implies (ii) by formula (4),since in (ii) we have X ( b ) .(a) = 0. Thus, it remains to prove that (i) implies (iii). Let (z,u ) be admissible with X ( b ) x(u) = x(b) and define the admissible pair (3,a) by (3). Then, by the assumption, . F ( T , a ) 2 0, so that formula (4)yields that

.F(x,u) 2 z T ( a ) A T ( bO) ( b ).(a) 2 A0 11x(a)1I2,

(5)

where A0 is the smallest eigenvalue of the symmetric matrix X T ( b ) o ( b ) . Then we distinguish the following cases. 0 Case I (A0 2 0). In this case S ( x , u ) 2 0, so that G(x,u) 2 0, being a sum of two nonnegative terms, for any a > 0. 0 Case I1 (A0 < 0). In this case we put a := -A0 > 0. Then inequality ( 5 ) yields that G(x,u)2 0. The proof is now complete. Let us remark that in general any a > lAol will satisfy the statement of condition (iii). 3. Positive Definiteness

The method used in the proof of Theorem 2.2 shows that a similar result regarding the positivity of 3and 9 can be proven in the same way. Namely, the following holds.

Theorem 3.1. The following statements are equivalent.

(i) .F is positive definite over zero endpoints x ( u ) = 0 = x(b), x $ 0 . (ii) 3 is positive definite over X ( b ) x ( u ) = 0 = z(b), x ( t ) $ % ( t )x ( u ) o n [a,b]. (iii) There exists a > 0 such that the perturbed quadratic functional is positive definite over X ( b ) z ( u ) = x(b), x $ 0 .

273

Proof. While each of the conditions (ii) and (iii) implies condition (i) triv-

ially, we prove the opposite implications. “(i) =+ (ii)” Let (z,u)be admissible with X ( b ) .(a) = 0 = z(b),z ( t )f X ( t ).(a) on [a,b]. Then the pair (3,ti) defined by (3) satisfies 3 f 0, so that formula (4)yields F ( z ,u)= F(3,21) > 0. Here we used X ( b ) z(u) = 0. “(i) =+ (iii)” Let (z,u)be admissible with X ( b ) .(a) = z(b), z f 0. If .(a) = 0, then also z ( b ) = 0 and, since z $ 0, condition (i) implies that G(z,u) = F ( z , u ) > 0, where a > 0 is arbitrary. Finally, assume that .(a) # 0. With ( 3 , t i ) defined by (3) we have F(?t,a)2 0 and, as in ( 5 ) , we get F ( z , u ) 2 A0 11z(~)11~. Hence, for any a > 1x01 and using ~ ~ z (>a 0) we ~ ~ conclude that

G(.,.)

= +,u)

2

(A0

+ a ll.(a)1I2

+ 1x01) 11(.4112

> F ( z , u )+ 1x01 11(.4112 2 0.

The proof is complete.

Remark 3.1. The method also shows that Result 1.3 can be derived in a similar way as condition (iii) of Theorem 3.1 by using the characterization of the positivity of F from [ll]. 4. Continuous-Time Case When T is the continuous-time (connected) interval [a,b], then the time scale symplectic system (S) reduces to the linear Hamiltonian system 2’

= A(t)z

+ B ( t )u,

U’

= C ( t )2 - AT(t)U ,

t

E

[a,b],

(H,)

and hence, Theorem 2.1 reduces to the following.

Corollary 4.1. The quadratic functional F,is nonnegative over zero endpoints z ( u ) = 0 = z(b) i f and only i f there exists a > 0 such that the perturbed quadratic functional 6, is nonnegative over X ( b ) z ( a ) = 0 and z(b) = 0, where (X,o)is the solution of the linear Hamiltonian system (H,) starting with X ( a ) = I and o ( a ) = 0 . The next example shows that the initial endpoint of the perturbed quadratic functional 6, cannot be free. We verify the nonnegativity of F,by using [14, Theorem 21, which uses a certain conjoined basis ( X a ,Ua) of (H) arising from the initial endpoint condition. For the case of zero endpoints one can take ( X a ,Ua)to be the principal solution ( X ,fi) of (H), that is, the conjoined basis given by X ( a ) = 0 and o ( a ) = I .

274

Example 4.1. Let [u,b]= [0,7r], n = 2, A ( t ) = 0, and B ( t ) = -C(t) = on [0, TI. Then the solution ( X , 0)from Corollary 4.1 and the principal solution ( X ,0)of (H) are

(A 1y2)

X(t) =

cost 0 0 cos(t/2)

(

)’

0(t) =

-

sint 0 0 -sin(t/2)

and X ( t ) = -o(t),U ( t ) = X(t). Then the admissibility condition for means that u1 = zl,, 212 = 2xL and, z(t)= and u(t) =

(::[:!)

(::[ti)

by [14, Theorem 21 with X,(t) = X ( t ) , the functional Fcis nonnegative over z(0) = 0 = z(7r). Note that Fc is not positive definite since for an admissible z(t) = ( si;t) and u(t) = ( ,Yt) we have F ( z ,u) = 0. We shall show that there is no Q > 0 such that the gunctional Gc is nonnegative over free z(0) and z(7r) = 0. Define the admissible pair z p ( t ) := ( n-t+Psin n-t t ) and up(t) := ( -‘+!Yt), where ,B E W is arbitrary. Then Qc(zcp,ucp)= 2Q7r2 - 2p7r

+ 37r - (1/2) 7r3

and we can see that for each Q > 0 there is p E R such that Q c ( z p ,u p ) < 0. Clearly, by Corollary 4.1, the value z(0) = cannot be in the kernel of X ( 0 ) = (i1;). On the other hand, there exists Q > 0 such that the functional Q, is nonnegative over the endpoint constraints z(0) = ( z2”(o,) E Ker X ( 7 r ) , where z2(0) is free, and z(7r) = 0.

(z)

Remark 4.1. In Example 2.1 the associate solution X ( t ) = cost is nonsingular (i.e. nonzero in this case) a t t = 7 r . This is the same situation as in the discrete time case in [12, Example 21. Thus, the only initial state .(a) in the kernel of X ( b ) is in this case z ( u ) = 0. In Example 4.1 above we show a situation where the kernel of Z(b)is a proper subspace. References W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin, 1995. G. Stefani and P. Zezza, Constrained regular LQ-control problems, S I A M J . Control Optim. 35 (1997), no. 3, 876-900. M. R. Hestenes, Applications of the theory of quadratic forms in Hilbert space to the calculus of variations, Pacific J. Math. 1 (1951), 525-581. D. H. Jacobson, D. H. Martin, M. Pachter and T. Geveci, Extensions of Linear-Quadratic Control Theory, Lecture Notes in Control and Information Sciences, Vol. 27, Springer-Verlag, Berlin, 1980.

275 5. V. Zeidan, Sufficient conditions for variational problems with variable endpoints: coupled points, Appl. Math. Optim. 27 (1993), no. 2, 191-209. 6. M. Bohner, Riccati matrix difference equations and linear Hamiltonian difference systems, Dynam. Contin. Discrete Impuls. Systems 2 (1996), no. 2, 147-159. 7. M. Bohner and A. Peterson, Dynamic Equations on T i m e Scales. A n Introduction with Applications, Birkhauser, Boston, 2001. 8. 0. DoSlL, S. Hilger and R. Hilscher, Symplectic dynamic systems, In: “Advances in Dynamic Equations on Time Scales”, M. Bohner and A. Peterson, editors, pp. 293-334. Birkhauser, Boston, 2003. 9. 0. DoSlf and R. Hilscher, Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales, J. Diger. Equations Appl. 7 (2001), 265-295. 10. R. Hilscher, Reid roundabout theorem for symplectic dynamic systems on time scales, Appl. Math. Optim. 43 (2001), no. 2, 129-146. 11. R. Hilscher and V. Zeidan, Time scale symplectic systems without normality, In preparation. 12. R. Hilscher and V. R3iiEkov.5, Riccati inequality and other results for discrete symplectic systems, J. Math. Anal. Appl., To appear. 13. R. Hilscher and V. Zeidan, Legendre, Jacobi, and Riccati type conditions for time scale variational problem with application, Submitted. 14. W. Kratz, Definiteness of quadratic functionals, Analysis (Munich) 23 (2003), no. 2, 163-183.

276

MONOTONICITY OF DYNAMICAL SYSTEMS AND THEIR DISCRETIZATIONS

z. HORVATH* Department of Mathematics, Sze'chenyi Istvcin University Egyetem te'r 1, Gy&, 9026, Hungary E-mail: hormathzQsze. hu

Monotonicity of dynamical systems is proven t o be a very powerful means t o discover long term behaviour of continuous-time and discrete-time dynamical systems. In this paper we consider monotone continuous-time dynamical systems and their discretizations with Runge-Kutta methods. We study the parameters of the discretization method that guarantee monotone discrete-time system as a result. In case of a general class of polyhedral order cones we conclude a simple and practically useful formula for the step sizes that ensure discrete monotonicity.

1. Motivation The mathematical model of physical (biological, chemical, etc.) time dependent processes is often based on a system of ordinary differential equations (ODES) for the unknown state variables

u'(t) = f ( u ( t ) ) , t 2 0. Modelling of the same physical process leads to difference equations when we are interested in, or have information on the development of the values of the modelled physical quantities at discrete time points only. Discrete models often arise from (or can be interpreted as) discretization of the continuous-time model, typically approximating the solution of (1) subject to an initial condition u(0)= u g at discrete time points. For example, the simplest way to derive a discrete-time model from (1) is the explicit Euler method where

*This work is partially supported by a grant of the Hungarian Scientific Research Found (OTKA) under contract T043177.

277

There are several kind of situations when we need a better discretization than (2). For example, if a chemical kinetics process with very fast and slow reactions is to be modelled then we know that the explicit Euler method is stable only when the h, step sizes are very small hence for an approximation at a certain time tfinal by (2) we need very large number of iterations. (For a simple test example see u'(t) = Xu(t) with X << 0, we need h, 5 l / l X l for stability hence at least tfinallXl >> 1 steps.) However, with the implicit Euler method of recursion u,+1 = 2172 + h,f(u,+d we do not have restrictions on the step sizes due to stability at the cost that in each step a system of algebraic equations is to be solved, which can be performed exactly for some models or can be well approximated if necessary. For the analysis of continuous- and discrete-time dynamical systems it is well-known that monotonicity properties, i.e. ordering preservation w.r.t. a fixed order cone, are often useful, e.g. in establishing convergence and stability of equilibria, see, for example, [7], [16] in the continuous-time and [4],[8]and [15] in the discrete-time models. A natural and, due to its applicability mentioned in the previous paragraph, important question arises for discrete-time models derived by approximating a monotone continuous-time model: under what conditions on the discretization procedure can we assert a priori that the resulting discrete model is monotone? In the literature, in [13] it was shown that for a large family of RungeKutta methods and the ordering generated by the positive orthant the discrete-time model is not monotone while the others are monotone for sufficiently small step sizes. Our aim is to give an a priori condition on the step sizes, namely a fairly simple and explicit formula for a step size constant H such that in case h, 5 H we could conclude monotonicity of the resulting discrete-time semiflow w.r.t. the same order cone. In this paper we consider monotone continuous-time semiflows when the order cone is polyhedral (this includes as special cases the classical situations when the order cone is an orthant of RN)and arbitrary Runge-Kutta methods for the time discretization. We shall see below that our construction applies to methods where a positive monotonicity step size constant exists at all.

278

2. Preliminaries; continuous-time problems

We denote the state space of the dynamical system by V . We assume that V = IRN, although most of the results of the paper hold true for general Banach spaces in the present form. Let f : V + V be a fixed continuously differentiable function and @ the semiflow generated by the ODE (1)above. For simplicity we suppose that (1)subject to the initial condition u(0)= uo has a unique solution u : [O,m) + V for all uo E V . Hence, with this solution, : [O,m) x V + V and @(t,ug):= u(t) (t E [O,m),ugE V ) . Further, we make use of the functions G t : V + V for all t 2 0 with @t(uo):= @(t,uo)(for all uo E V ) as well. Let C be a fixed order cone of V , i.e. C is a closed, convex cone (latter means that for all u, w E C and Q E [0, 00) we have QU E C and u + u E C). Then C defines an ordering 5 on V with w 5 u iff u--21 E C (for all u,w E V ) . We say that the semiflow @J is monotone w.r.t. C whenever vu, 21 E VVt E [O, m) : (v

I u =+ q t , w ) L q t , u ) ) .

In other words, @ t is monotone on V for all t E [0, m). There are several ways t o characterize monotone semiflows, see e.g. [l] and [7] where, respectively, the tangent cones and the quasi-monotonicity property were used. In this paper we apply the latter one. In order to formulate this we have to recall the notion of the dual cone. The dual cone of C , denoted by C*, consists of the linear functionals of V = RN that are non-negative on C. Identifying the $J linear functional and the uniquely existing vector cp for which $J(v)= cpTu (for all v E V ) we may write C* = {cp E mNI vx E c : cpTx 2 0). The quasi-monotonicity condition on f and C can now be formulated as (cf. [71, P. 48)

vw E v,vc E c, vcp E C’

:

(cpTC

= 0 =+ c p T ( f ( w

+ c) - f(w))

2 0).

(3)

It is proved Zoc. cit. that Q, is monotone w.r.t. C iff (3) holds for f and C. 3. Discrete-time problems

Suppose that we are given a time grid 0 = to < tl < ... with time step sizes h, := tn+l - t,. Let us approximate the trajectories of the dynamical system at time points t, emanating from uo E V as 11, M @(tn,uo),

12 = 0,1,2,

....

We consider “onestep” discretizations, i.e. approximations when

un+l = !@(h,,u,),

n

2 0,

uo given.

(4)

279

Here Q : [O,Hdef] x V 4 V depends on the discretization method and f. We assume that the step size threshold Hdef could be chosen such that Q is well-defined over [o, H d e f ] X

v.

Definition 3.1. The one-step method with definition (4), or shortly: Q is (discrete) monotone w.r.t. c with step size constant H E (0, Hdef] iff Vuo, uo E V : (uoI uo and V n : h, E [0,HI

+ V n : u, I u,)

whenever the sequences u,, u, are defined by (4). Or, equivalently, Q(h,.) is monotone w.r.t. C for all h E [0,HI:

V W E V , V C E C , V ~ E [ O , :HQ] ( h , u ) < Q ( h , w + c ) . 3.1. Time discretization of

with Runge-Kutta methods

Runge-Kutta (RK) methods are well-known and widely used one-step methods for discretization of semiflows, see e.g. [3]. The RK method with coefficient arrays A = ( a i j ) E ELsxs and b = (bi) E IR"is given by (4) with

i=l

where yi, i = 1,.. ., s form the (unique) solution of the system of algebraic equations

c S

yi = u

+h

aij

f(yj),

i = 1,..., s.

(6)

j=1

This method being determined completely by its arrays is denoted by RK(A, b). During this paper we assume that system (6) possesses a unique solution whenever u E V and h E [O,Hdef]are arbitrary. For more details about unique solvability and formulas for H d e f in terms of some characteristics, esp. measures of dissipativity of f see e.g. [3], [14]. We remark that A and b are usually chosen according to the stability and accuracy of the method. We shall see below in the next sections that the scheme functions and the absolute monotonicity radius of RK(A, b ) play an important role in our analysis. (For more details consult [14], see also [6], [9].)

Definition 3.2. The scheme functions of RK(A, b) are defined as

K A ( z ):= (I - zA)-'e,

+

Kb(z) := 1 bTz(I - zA)-le,

JA(z):= A ( I - zA)-l, J b ( z ) := bT(I - zA)-l

280

where z is a real variable, e = (1, ..., l)TE ELs and I denotes here and throughout the paper the identity matrix of appropriate order (here it is of s x s order). Further, the absolute monotonicity radius of RK(A,b)

R(A,b) := SUp{p 2 0 I V T

E

[o,p] : K A ( - T ) ,

JA(-T), Kb(-T),

Jb(-T)

20) (7)

(the supremum of the empty set is taken -m).

Remark 3.1. From the definition it is clear that R(A,b)> 0 implies 0 5 Jb(0) = bT and 0 5 JA(O) = A, hence A, b 2 0 is necessary for R(A,b) > 0. It happens that this condition is violated by many methods that are used in practice (see e.g. [3], [12]). Moreover, R(A,b) can be easily computed for each RK scheme (A,b) just from the definition. For example, R(A,b) = +m for the implicit Euler method. (Unfortunately, this very favourable property holds only for some first order methods.) 4. Discrete monotonicity of RK methods Now we are in the position to formulate the main goal of the paper precisely as follows: Given @ , C , RK(A,b) as above and 9 is from the RK(A,b) discretization. Suppose that GJis monotone w.r.t. C . Goal: construct a formula H = H(f;C;A,b)such that 9 is monotone w.r.t. C with step size constant H .

Remark 4.1. This problem is considered in the literature by several authors in special cases: for linear problems and C = [0,C O ) ~see [2] (in terms of positive invariance of C ) ; for nonlinear problems and C = [ O , C O ) ~ but concerning only the existence of a suitable H > 0 see [13]. We remark further that similar problems for RK methods were considered in the literature, namely contractivity (see [14]), positive invariance (see [9], [ll],[12]); total variation diminishing methods (see [S]) (in each case see also the references in the papers Zoc. cit.). In this paper, to be able to give a constructive and simple formula for

H , we consider the case when C is a convex, polyhedral with non-obtuse face angles i.e. it is the intersection of finitely many halfspaces with given non-acute inner normal vectors (pi, i = 1, ..., K

C = {x E VI(pTx 2 0, i = 1,..., K }

with V i # j : (p7p.j 5 0

(8)

281

For example, the positive orthant of V and the other orthants are of form (8) with choosing appropriately cp = ei or cp = -ei for all i where ei is the ith coordinate unit vector of V . It follows from the Farkas' theorem (see [5] or any linear programming textbook) that C* = cricpi I for all i: cri 2 0).

(xi

4.1. Discrete monotonicity of the explicit Euler method For practical applications it is advantegeous to know the size of the monotonicity step size constant E for the explicit Euler method. This is important for this method itself and also from the point of view that the monotonicity step size constant of a general RK method equals a scheme dependent factor times this E , see Theorem 4.1 below. So, let us consider first the discrete monotonicity of the explicit Euler method. As we mentioned above, for a constructive form of the monotonicity step size constant in this paper we restrict ourselves to polyhedral cones with non-obtuse face angles of form (8). Hence this lemma applies for several important cases such as the non-negative orthant (and the other orthants).

Lemma 4.1. Let C be polyhedral with representation given in (8). Assume further that @ is monotone w.r.t. C. Then E = & is a monotonicity step size constant of the explacit Euler method w.r.t. C whenever l+:= max(4,O) and

e

:= -

inf

cp?f'(+i

z E V , i E { l , ...,K}

< 03.

(9)

Proof. Let Q = Q E E according t o the explicit Euler method (see (2)), h > 0 , u , v E V with v 5 u and i E (1, ..., K } arbitrary and fixed; define c:=u-v~Cand m

First we show that cpTZ = 0 and Z E C. Indeed, cpTZ = 0 is trivial and for any j # i there holds cpTZ = ($c) - (cpTc)(cpTcpi)/(cpTcpi)2 0 since the first two terms in the round brackets are non-negative due to c E C and the third one is non-positive by (8). Therefore Z E dC and thus, by the monotonicity of @ w.r.t. C and (3), we have

cpT(f(v

+ q - f(v)) 2 0.

(10)

282

+

+

Further, there exists x from the segment joining v c and v E such that pTf(w c ) - (pyf(v - t E ) = cpTf’(x)(c- 3, due to the mean value theorem. Thus we have with such an x and by (10) that

+

v)) = (P?) + h ( P m J+ 4 - P m J ) ) = ( C P Y C ) + h (PYfb + 3 - P W J ) ) + h ( P m + 4 - ( P

cpW(h7 u)-

v

7

which is non-negative if h 5

m

+ q)

&. This proves the monotonicity of 9 ( h ,.).o

Remark 4.2. Observe the Rayleigh quotient of f ’ ( x ) in (9). Hence the infimum of the smallest eigenvalues of the Jacobian matrices f ’ ( x ) ,x E V is a lower bound for -4.However, this estimate is usually not strict because the infimum of the Rayleigh quotient over a cone C* can be much larger than the minimal eigenvalue. Theorem 4.1. Let C be polyhedral with representation given in (8). Assume further that the interior of C is non-empty, @ is monotone w.r.t. C and let E > 0 be chosen according to Lemma 4.1. Further let RK(A, b) be given with R(A, b) > 0. Then RK(A, b ) is discrete monotone w.r.t. C with step size constant H where

H = min{ER(A, b),

Hdef}.

(11)

Proof. Let v E V , c E C, H be given by (11) and h E [0, HI be arbitrary; we need to prove that v1 5 u1 where u := v+c, u1 := @ ( h , u )w1 , = 9 ( h ,w). Applying (5) and ( 6 ) we have that ~1

=u+hCbif(yi),

~1

=v+ hCbif(zi)

i

(12)

i

where for all i there holds

yi =

+hCaijf(yj),

zi

=

+hCaijf(zj).

i

(13)

j

We shall prove wi := yi - zi 2 0 for all i as well. The main idea is to use the following quasi-linear representation 1 1 1 1 f ( Y j ) = --Yj - (Yj E f ( Y j ) ) = --Yj -Qm(E,Yj). &

+&

+

&

+

&

283

Taking similarly for f(zj) and denoting we obtain

Q E E ( E , yj) - Q E E ( E , z j )

by p(yj, z j )

We know by Lemma 4.1 that p(yj,zj) 2 0 whenever w j 2 0. Taking the difference of yi and zi by (13) and inserting (14) into this we obtain

or in more compact form with the additional notations

we have w = (e @ I ) c

+ -h& (A @ I ) (-w + P )

and, solving it formally to w we obtain by the term IT := I @ I with identity matrices of order s x s and N x N , respectively

( ( I + h& a ) - ' e ) @ I

( A ( I + FA)-') & @I

Writing out the components of w and using the notation of the scheme functions introduced in Definition 3.2 we arrive at

Thus, in (15) and (16) wi and u1- v1 are written as linear combinations of the terms c and p(yj, z j ) , j = 1,..., s. Here the coefficients are non-negative due to 5 R(A,b). Moreover, the vectors in the linear combinations fulfil c E C and p(yj, z j ) E C whenever w j 2 0 (for all j). Hence the linear combinations, the left hand side vectors of (15) and (16) belong to C supposing w j 2 0 for all j. So, finally, application of

3

284

some continuation techniques, e.g. almost word-by-word application that of Theorem 1 in [9] results in the statement of the theorem. 0

Corollary 4.1. Theorem 4.1 provides us with a quite simple formula, essentially h 5 E R ( Ab, ) to determine step sizes a priori that guarantee discrete monotonicity of RK(A,b) w.r.t. c. Moreover the terms E and R(A,b) can be computed by applying Lemma 4.1 and (7),respectively.

Remark 4.3. Comparing our result with that of [2] for linear problems and C = [0,0 0 ) ~we can see that H in (11) can not be improved in general. Moreover, as a consequence of Theorem 4.1, the implicit Euler method is discrete monotone with H = 00, i.e. without step size restriction whenever Hdef = 00 and if l in (9) is finite (these hold true for dissipative problems and when f is Lipschitzian, respectively). As a counterpart, for the explicit Euler method we have R(A,b) = 1 and H = E , which is inverse proportional to the Lipschitz constant and so H << 1 for many problems (such as chemical kinetics problems with slow and fast reactions). Hence the implicit Euler discretization is much more preferable from the point of view of monotonicity than the explicit Euler method. Finally we remark that the Runge-Kutta discretizations preserve monotonicity of the continuous-time linear problems with much less severe conditions on the time step sizes w.r.t. some special sub-cones of C (see [lo]). References 1. Angeli, J.D. and Sontag, E.D., Monotone control systems, I E E E Trans. Autom. Control 48 pp. 1684-1698 (2003). 2. Bolley, C. and Crouzeix, M., Conservation de la positivite lors de la discretisation des problhmes d’6volution paraboliques, R.A.I. R. 0. Analyse num6rique 12(3),p. 237 B 245 1978. 3. Butcher, J.C., Numerical methods for ordinary differential equations. Chichester: Wiley (2003). 4. Dancer, E.N. and Hess, P., Stability of fixed points for order-preserving discrete-time dynamical systems. J. Reine Angew. Math. 419, pp. 125-139 (1991). 5. Far&, J., Die algebraische Grundlage der Anwendungen des mechanischen Princips von Fourier, Mathematische und Naturwissenschaftliche Berichte aus Ungarn, 16 154-157. (1899) 6. Ferracina, L. and Spijker, M.N., Stepsize restrictions for the total-variationdiminishing property in general Runge-Kutta methods. S I A M J. Numer. Anal. 42, No.3, 1073-1093 (2004).

285 7. Hirsch, M.W. and Smith, H.L., Monotone Dynamical Systems. Handbook of Differential Equations, Ordinary Differential Equations (second volume), Elsevier (2005). 8. Hirsch, M.W. and Smith, H.L., Monotone maps: a review. J. Dzfference Equ. Appl. 11,No.4-5, 379-398 (2005). 9. Horvtith, Z., Positivity of Runge-Kutta and diagonally split Runge-Kutta methods. Applied Numerical Mathematics, 28, No.2-4, 309-326 (1998). 10. Horvtith, Z., On the positivity of matrix-vector products. Linear Algebra and its Applications. 393C pp. 253-258 (2004). 11. HorvAth, Z., On the positivity step size threshold of Runge-Kutta methods. Applied Numerical Mathematics, 53 pp. 341-356 (2005). 12. Hundsdorfer, W. and Verwer,J., Numerical solution of time-dependent advection-diffusion-reaction equations. Springer Series in Computational Mathematics. 33. Berlin: Springer, 2003 13. Kloeden, P.E. and Schropp, J., Runge-Kutta methods for monotone differential and delay equations. BIT 43, pp. 571-586 (2003). 14. Kraaijevanger, J.F.B.M., Contractivity of Runge-Kutta methods, BIT 31, pp. 482-528 (1991) 15. Matano, H., Strongly order-preserving local semi-dynamical systems - theory and applications. Semigroups, theory and applications, Vol. 1, Pap. Autumn Course Int. Cent. Theor. Phys., Miramare (Trieste)/Italy 1984, Pitman Res. Notes Math. Ser. 141, 178-189 (1986). 16. Smith, H.L., Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Mathematical Surveys and Monographs. 41. Providence, RI: AMS (1995).

286

CONSTRUCTION OF AN INTEGRAL MANIFOLD FOR A LINEAR DELAY SYSTEM

K.R. JANGLAJEW Institute of Mathematics, University of Bialystok 15-267 Rialystok, Poland E-mail: [email protected] A linear difference-differentia1 system with constant coefficients and containing a small parameter is investigated. We establish sufficient conditions for the existence of an asymptotic integral manifold of solutions described by a linear system of differential equations without delays.

Key words and phrases. Linear system, Delay equations, Integral manifold of solutions, Exponential dichotomy 2000 Subject Classification: 34K06

1. Introduction and preliminaries

Delay differential equations, also known as difference-differential equations, are a special class of functional equations. The theory of linear delay differential equations is well established. The classical works here are [2], [ 5 ] . This paper is devoted to the study of asymptotic properties of linear systems with delays. There are numerous important papers on the subject, and we only mention a few [l],[6]. We are interested in finding an asymptotic integral manifold of solutions. Such kind of manifolds may be used to reduce the dimension of a dynamical system (see for instance [3], [4] ). Consider the linear system of difference-differential equations

where t E ( - 0 0 , + 0 0 ) , p is a small parameter, ~k 2 0 , A , B , Ak, B k , Ck, Dk are constant matrices, d i m X ( t ) = p , dimY(t) = q.

287

It is assumed that the spectrum of the matrix A lies in the half-plane Xz > u, and the spectrum of the matrix B lies in the half-plane Xz < u. For p = 0 System (1)possesses an exponential dichotomy with an exponent 0 > o( [71). We will construct an integral manifold of solutions of System (1) in the form of ( [S]) dX (t) = H X ( t ) , Y ( t )= K X ( t ) , dt where H = H ( p ) , K = K ( p ) . If this integral manifold exists, then

x(t)= eH(t-tO)X(to), X ( t - ~ k ) )= e Y ( t- ~

k = ) KX(t-~ k = ) Ke

-HTk

-HTk

x (t)

1

X ( t ) , Ic = 1,...,12.

It follows that the matrices H , K satisfy the matrix equations n

H =A

+ p C ( A k + BkK)e-rkH,

(3)

k=l n

KH =BK

+p c ( C k +DkK)ePTkH.

(4)

k=l

Equation (4)could be written in the form n

BK - K A = p K c ( A k

+ BkK)e-TkH-

k=l n

p C ( C k + DkK)e-TkH. k=l

(5)

The matrix equations ( 3 ) , ( 5 )may be solved by the method of successive approximations. Let F denote the expression on the right-hand side of (5). A solution of the auxiliary matrix equation

BK - K A = F is given by the matrix oc)

K =-

eBtFe-Atdt.

This can be easy verified by direct substitution. In addition, under the following conditions:

(leBtll 5 cleTxt, Ile-Atll 5 czect, x > c

(7)

288

from (6) we obtain

Equations (3) and (4) can be put in the form n

H = A + p x ( A k + BkK)e-Tk*H, k=l n

eBtx ( C k

K =p

+ DkK - KAk - KBkK)e-HTke-Atdt.

(8)

k=l

Our purpose here is to give a proof that an integral manifold of solutions of System (1) exists in the form (2). If System (8) has a solution ( H ,K ) then System (1) has an integral manifold of the form (2).

2. Successive approximations

The system of matrix equations (8) defines the matrices H, K and one can be solved by the method of successive approximations. We start this process by letting Ho = 0, KO= 0 and n

Hj+l = A + p X ( A k + BkKj)e-TkHj, k=l

for j=0,1,2, ... . In order to get simple sufficient conditions for the existence of a solution of (8), it is assumed that n

maxkrk = 7

(k = 1, *.en), IlAll 5 a07

llAkl1 k=l

n.

n.

n.

k=l

k=l

k=l

From (9) we find the estimates

I a,

289

We shall have established the boundedness for the sequences IlHjll and llKJll. It suffices to show that the non-linear equations a0

+ Ipl ~ (+ kl ) e T h= h,

h = maz IlHjII

, k

= maz llKjII

lpl a p ( l + Icl2eTh= k ,

,

c1 c2

P = -. A--E

(11)

have positive solutions h > 0, k > 0. Excluding the expression 1pl a(1 k)eTh from the second equation of (11) we obtain

+

The equation for k take the form

and it can be solved by the method of successive approximations. We are looking for the conditions under which (12) has a positive solution k. Equation (12) has a multiple positive root if the following equality holds:

From equations (12) and (13) we have

k 2 + kT0-I - 1 = 0 and the positive root is

From Equation (12) we obtain a critical value of the small parameter p

For Ipl 5 po the sequences of llHjll, j = 0 , 1 , 2 ,... and we have

IlHjII < Qo

2

+ + 2P +@>-& .J

llE(jll

are bounded for all values

IlKjll <

+

d2P y+ 4P27

(16)

where P = clcz/(A - 6 ) . It follows that for lpl 5 po the sequences of matrices Hj and Kj (j=0,1,2, ...) are well defined and bounded in norm.

290

3. Convergence

Let us turn to prove that the sequences of the matrices Hj , Kj (j=0,1,2, ...), given by System (9), converge. First, we estimate the differences n

Hj+l - Hj = p

((Ak

4- BkKj)e-'"J

- (Ak

+ BkKj-l)e-rkHj-l),

k=l

2 1p1 a P ( l + k)eh' llKj - Kj-111.

(18) Now we take 1p1 = po, therefore h and k achieve the maximum values defined by (13),(14)

Then System (18) takes the form llHj+l -

291

The matrix of coefficients of the expression on the right hand side of (19) is

Matrix R has eigenvalues, defined by the equation P2 - /I( -k.2p)(1 -k

k) -k pT(1-k k)2 = 0

and the largest eigenvalue of R is

This clearly forces that the largest in modulus eigenvalue of the matrix G is

For

14 < Po Pmax(G)

<1

and, therefore the terms of the sequences IIHj+l - Hjll, IIKj+l - Kjll, ( j = 0,1,2, ...) are bounded from above by the terms of the decreasing geometric progression. It follows that the series

c 00

H(P) =

c 03

(Hj+l(P) - Hj(P))

7

j=O

converge uniformly for Ipl 5 p1

K(P)=

(Kj+l(P) - K j ( P ) ) (20)

j=O

< po.

We have thus proved Theorem 3.1. If the delay System (1) is dichotomic for p = 0 o n the strength of inequalities (7), then f o r Ipl < PO, with PO defined by (15), System (1) has an integral manifold given by System (2) without delays. Moreover the matrices H(,LL),K ( p ) of System (2) depend analytically o n P.

Remark 3.1. Matrices H ( p ) , K ( p ) of System (2) can be found as power series of p , provided that < PO.

292

Without proof we notice that System (2) is asymptotic to System (1) and every solution of System (1) tends asymptotically to a solution of System (2) as t + +m.

Remark 3.2. In System (1) the values Tk may have different signs. Under Ipl < po there exists also an asymptotic integral manifold in the form of System (2) with matrices H ( p ) , K ( p ) depended analytically on p. Example 3.1. Consider the delay system

We seek an integral manifold of solutions of System (21) in the form (2)

System of nonlinear equations (3),(5) take the form h(p) = E (1

+ pak(p)e-T1h(p),

+ ~ ) k ( p=) -pbe-T2h('l) - pUak(p)2e-T1h(p).

(23)

Solving the last system by the method of successive approximations, we find

We shall be able to prove that the asymptotic behavior of solutions of System (21) is defined by the differential equation without delays

For the sufficiently small values of 1p1 > 0, 161 > 0 all solution of System (21) are asymptotically stable under the condition

The functions k and h to System (22) expand into convergent power series of p , under lpl < po, where po is given by (15).

Acknowledgements The author is grateful to the reviewers for the valuable comments and suggestions.

293

References 1. Elaydi, S., Gyori, I., Asymptotic theory for delay difference equations, J . Dig.

Eq. and Appl. I., (1995), 99-116. 2. Hale, J. K., Theory of Functional Differential Equations. Springer-Verlag, New York,1977. 3. Janglajew, K., On the existence of integral manifolds for a system of difference equations, Advances an diflerence equations (Veszprem, 1995),Gordon and Breach, 1997, 321-326. 4. Janglajew, K., On the reduction principle of difference equations, Dynam. Contin. Discrete Impuls. Systems 6., (1999), 381-388. 5. Myshkis, A. D., Linear differential equations with a retarded argument. (In Russian) Nauka, Moscow, 1972. 6. Ryabov, Yu. A., Certain asymptotic properties of linear systems with small time delay,Sov. Math. Dokl. 4.,(1963), 928-930. 7. Valeev, K. G., Splitting of matrix spectra. (In Russian)’Vishcha Shkola’, Kiev,1986. 8. Valeev, K. G., Zhautykov, 0. A., Infinite Systems of differential equations.(In Russian) ‘Nauka’, KazahSSR, Alma-Ata,1974.

294

CHAOTIC BEHAVIOR IN A TWO-DIMENSIONAL BUSINESS CYCLE MODEL

CRISTINA JANUARIO Department of Chemistry, Mathematics hat, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emidio Nauarro, 1949-014 Lisboa, Portugal E-mail: cjanuarioOdeq.isel.ip1.pt CLARA GRACIO Department of Mathematics, Universi$ade de Euora, Rua Rornio Ramalho, 59, 7000-585 Euora, Portugal E-mail: mgmcioQueuora.pt J. SOUSA RAMOS Department of Mathematics, Instituto Superior Te'cniw, Au. Rouisco Pais 1, 1049-001 Lisboa, Portugal E-mail: sramosOmath.ist.ut1.pt We consider a discrete-time economic model which is a particular case of the Kaldor-type business cycle model and it is described by a two-dimensional dynamical system. Under certain conditions the map can be reduced t o a skew map whose components, the base and the fiber map, both have entropy. Our proposal is to study and measure the complexity of the system using symbolic dynamics techniques and the topological entropy.

1. Introduction

The manifestation of nonlinear effects can be easily detected in almost all real world systems. In the particular case of economics, many of its important fluctuations arise out of nonlinear dynamic phenomena. The most interesting theories of business cycles in the Keynesian vein are the ones that utilizes non-linear functions. In this work we present a variation of the model considered by Bischi et aL2 and Dieci et aL3. It departs from a discrete time-version of the Kaldor model described by a two-dimensional system. We intend to introduce nonlinearity in a variable of the model in order to study the effect it produces.

295

Firstly, consider the Kaldor-type business cycle model

K+l - yt = a(&- St) Kt+l = (1 - 6)Kt +It which depends on four variables, Y, K , I and S, representing, respectively, income, capital stock, investment and savings. We have also two parameters, Q: and 6, verifying the inequalities, Q! > 0 and 0 < 6 < 1. The parameter Q! represents the speed of reaction to the excess demand and 6 is the capital stock depreciation rate. The function describing investment is assumed to take the form of an increasing arc tangent type function of income, like suggested in Bischi et aL2 and Dieci et aL3. Since Kaldor5 concluded it might be sensible to assume that the S and I curves are nonlinear, in general, he assumed I = I(Y,K ) and S = S(Y,K ) , where investment and savings are nonlinear functions of income and capital. In order to introduce nonlinearity to the capital stock variable, instead of considering it just a linear decreasing function, we shall propose it as the sum of two functions: one linear decreasing and other a trigonometric function of the sin type. This sum, depending on the parameters, is monotone decreasing or has consecutive decreasing local maxima (or minima) when K increases (see Fig.1). Therefore, let ~t = op

+ y1 (y - ~ t + )arctan(& - p ) - psin(0.2 ~t

where the parameters o and 6 are such that 0 < 0 , 6 P, 71,P, 0 are positive (P, 71,P, 0 > 0).

I

-p),

(2)

< 1 and the parameters

I

Figure 1. Effect of the nonlinearity in the capital stock variable, K . In plot a) 2.0, B = 10, /I = 100 and in plot b) 71= 0.8, B = 9.5, p = 100.

=

296

Concerning the savings function, we also modify the assumption of proportionality to the income assuming, as in many versions of the Kaldor business cycle models, that savings are nonlinear. We consider that it depends also on the capital stock in a decreasing way:

St = a x

+ y2 (y- K t ) - psin(0.2 Kt - p ) ,

(3)

with y2 a positive parameter. Replacing expressions (2) and (3) in (1) and considering the particularly interesting case when y1 = y2 = y we get the following two dimensional system in income and capital stock variables:

K + l = yt + aap + aarctan(K - p ) - aaYt

{ Kt+l

= (1 - 6)Kt

+ up + y ( y - Kt) + arctan(K - p ) - psin(0.2 Kt - p )

(4) This paper is organized as follows: in Section 2 we present the model and the main results concerning triangular maps and the topological entropy. In Section 3 we explore its complex dynamic behavior. We also illustrate the computation of the topological invariant in an example and finally in Section 4 we make some final considerations. 2. The model and some considerations about triangular maps

Let us change the notation in the model (4) to: y := Y and k := K. We consider a family of two-dimensional continuously differentiable maps F : R2 + R2 given by

F:

- (;) (( ~ - d ) k + a py++y ( ? -+ka) arctan(y +arctan(y-p)-psin(0.2 =

aap

p)

aay

k-p) (5) where a,6, p, y,p, a are real parameters such that a > 0,O < a < 1, p > 0,o < 6 < 1,y > 0,p > 0. This dynamical system is generated by a skew map (triangular map) since it has the form F (y, k) = (f (y) ,g (y, k)), that is, the first component doesn’t depend on the second variable. So, from the economic point of view, the dynamics of income is only affected by income itself. The map f is called the basis map and g is called the fiber map. Due to this triangular structure it is possible to apply mathematical methods to compute relevant quantities that characterize the system as chaotic or non chaotic. In Fig.2 we present the graphical representation of the map F for some values of the parameters.

297 b)

a) 90 68.5

k

88

68

k

86 84

67.5

82 67 93

94

95

96

97

98

99

8o 94

100

Y

96

98

100 102 104 106

Y

Figure 2. Graphical representation of the map F for some values of the parameters. In plot a) a = 10, /I = 100, u = 0.3, 6 = 0.4, j3 = 9.5, 7 = 1.0 and in plot b) a = 10, /I = 100,u = 0.32, 6 = 0.4, j3 = 9.5, 7 = 0.8.

Let P = (20, 2 1 , ..., z p - l } be a periodic orbit of period p of the map for i = 0, ...,p - 2 and f ( ~ ~ - 1=)20. We define f such that f (xi)= the map gp : Y +Y as

gp ( y ) = 9 ( Z p - 1 , 9 (zp-2, ee.9

(21,

9 (207 Y ) ) .-))

-

(6)

If Q = { y o , y l , ..., yq-l} is a periodic orbit of period q of the map g p such that gp ( y i ) = yi+l for i = 0, ...,q - 2 and gp (yp--l) = yo, we can define the product P.Q as the set containing the p.q pairs:

(20,yo)

(z1,9(zo,yo))

*

*

*

(Z0,Yl)

( ~ l , g ( z O , Y l ) )*

*

*

(z0,yq-1)

(zl,g(Zo,Yq-l))

*

...1) ( ~ p - 1 , 9 ( ~ p - 2 , . . . 9 ( ~ 1 , 9 ( ~ O , Y l ) )...1) (~p-l,g(~p-2,...9(~1,9(~o,Yo))

*

...1)

(~p-l,g(2p-2,...9(~1,9(~o,Yq-l~~

The orbits of the one-dimensional maps f and gp determine the orbits of the triangular map T , as we show in the following Lemma: L e m m a 2.1. Let T = (f,9 ) : X x Y

+X

x Y be a continuous triangular

map. T h e n the following hold: (1) I f f has a periodic orbit P and gp has a periodic orbit Q , then P.Q i s a periodic orbit of T. (2) Conversely, each periodic orbit of T can be obtained as a product of a periodic orbit P o f f by a periodic orbit of gp.

*

298

Proof. See AlsedB-Llibre'.

0

The topological entropy is a measure of complexity of a dynamical system. Let T be a triangular map like defined in the earlier Lemma. The Bowen's formula for the inferior and superior values of the topological enis valid, that is, tropy of T , htop(T),

where h t o p ( f )and htop(gp)represent, respectively, the topological entropy of the basis map, f, and the topological entropy of the fiber map associated to the orbit P, g p . 3. Chaotic behavior in the map F

It is possible to prove the existence of chaotic behavior in the map F if there are parameter values that correspond to a positive value of the topological entropy. Let us consider first the basis map f. It is an one-dimensional map and it can be easily proved that f is a bimodal map when cra - 1 > 0. In this case there are two turning points, whose expressions are: c1=p-

JZ ~-

and c 2 = p +

JZ. ~-

The first one corresponds to the relative minimum and the second corresponds to the relative maximum. Fixing the parameters p = 100 and c = 0.3, f depends only on a which is the most interesting parameter from the economics point of view. In Fig.3 we present the long term behavior of the map f when a changes between 7 and 14. For a = 11.1726, f has an orbit, P = {yo, y1, 32, y3}, of period 4, such that f (f (c1)) = c2, described in Fig.4. The corresponding kneading data (ALBR,BRAL) generates a three interval Markov partition of the line y, which corresponds to the transition matrix:

[:1:I

M ( f )= 1 1 1

The maximum eigenvalue is A, = 2.41421 and the value of the topological entropy of the basis map is htop(f)= log (A,) = 0.881374....

299

92.5-

7

8

9

10

11

12

1

13

cx Figure 3. Bifurcation diagram of the basis map f as a function of a,with initial conditions ci and cz.

110-

95

100

Figure 4. Period-4 orbit of the map f, when

Y

105 (Y

= 11.1726, p = 100 and u = 0.3.

Let us consider now the points of the orbit P ordered like: yo = 92, ...;

91 = 101, ...;

313

= 107, ...;

34

= 98, ....

Consider also the function gm ( I c ) , depending on the variable Ic and defined, according to (6), by gpo

(L) = 9 (313, 9 (312, 9 (31, 9 (310, k))))

If we fix /3 = 9.5 and 6 = 0.52, the bifurcation diagram of gpo as a function of y, the free parameter, is presented in Fig.5. The study of this function can give us significant information about the long term behavior of the original map F . Let’s consider, for example, the case y = 0.8518. The function

300

65

0.4

0.6

1

0.8

1.2

Y

Figure 5. Bifurcation diagram of the map gpa (k) as a function of 7.

gpa (k), whose graphical representation is showed in Fig.6 is continuous, limited and has many critical points but its dynamics reduces to the values where there is an intersection with the diagonal line. If we iterate all the critical points of gpa (k) it is all reduced to an orbit of period 4 or to a fix point (see Fig.7).

70

65

60 55 50 45 t

20

I 40

. 60

k 80 1 0 0 1 2 0 1 4 0

Figure 6. Graphical representation of the map gPa (k) for 7 = 0.8518.

It is very interesting to notice what happens if we change the order of appearance of the points yo, y1, y2, y3, of the orbit P , in the construction of the function gP (k). We have three more possibilities. We can also define

301

Figure 7. A magnification of the graphical representation of the map g p o (k) for 7 = 0.8518.

gpl ( k ) , gpz ( k ) and gp3 ( k ) in the following way:

(k) = 9(Yo, Ll(Y3, 9(Y2, 9 ( Y l , k)))) g p 2 (k) = 9 (Yl, 9 (Yo, 9 (33, 9 (927 k)))) and g p 3 = 9 (32, 9 ( Y l , 9 (Yo, 9 (Y3, k)))). QPl

7

7

We obtain functions, all very different from each other, but the dynamic behavior of all these maps asymptotically goes to a period 4 orbit or to a fix point. In Fig.10 are shown the stable orbits for the considered parameter values. The periods of the orbits are in agreement with the considerations made in Section 2: the period of the orbit of F is equal to 16 or to 4 depending on the initial conditions. In each fiber we can see the orbit of period 4, in black, and the fix point, in grey. The first fiber corresponds to the stable orbits of the function gpo ( k ) and the others respectively to the stable orbits of functions gp3 ( k ) , gpl (k) and gpz ( k ) . The transition matrix is:

[:I

M(f)= 001 which corresponds to a topological entropy given by: htop(gpo)= log (1.83929...) = 0.609378....

302

Figure 8.

45

50

55

Graphical representation of the functions gpl and gP2.

6‘0

65

70

45

50

55

60

65

70

Figure 9. In plot a) graphical representation of the function g p 3 and in plot b ) graphical representation of the functions g p i , i = 0,1,2,3in the same plot.

The earlier value doesn’t depend on the function gpi , that is, doesn’t depend on the fiber. So we can conclude that the topological entropy of the original triangular map F is such that:

0.881374 ... 5 ht,,(F) 2 1.574521....

303

Y

Figure 10. The stable orbits of the triangular map F for a = 11.1726, @ = 9.5, d = 0.52 and 7 = 0.8515.

,u = 100,~7= 0.3,

The map F certainly has a value of the topological entropy that is positive, and therefore, the system is chaotic for these parameter values. 4. Final conclusions

In this paper we presented a Kaldor-type business cycle model described by a twedimensional map. The model is a modification of the one proposed in Bischi e t aL2 and Dieci e t aL3 and can be reduced to a map of triangular type with income and capital stock as variables. Due to its triangular structure it is possible to compute the topological entropy, for certain values of the parameters, as a measure of complexity of the system. The introduction of nonlinearity in the capital stock variable, into the investment and savings functions allows us to have chaotic basis map and chaotic fiber map, since we found a positive value for the topological entropy in both cases.

References 1. L. Alsedi; J. Llibre; Periods for triangular maps, Bull. Austral. Math. SOC. (1993), no. 47, 41-53. 2. GI. Bischi; R. Dieci; R. Rodano; S. Saltari; Multiple attractors and global bifurcations an a Kaldor-type business cycle model, J. Evol Econ (2001), no. 11, 527-554. 3. R.Dieci; GI. Bischi; L. Gardini; From bi-stability to chaotic oscillations in a macroeconomic model, Chaos Solitons Fkactals 21 (2004), no. 2, 403-412. 4. D. Mendes; J. Sousa Ramos; Kneading theory for triangular maps, Int. J. Pure Appl. Math. 10 (2) (2004) 421-450. 5. N. Kaldor; The model of the trade cycle, Economic Journal, 50 (1940), 78-92.

304

6. J. P. Lampreia; J. Sousa Ramos; Symbolic dynamics of bimodal maps, Portugal. Math. 54 (1997), no. 1, 1-18. 7. C. Mira; L. Gardini; A. Barugola; J.-C. Cathala; Chaotic dynamics in twodimensional noninvertible maps. World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 20. World Scientific Publishing Co., Inc., River Edge, NJ, 1996. xiv+607 pp. 8. J. Milnor; W. Thurston; On iterated maps of the interval. Dynamical systems (College Park, MD, 1986-87), 465-563, Lecture Notes in Math., 1342, Springer, Berlin, 1988. 9. T. Puu; Attractors, bifurcations, and chaos. Nonlinear phenomena in economics. Springer-Verlag, Berlin, 2000. xii+507 pp.

305

MULTISCALE EXPANSION OF THE LATTICE SINE-GORDON EQUATION

XIAODA JI Department of Mathematics, University of Science and Technology of China, Hefei, Peoples Republic of China E-mail: [email protected] D. LEV1 Dipartimento d i Ingegneria Elettronica, Universitd degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84,00142 Roma, I T A L Y E-mail: [email protected]

M. PETRERA Dipartimento di Fisica E. Amaldi, Universitd degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84,00142 Roma, I T A L Y E-mail: [email protected]

We expand a discrete-time lattice sine-Gordon equation on multiple lattices and obtain the partial difference equation which governs its far field behaviour. Such reduction allow us to obtain a new completely discrete nonlinear Schroedinger (NLS) type equation.

1. Introduction Reductive perturbation techniquel0)l1 has proved t o be an important tool to find approximate solutions for many important physical problems by reducing the given nonlinear partial differential equation in the far field often t o an integrable one. Recently this approach has been extended t o the case of equations living on lattices6y8. Here we apply it t o the case of a discrete-time lattice sine-Gordon equation. In Section 2 we briefly describe the discrete perturbation technique and in Section 3 we apply it to a discrete-time lattice sine-Gordon equation.

306

2. The discrete perturbation technique

The aim of this section is to fix the notation and to introduce the formulae necessary to reduce lattice equations in the framework of the discrete reductive perturbation technique6>*. Given a lattice defined by a constant spacing h, we will denote by n the running index of the points separated by h. In correspondence with the lattice variable n , we can introduce the real variables x = h n . We can define on the same lattice a set of slow varying variables by introducing a large integer number N , defining a small parameter E = N-l and requiring that nj =

n.

(1)

This correspond to sampling points from the original lattice which are situated at a distance of N j h between them. If we set them on a lattice of spacing h, the corresponding slowly varying real variables xj are related to the variable x by the equation xj = d x. Let us consider a function f = f n defined on the points of a lattice variable n and let us assume that f n = g n l , n Z , . . . , n Ki.e. , f depends on a finite number K of slow varying lattice variables nj j = 1 ,2 ,. . . ,K defined as in eq. (1). We are looking for explicit expressions for, say, fn+l in terms of gnl,nz,...,n K evaluated on the points of the n l , 712, . . . , nK lattices. At first let us consider the case, studied by Jordan4, when we have only two different lattices, i.e. K = 1. Using the results obtained in this case we will then consider the case corresponding to K = 2. The general case will than be obvious.

I) K = 1 (fn = g n l ) . In Jordan4 we find the following formula:

Here the coefficients P ( i ,k) are given by i

a=k

where w is the ratio of the increment in the lattice variable n with respect to that of variable nl. In this case, taking into account eq. (l),w = N . The coefficients Sr and (5; are the Stirling numbers of the first and second kind respectively'. Formula (2) allow us to express a difference of order k

307

in the lattice variable n1 in terms of an infinite number of differences on the lattice variable n. The result (2) can be inverted and we get:

k! C 7a . Q ( i k, ) Aignl i=k cm

Ak fn =

(4)

where the coefficients Q(i,k ) are given by (3) with w = N-l = c. To get from eq. (2) a finite approximation of the variation of gnl = fn we need to truncate the expansion in the r.h.s. by requiring a slow varying condition for the function f n . Let us introduce the following definition:

Definition 2.1.

fn

is a slow varying function of order p iff AP+l

fn

= 0.

From Definition (2.1) it follows that a slow varying function of order p is a polynomial of degree p in n. From eq. (2) we see that if fn is a slow varying function of order p then AP+l gnl = 0, namely gnl is also of order p . Eq. (4)provide us with the formulae for fn+l in terms of gnl and its neighboring points in the case of slow varying functions of any order p . Let us write down explicitly these expressions in the case of gnl of order 1 and 2. 0

p = 1. Formula (4) reduces to

1 Afn = Z A g n l , i.e.

reads

fn+l

fn+l = gnl 0

1

+ -(gnl+l N

-gnl)

+O(N-~).

p = 2. From eq. (4) we get

and thus

fn+l

reads

fn+l = gnl

1 + -(-gn1+2 2N

1 +-(gn1+2

+ 4gn1+1 - 3gn1) +

- 2gn1+1+ gn1)

+ O(N-3).

(5)

In the next section we will consider the reduction of an integrable lattice sine-Gordon equation. It is known1' that a scalar differential-difference equation can possess higher conservation laws and thus be integrable only if it depends symmetrically on the discrete variable. The results contained in (4) do not provide us with symmetric formulae. To get symmetric formulae we take into account the following remarks:

308

( 1 ) Formula ( 2 ) holds also if h is negative; ( 2 ) For a slow varying function of order p , we have A P f n = A p fn+e, for all C E Z.

When f n is a slow varying function of odd order we are not able to construct completely symmetric derivatives using just an odd number of points centered around the n1 point and thus f n k l can never be expressed in a symmetric form. From eq. (4), using the above remarks we can easily construct the symmetric version of (5). We get: 1 1 fn+l = g n l + -2(N( ~ n l + l - ~ n l - l ) + ~ ( ~ n l + l - 2 ~ n+lg n l - 1 ) + O ( N - 3 ) .

(6) 11) K = 2 ( f n = g n l , n z ) . The derivation of the formulae in this case is done in the same spirit as for the symmetric expansion presented above, resulting in eq. ( 6 ) . Let us just consider the case when p = 2 , as this is the lowest value of p for which we can consider f n as a function of the two scales nl and 722. From eq. (4)we get:

Qnl,nz+l = gn1,nz

+ N2A2fn,n+ s1N 2 ( N 2- l)Agfn,n.

(8)

Here the symbols A1 and A2 denote difference operators which acts on the first and respectively on the second index of the function f n , n G gnl , n 2 , e.g. Alfn,n fn+l,n - f n , n . and A 2 f n , n fn,n+l - f n , n . Let us now consider a function gnl,nz where one shifts both indices by 1. From eq. (7), taking into account that, from eq. ( l ) ,for example, gnl+l,nz = fn+N,n, one has: 1 gni+l,nz+l = gni,nz+l+ NAlfn,n+NZ + - N ( N - l ) A f f n , n + N Z . (9) 2 Using the result (8) we can write eq. (9) as 1 Qnl+l,nz+l - gn1,nz N2A2fn,n 5 N 2 ( N 2- l)Aifn,n

+

+

1

+NAlfn,n + N3A1A2fn,n+ 5 N 3 ( N 2- l)AlAifn,n 1 + - N ( N - 1 ) A f f n , n N 3 ( N - l)A;A2fn,, 2 1 + - N 3 ( N 2 - 1 ) ( N - 1 ) A q A ifn,n. 4

+

(10)

309

As, using the second remark, the second difference of fn,n depends just on its nearest neighboring points, the right hand side of eq. (10) depends, apart from fn,n = 9n1,nz, on fn,n+l, fn,n-1, fn+l,n, fn-l,n, and f n - ~ , n - ~ ,i.e. 8 unknowns. Starting fn+1++l, fn+1+-l, f n - - l , n + ~ , from eqs. (7), (8) and (10) we can write down 8 equations, using the first remark, which define gni 9 gnl - 1,122 7 gni ,nz+l>gni , n z - 17 gnl+1 ,n2+19 gnl+l,nz-l, 9n1-l,nz+l, and 9n1-1,nz-1 in terms of the flmtions fn+i,n+j with (i,j ) = 0, f l . Inverting this system of equations we get fn*l in term of gnl,nz and its shifted values:

It is worthwhile to notice that the two lowest order (in N-l) terms of the expansion (11) are just the sum of the first symmetric differences of gnl and gnz. Thus in the continuous limit, when we divide by h and send h to zero in such a way that x = h n, x1 = h nl and 2 2 = h n2 be finite, we will have f,x = egxl + e 2 gxz. Extra terms appear at the order N-3 and contain shifts in both n 1 and 7x2. When fn is a slow varying function of order 2 in n 1 it can also be of order 1 in 712. In such a case eq. (8) is given by

Starting from eqs. (7), (12) and a modified (10) we can get a set of 8 equations which allows us to get fn*l in terms of gnl,nz and its shifted values. In such a case fn*l reads fnkl

= 9n1,nz

*

1 %(9n1+l,n2

- 9n1-1,nz)

1

+ &9n1,nzfl

- 9n1,nz)

It is possible to introduce two parameters in the definition of n1, 722 in terms of n. Let us define n1 = (n M 1 ) / N , 122 =& (n M2)/N2, where MI and M2 are divisors of N and N2 so that n1 and 722 are integers numbers. In

310

such a case eq. (11) reads

M?

- 2gn1,nz

+*(9n1+1,nz

+- 4N3

M 1 M2

(gnl+l,nz+l

+ Snl-l,nz)

- gnl-l,nZ+l

- gnl+l,nz-l

+ gnl-1,nz-l)

+o(N-~) and eq. (13) accordingly. Let us consider the case of two independent lattices and a function f n , m defined on them. As the two lattices are independent the formulae presented above apply independently on each of the lattice variables. So, for instance, the variation fn+l,mwhen the function f n , m is a slowly varying function of order 2 of a lattice variable nl reads fn+l,m

1 2N

= gn1,m + --(gn1+1,m

1 +-(gnl+l,m 2 N2

- Snl-l,m) - 2gn1,,

+ gnl-l,m) + o ( N - ~ ) .

A slightly less obvious situation appears when we consider f n + l , m + las , new terms will appear, see Levi et a1.638for the formulae in this case. 3. Reduction of the lattice sine-Gordon equation A discrete analogue of the sineGordon equation is given by the following nonlinear PAE': 1

'Iln+l,m+l

=-

%+l,rn %,m+1

- P4

%wl 1 - q4 %+l,m 'Iln,m+l'

This equation involves just four points which lay on two orthogonal infinite lattices and are the vertices of an elementary square. When written in polynomial form has quartic nonlinearity. In eq. (14) un,mis the dynamical (real) field variable at site ( m , n ) E Z x Z and p , q E IR are the lattice parameters. These are assumed different from zero and will go t o zero in the continuous limit so as t o get the continuous sine-Gordon equation. To get a nonlinear dispersion relation we carry out the change of variable u ~H p, / q + ~ u ~ ,The ~ linear . part of the resulting equation is given by: (0- l ) ( u n , m

where o = p 2 q 2 .

+ 'Iln+l,m+l) + (0+ l ) ( % + l , m + U n , m + l )

= 0,

(15)

311

The general solution of eq. (15) is written as a superposition of linear waves En,m = exp[i( k n - w ( k )m)] = zn 0". The dispersion relation for these linear waves is given by

0 = e-'w = - (u + 1)z + o - 1 (o-l)z+u+l'

(16)

namely 2usink

w = - arctan (02

+ 1)cos k +

02

-1

From eq. (16), by differentiation with respect to k, we get the group velocity: W,k

=-

+

[(o 1)z

+

4uz o - 1][(o- 1)z

+ o + 11'

(17)

We now look for real solutions of the nonlinear equation (14) written as a combination of modulated waves:

s=o

s=1

where the functions @tLare slowly varying functions on the lattice, i.e. and €7 = N - l . By 6 we mean the complex conjugate of = a complex quantity b so that, for example, The positive numbers ps are to be determined in such a way that :

q!~kL= $e),ml,mz

(1) p1 5 ps V s = 0 , 2 , 3 , .. . ,oo.In general it is possible to set 81 = 1. (2) In the equation for +tk = $ J ~ the , ~ lowest , order nonlinear terms should match the slow time derivative of the linear part after having solved all linear equations. This will provide a relation between y and the pS. Introducing the expansion (18) in the PAE obtained from equation (14) after the change of variable un,m c)p/q+u,,,, we analize the coefficients of the various harmonics (En,,)' for s = 1, s = 2 and s = 0 and, as assuming that ps increases with s, the nonlinear terms will depend only on the lowest s terms, we came t o the conclusion that we can choose y = 1,p0= 2 , p s = s, s 2 1. The discrete slow varying variables n1, ml and m2 are defined in terms of n and m by:

. Mln n1 = -,

. Mzm m1=- N ,

. n N 2'

m2=-

312

For s = 1 we get, at the lowest order in E , a linear equation which is identically solved by the dispersion relation (16). At c2 we get the linear equation

+ + 11(+nl+l,ml,mz - + n l - l , m l , m z ) + M2 0 [(u- 1)z + 0 + 11(+nl,ml+l,mz - +n1,m1-1,m2) = 0 , (19) nz = nl - m l , provided whose solution is given by IJlnl,ml,mz = 4n2,m2, MlZ

[(u- 1) R

that the integers MI and MI

=

M2

are choosen as

s R [(u- 1)z + u + 11,

M2

= s z [(u- 1)52

+ u + 11,

(20) where S E C is a constant. Defining S = pexp(i6), p E R+ and -7r 5 6 < T , we can choose 6 and p in such a way that M I is an integer number:

6 = - arctan

p = (-l)%1

(a

(u + 1) sin k + 1 ) c o s k t u - 1]+h,C E Z ,

1

- [(uZ- 1) cosk + uz + 1]-1/2.

Jz

The request that also M2 is an integer impose a constraint on k as M2 = u,kM1, i.e. u , E ~ Q, see eqs. (17) and (20). Let us notice that also n2 = 711 ml solves eq. (19) by an appropriate choice of M I and M z . At c3 we get a nonlinear equation for q5nz,mz which depends on and

+ $JZ),~~:

+ c1(4n2+2,mz + 4 n z - 2 , m z - 2 4n2,mz> + c2 ( 4 n z + l , m z + 4nz-1,mz - 2 4 n 2 , m z ) + c3 4nz,mzl4nz,m2l2 4 n z , m z + l - 4n2,mz

+ c4 @gm2 4nz,m, + c5 +g),mz 8nz,mz = 0 ,

(21)

where the ci’s, 1 5 i 5 5, are known coefficients depending on z , S and the lattice parameters p, q . The functions and that appear in equation (21) are obtained considering the equations for the harmonics s = 0 , at the third order in 6 , and s = 2 a t the second one. We get:

+g),mz @it),mz

+it),mz

Inserting given by eqs. (22) in equation (21) we obtain and the following nonlinear lattice equation: i (4nz,mz+l - 4n2,mz) + 21 (4n2+2,mz

+ 2 2 (4nz+l,mz + 4nz-1,mz

- 24nz,mz)

+ 4nz-2,mz

- 2 4nz,mz)

+ e 3 4nz,mzl4nz,mzI2 = 0, (23)

313

where the coefficients &’s, 1 z = exp(i k). They read:

5 i 5 3, can be computed also recalling that

.M ~ ( a - l ) [ ( a + l ) ( c o ~ k + i s i n k ) + a + l ]

c1 = 1 c2

c3

16 a . M i (a - 1)[(0 1) C O = -1 4a 2 q4 (a2- 1 ) sin3 k = [(a2- 1 ) c o s I c + a 2 + 1 ) ] ’

+

7

+ + 13

S ~

(T

7

The coefficients (24) depend just on the integer constant M z . Taking into account that +nz,mz is a slow varying function of order 2 in n2 we can, using the remark 2, substitute 4n2+2,m2 by 3+n2+l,mz 3 4nz,ma + +nz-l,mz and 4 n z - 2 , m z by 3 +nz-l,mz - 3 +nz,mz + 4nz+l,mz. In such a way eq. (23) becomes

where 4c1

+ E2 = - M; (a24-0 1)sin k

The P A E (25) is a completely discrete and local NLS equation depending just on neighboring lattice points. At difference from the Ablowitz and Ladik discrete NLS, the nonlinear term in equation (25) is completely local. The P A E (25) has a natural continuous limit when m2 + co and 722 + 00 which, as the coefficients (26) and 23 are real, is just the well known integrable NLS equation. Acknowledgments

The author D.L. thanks the Department of Mathematics, University of Science and Technology of China, Hefei (China) for its kind hospitality and the ICTP (Trieste, Italy) which made its visit to China possible by providing a Visiting Scholars/Consultants fellowship. D.L. was partially supported by PRIN Project “SINTESI-2004” of the Italian Minister for Education and Scientific Research and from the Projects Sistemi dinamici nonlineari discreti: simmetrie ed integrabiliti and Simmetria e riduzione di equazioni differenziali di interesse fisico-matematico of GNFM-INdAM.

314

References 1. M.J. Ablowitz and J.K. Ladik, Stud. Appl. Math. 5 5 , 213 (1976). 2. M. Abramowitz and I.A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables (Dover Publications, Inc., New York, 1992). 3. F. Calogero and W. Eckhaus, Inverse Problems 3,L27 (1987). 4. C. Jordan, Calculus of finite diflerences (Rottig and Romwalter, Sopron, 1939). 5. J. Leon and M. Manna, Jour. Phys. A: Math. Gen. 32,2845 (1999). 6. D. Levi, Jour. Phys. A: Math. Gen. 38, 7677 (2005). 7. D. Levi and H. Heredero, Jour. Nonlinear Math. Phys. 12,suppl. 1,440 (2005). 8. D. Levi and M. Petrera, math-ph/0510084, submitted t o Jour. Math. Phys.. 9. R. Sahadevan and H.W. Capel, Phisica A 330,373 (2003). 10. T. Taniuti, Prog. Theor. Phys. 55 , 1 (1974). 11. T. Taniuti and K. Nishihara, Nonlinear waves (Monographs and Studies in Mathematics, Pitman, Boston,1983). 12. R. Yamilov, review article in preparation. 13. V.E. Zhakarov and E.A. Kuznetsov, Physica 18D, 455 (1986).

315

STABILITY OF DELAY DIFFERENCE AND DIFFERENTIAL EQUATIONS: SIMILARITIES AND DISTINCTIONS

M. M. KIPNIS AND I. S. LEVITSKAYA* Chelyabinsk State Pedagogical University, 69 Lenin Avenue, Chelyabinsk 4 54 080, Russia E-mail: kipnis Omail.ru

We compare the stability domains in the space of the parameters for the pare of differential and difference equations k ( t ) = A z ( t - 7)and Z n - Zn-1 = Azn-k in Rm, as well as the pair of scalar equations with two delays k ( t ) = a z ( t - 71) b z ( t - 7 2 ) and xn - xn-l = b x n-k.

+

+

1. Equation k ( t ) = a z ( t - 7 )

Equation Zn - Zn-1 = U Z n - k Euler - Levin and May (1976) V8

It is well known that the zero solution of delay differential equation

k ( t ) = U z ( t - T)

(1)

is asymptotically stable if and only if

0 > a7

7r

> --.2

(2)

Probably L. Euler knew this result. Related difference equation is the following: zn

- zn-1

= a zn-k,

(3)

where a E R, k E N. The zero solution of equation (3) is asymptotically stable if and only if O>a>-2sin

7r

2(2k - 1) *

(4)

*Work is partially supported by grant 04-01-96069 of the Russian Foundation of Basic Research.

316

The result has been obtained by Levin and May [l].Let us compare the domains (2) and (4). If k tends to infinity, the inequality (4)transforms A A into 0 > a > --, or, equivalently, 0 > a k > -2k 2' Consider the equation

X(t) = a x ( t - 1)

(5)

with the asymptotic stability domain A

O>a>--. 2 Let us apply the Euler method for ( 5 ) . Let t = nh, x ( n h ) = xn. Let

X(t) N-

x ( t ) - x ( t - h) h

1 Let h be a unit fraction: h = -, k E k

xn

- xn-1

(6)

(7)

N. Then =

a

xn-k.

(8)

According t o the Levin-May results we get the following asymptotic stability domain of (8):

0 > a > -2ksin

A

(9)

2(2k - 1)'

We compare the stability domains (6) and (9) of difference equation (8) and differential one ( 5 ) in the following table. We have seen that the stability Table 1. The length of the stability domains for (8) compared with k 2k sin &-q [2ksin .&I

2

1

-4

3

4

5

4.

10

2

2

1.854

1.780

1.736

1.652

0.429

0.429

0.283

0.209

0.166

0.081

domain of difference equation (8) is more wide than that of the differential equation ( 5 ) . If k + 00 then the stability domains of ( 5 ) and (8) coincide. 2. System & ( t )= A s ( t

-T)

System x n - xn-1 = A5n-k Rekhlitskii (1956) - Levitskaya (2005) US

Z. Rekhlitskii [2] considered the delay system of differential equations

* ( t )= A x ( t - T ) ,

(10)

317

where A is a m x m matrix, x(t) : [0,00)+ Rm, T > 0. He constructed an oval in the complex plane, such that the equation (10) is asymptotically stable if and only if any eigenvalue of the matrix A lies inside the oval. The boundary of the oval is a curve

The oval has been rediscovered by many later writers [3-41.Levitskaya [5] explored a similar system of delay difference equations

where A is a m x m real matrix, xn : N + Rm. A positive integer k is a delay. The stability oval for the system (12) is the following:

z = (a

+ ip) E c : a = -

p = 2sin-

2 s i n L sincp; 2k - 1

-lr

2k - 1

To explain the interaction between ovals (11)and (13),consider the system of differential equations

k ( t ) = A ~ (-t1).

(14)

Let us use the Euler method. Similarly t o (8) we put

Equation (15) is asymptotically stable if and only if all eigenvalues of the matrix A lie inside the oval bounded by the curve z = (a +ip) E C : a = - 2 k s i n L sincp;

2k - 1

-lr

p=2ksinLcoscp; 2
i}.

If k + 00 then rz (the dotted line in Fig. 1) tends t o the Rekhlitskii oval rl with T = 1.

318

Figure 1. The stability ovals (16) (dotted line, k = 2) and (11) (solid line, T = 1) for (15) and (14).

3. Equation $ ( t ) = a x ( t - 7 1 ) 4-b x ( t - 7 2 ) US Equation X n - X n - i = a X n - m bXn-k Levitskaya (2005) - Kipnis and Levitskaya (2005)

+

3.1. Preliminaries The difference equation presented in the title of section 3 presents the natural completion of the following series of equations: Levin and May [l]1976, z, = zn-l b 2 n - k z, = az,-l b z n - s Kuruklis [6] 1994, Papanicolaou [7] 1996, Dannan and Elaydi [8] 2001, Z , = az,-k+l bx,-k z, = a zn-, 2(a - 1) xn-k Kipnis and Nigmatulin [9] 2003, Nikolaev [lo] 2002, Dannan [ll]2004, Kipnis and bx,-b z, = Nigmatulin [12] 2004, again Nikolaev [13] 2004. All these equations were investigated by exact mathematical methods. For any equation of this list we know formulas, which give the boundaries of stability domains in the (a,b) plane for any delays. The situation changes drastically in the following case:

+ +

+ + +

Z , = Zn-l

4-a x n - ,

bXn-k.

(17)

More exactly, we know that the boundary of the stability domain of this equation consists of the curve

a=

2 sin 5 cos y w 2 sin f cos w , b= sin(k - m)w sin(lc - m)w

and certain straight lines. The parametric equation of the curve was found by the D-decomposition (parameter plane) method [14]. However, the above formula has the essential feature: we do not know the boundaries

319

of the intervals of the parameter w . So any curve in Fig. 2 and in the right-hand side of Figs. 2-5 has the equation written above (except of the straight lines). However we cannot find theoretically the intervals within which the variable w changes. Thus the general statements about the behavior of the stability domains are deducted from numerical experiments, rather than theoretical investigations. The situation with the differential equation corresponding with the difference one is similar. This equation has the form

It has been studied by Levitskaya [15] (see also [16-181). For the differential equation we have got the curve which gives the boundary of the stability domain: a=

w cosr2w --WCOS~1W b= sin(72 - r1)w’ sin(r2 - r1)w

*

In (17) and (18) z(t): [0, 00) + lR, TI > 0,72 > 0, a, b E lR, m, k E N. We suppose that r1 < 72 and m < k. And again we cannot find theoretically the admissible intervals for the variable w , and the general statements about the behavior of the stability domains are deducted from numerical experiments, rather than theoretical investigations.

3.2. Results

It appears that for the differential equation (18) the shape of the stability domain changes when the ratio of the greater delay 7-2 to the smaller one 71 passes through the numbers 1,3,5,7.. . Similarly, for the difference equation (17) the shape of the stability domain changes when the greater delay k passes through the values m, 3 m - 1,5771 - 2,7m - 3.. . (2r l ) m - r . . . The dynamics of the of the stability domains of (18) and (17) is shown in Figs. 2-5. Let us denote the area of the stability domain in plane (a,b ) as S ( r 1 , r Z ) and S(m,k) for the equations (18) and (17) accordingly. The

+

7-2

k

71

m

least value of S is achieved when - = - = 1.817.. . In case of difference equation it is true when k,m are sufficiently large. In Fig. 6 the least value

k

8

of S is achieved when - = - = 1.6. m 5

320

7 1 < 7 2 < 3r1 m < k < 3m- 1

7-2 = 371

k=3m-1

371

< 7-2 < 57-1

3m- 1 < k < 5m-2

571 < 7-2 < 771 5m-2 < k < 7 m - 3

7-2 = 571

k=5m-2

Figure 2. The shapes of the stability domains for equations (17),(18).

3.3. Explicit Asymptotic Stability Conditions Gopalsamy (1 990), Vaguina and Kipnia (2003), Cook and Gyori (1990), Kipnis and Levitskaya (2005) Here the explicit sufficient asymptotic stability conditions are indicated for the equations in question. K. Gopalsamy [17] proved the sufficiency of the inequalities a < 0, b < 0 , 0 > a71 b-rz > -1 for the asymptotic stability of the differential equation (18). Later M.Yu. Vaguina and M. Kipnis n [18] have replaced 1 by -. On the other hand Cook and Gyori [19] have 2 asserted that the conditions a < 0, b < 0 , 0 > a(m - 1) b(k - 1) > -1 are sufficient for the asymptotic stability of difference equation (17). Our numerical experiments show that the latter condition may be replaced by n 0 > am + bk > --. If m > 2, k > 2 then this condition guarantees the 2 greater stability domain.

+

+

3.4. Summary

There are obvious SIMILARITIES in the stability domains of differential equation (18) and difference one (17).

321 .

4

.

.

b

.

.... . . . . . . . . . . .

.....

a .... .... .....

-0.1 . . . . . .

-8

-10

-6

-4

.

-2

2

0

--.-

4

-0.5-0.4-0.3-0.2-0.1

0

0.1

0.2

............... .

.

.

a .

.

.

.

.

.

-0.2 4.5-0.4-0.3-0.2-0.1

. m =8 . .

0

-2

.

.:

.

-4 -10

-8

. -6

0

.

.

. ..

. .

0.1

0.2

1

. .

.

....

.

.. . . . .. . . . . . . . . . . . ... , .

...

-4

-2

0

2

..

..

.

.

-0.2 -0.5-0.4-0.3-0.2-0.1

4

0

0.1

0.2

<

Figure 3. Stability domains of (18) with n < n 371 (left-hand side) and (17) with m < k 3m - 1 (right-hand side). Special cases: 7 2 = 371 and k = 3m - 1.

<

(1) We see the similar shapes of the domains with the following relations between the delays: 371 > 72 > 71 571 > 72 > 371 771 > 7 2 > 571.. . 3 m - 1 > k > m 5 m - 2 > k > 3 m - 1 7 m - 3 > k > 5 m - 2 ... (2) We see a particular contours (containing a n additional straight line besides the line a b = 0) of the stability domains with the special relations between two delays: 72 = 71 72 = 371 7 2 = 571 7 2 = 771 ... k = m k = 3 m - 1 k = 5 m - 2 k = 7 m - 3 ...

+

(3) If we fix the greater delay

72

in differential equaion, then the min-

322

a -0.1 . . . . .: . .

-0.2,-

" 1

..:. IIn

...

n-

-.

_.

~

n^

.....

a .....

. -0.2

.

'

-0.5-0.4-0.3-0.2-0.1

b

4,

0.1

0.2

1

I

7

0

=0.2 a

-10

:...

-0.1 . . . .

-A -2

-8

-6

-4

-2

0

2

4

-0.5-0.4-0.3-0.2-0.1

0

0.1

0.2

<

Figure 4. Stability domains of (18) with 371 < TZ 571 (left-hand side) and (17) with 3m - 1 < k 5m - 2 (right-hand side). Special cases: 7 2 = 571 and k = 5m - 2.

<

imal area of the stability domain is achieved when r1 = 0.5504~~. Similarly, if we fix the greater delay k in difference equaion, then the minimal area of the stability domain is achieved when m = 0.5504k (with the large k). However we see some DISTINCTIONS. (1) If we fix the greater delay 7-2 in differential equation, then all the history of the stability domains development may be written by changing the least delay 7-2 from 7-1 t o 0. Contrarywise, if we fix the greater delay k in difference equation, then some stages of the history disappear. So, if 7-2 = 1, then we can easily find the value of 7-1, such that 771 = 7-2. However, if k = 23, there exists no m E N, such that k = 7m - 3. So, there exists no special case k = 7m - 3 for (17) with k = 23.

323

a

a

-2 -4

-12 -10

-8

-6

-12 -10

-8

-6

-4

-4

-2

-2

0

2

0

4

2

4

<

Figure 5. Stability domains of (18) with 571 < 7 2 771 (left-hand side) and (17) with 5m - 2 < k 6 7m - 3 (right-hand side). Special case for (18): 72 = 771. There is no special case k = 23 = 7m - 3 for (17).

0,35

5

0,30 0,25 0,20 0,15 0,10 0,05

m 1 2 3 4 5 6 7 8

Figure 6. The area of the stability domain S as the function of the least delay m. The greatest delay k is equal to 8.

(2) In the exclusive case m = 1 the stability domain contour depends on whether k is odd or k is even. We failed in searching of analogous

phenomenon in differential equations. (3) In the other exclusive case m = 0, according t o results of Kuruklis [6],the stability domain of the difference equation (17) consists of two simply connected domains, while the stability domain of the differential equation (18) is simply connected with 71 = 0.

324

References 1. S.A. Levin and R. May, A note on differencedelay equations, Theor. Pop. Biol. 9, 178-187 (1976). 2. Z. I. Rekhlitskii, On stability of solutions of some linear differential equations in Banach space, DAN USSR 111, 29-32 (1956). 3. T. Mori and E. Noldus, Stability criteria for linear differential difference systems, Int. J. Systems Sci. 15, 87-94 (1984). 4. M. BusIowicz, Simple stability criterion for a class of delay differential systems, Int. J. Syst. Sci. 18, 993-995 (1987). 5. I.S. Levitskaya, A note on the stability oval for zn+l = zn Azn-k, J. Diflerence Equ. Appl. 11, 701-706 (2005). 6. S. Kuruklis, The asymptotic stability of z(n 1) - az(n) bz(n - k ) = 0 , J . Math. Anal. Appl. 188, 719-731 (1994). 7. V.G. Papanicolaou, On the asymptotic stability of a class of linear difference equations, Mathematics magazine 69, 34-43 (1996). 8. F.M. Dannan and S. Elaydi, Asymptotic stability of linear difference equation of advanced type, htpp://www.trinity.edu/ departments/ mathematics (2001). 9. M.M. Kipnis and R.M. Nigmatulin, Stability of some difference equations with two delays, Autom. and Remote Control 64, 782-790 (2003). 10. Yu.P. Nikolaev, On studying the geometry of the set of the stable polinomials of the linear descrete systems, Autom. and Remote Control 63, 1080-1088 (2002). 11. F.M. Dannan, The asymptotic stability of z(n k ) az(n) bz(n - I ) = 0 , J. Diflerence Equ. Appl. 10, 589-599 (2004). 12. M.M. Kipnis and R.M. Nigmatulin, Stability of the trinomial linear difference equations with two delays, Autom. and Remote Control 65, 1710-1723 (2004). 13. Yu.P. Nikolaev, The geometry of D-decomposition of a two-dimensional plane of arbitrary coefficients of the characteristic polinomial of a descrete systems Autom. and Remote Control 65, 1904-1914 (2004). 14. E.N. Gryazina and B.T. Polyak, Stability regions in the parameter space: D-decomposition revisited, Automatica 42, 13-26 (2006). 15. I.S. Levitskaya, Stability domain of a linear differential equation with two delays, Computers and Math. with Appl. (2006, t o appear). 16. M.Yu. Vaguina, Linear stability of the delay logistic equation, Appl. Math. Letters 17, 1069-1072 (2004). 17. K. Gopalsamy, Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic Publishers (1992). 18. M.Yu. Vaguina and M.M. Kipnis, Stability of a zero solution of a differential equation with delays, Math. Notes 74, 740-743 (2003). 19. K.L. Cook and I. Gyori, Numerical approximation of t h e solutions of delay differential equations on an infinite interval using piecewise constant arguments, IMA Preprint Series, 633 (1990).

+

+

+

+ +

+

325

COMPUTER ALGEBRA METHODS FOR ORTHOGONAL POLYNOMIALS

W. KOEPF University of Kassel Dept. of Mathematics and Computer Science Heinrich-Plett-Str. 4 0 34132 Kassel, Germany E-mail: [email protected]

In this article we will show how computer algebra can be used in the study of orthogonal polynomials and special functions. The classical orthogonal polynomials named after Jacobi, Gegenbauer, Chebyshev, Legendre, Laguerre, Hermite and Bessel can be classified as the polynomial solutions of second order differential equations. Similarly the classical “discrete” orthogonal polynomials named after Hahn, Krawtchouk, Meixner and Charlier are classified as the polynomial solutions of second order difference equations. Using computer algebra one can compute the recurrence equations and hypergeometric representations of these systems, one can convert this process by computing differential and difference equations from the hypergeometric representations automatically, and one can decide whether a recurrence equation has classical orthogonal polynomial solutions. We will discuss these and related algorithms, and give some demonstrations with Maple.

1. Demonstrations with Computer Algebra I will use the computer algebra system Maple to demonstrate and program the algorithms presented. Of course, we could also easily use any other general purpose system like Mathematica, MuPAD or Reduce. Internally the following algorithms are most prominently used: linear algebra techniques, multivariate polynomial factorization and the solution of nonlinear equations, e . g. by Grobner basis techniques2. As an appetizer we consider the conversion between a recurrence equation and a difference equation. In this article a difference equation is an equation involving the forward difference operator

326

We ask the question: How can one convert a linear recurrence equation up f(x

+ p ) +. . . +

a1

f(x

+ 1)+ a0 f(x) = 0

(1)

(involving the shift operator) to a difference equation (involving the forward difference operator)? Here is a Maple worksheet for this purpose. First we load the LREtools package >

with(LREtoo1s) ;

[AnalyticityConditions, HypergeometricTerm, IsDesingularizable, REcontent, REcreate, REplot, REprimpart, REreduceorder, R E t o D E , REtodelta, REtoproc, ValuesAtPoint, autodispersion, constcoeffsol, dAlembertiansols, 6, dispersion, divconq, firstlin, hypergeomsols, polysols, ratpolysols, riccati, shift] Let us start with the left hand side of the recurrence equation n fn+2 ( n - 1)( n 1)f n + 1 f n = 0 of type (1): > RE:=n*f (n+2)-(n-l)*(n+l)*f (n+l)+f (n) ; RE := n f ( n 2) - ( n - 1)( n 1)f(n 1) f(n)

+

+

+

+

+ +

+

The desired conversion rewrites the highest term up f(x p ) of Eq. (1) in terms of apAPf(x)which changes the coefficient of f ( x + p - l),and iterates this process. This procedure is already implemented in the LREtools package and can be invoked by the command REtodelta: >

L:=REtodelta(RE,f (n) ,{}I ;

+

+ +

+ +

L := LREtoohA, 2 (-n2 1 2 n) LREtoolsA, n 2 - n2 This results in an operator L such that L f ( z ) = 0 is equivalent t o the starting recurrence. The LREtools package uses its own Delta operator LREtoolsA which may be replaced by the usual A: >

subs(LREtoo1s [Delta] [n]=Delta,deltaexpr) ;

nA2

+ (2n- n2 + 1)A + 2 + n

-

n2

so that the resulting operator looks more familiar. Now let's ask the opposite question. How can we compute the original recurrence equation from the operator L? This is easy: Replacing A and its powers by their defining equations does the job. For this purpose we implement the program deltatoRE and some subroutines by" deltaop := (f ,x) ->subs (x=x+l ,f ) -f : aYou don't have to study this code in detail

327

expanddelta := (f ,x ,n) -> if n=O then f(x) elif n=l then f(x+l)-f(x) else deltaop (expanddelta(f ,x ,n-1) ,x) fi: deltatoRE:=proc(deltaexpr,an) local a,n,term,deg,k; a :=op (0,an) ; n:=op( 1,an) ; term:=deltaexpr-coeff (deltaexpr,LREtools[Delta] [nl , O > + coeff (deltaexpr,LREtools[Deltal [n] ,O>*a(n>; deg:=degree(deltaexpr,LREtools[Deltal[n]); for k from deg by -1 to I do term:=subs(LREtools [Delta] [n]-k=expanddelta(a,n,k) ,term) ; od; collect(term,a,factor); end : Our function deltatoRE uses L and creates the original recurrence equation, again: >

deltatoRE(L,f (n)) ; nf(n

+ 2) - ( n- 1)( n+ 1)f(n + 1)+ f(n)

2. Classical Orthogonal Polynomials Assume, a scalar product b

( f , d :=

f(z)g(z)&(Z)

with non-negative Bore1 measure p(x) supported in a real interval [a,b] is given. One considers essentially the following particular cases: 0

0 0

absolutely continuous measure dp(z) = p(x) dx with weight function p ( z ) , discrete measure p ( z ) = p ( z ) with support in Z. discrete measure p ( z ) = p ( z ) with support in q".

328

In the sequel we will only consider the first two cases, whereas the third instance is called the basic case, q denoting the base. A family Pn(z)of polynomials ~ ~ (=2knxn )

+ IC;Z~-’ + k:xn-’ + . .. ,

kn

#o

(2)

is called orthogonal w. r. t. the positive definite measure p(x), if

The classical orthogonal polynomials can be defined as the polynomial solutions of the differential equation

Since one searches for solution families Pn(x) according to Eq. (2) one gets the following conclusions: 0 0 0

n=l n=2 coefficient of xn

+ + + +

implies r(x) = dx e, d # 0 implies o(x)= ax2 bx c implies An = -n(a(n - 1) d )

Therefore these differential equations are easily classified according to the five parameters a , b, c, d and e, the coefficients of the polynomials ~ ( xand ) By this procedure the classical systems can be classified according to the following scheme (Bochnerl): 0 0 0

0

o(x)= 0

.(x) = 1 .(x) = x .(x) = z2 o(x)= x2 - 1

powers xn Hermite polynomials Laguerre polynomials powers, Bessel polynomials Jacobi polynomials

Every solution family Pn(x) of Eq. (3) is a translate of one of the above systems. The powers are not orthogonal, whereas all other systems are. The Bessel polynomials are, however, not orthogonal in a real interval but on the unit circle of the complex plane. It turns out that the corresponding Bore1 measure dp(x) is absolutely continuous, and the corresponding weight function p(x) fitting to the differential equation (3) satisfies Pearson’s differential equation

329

Hence it is given as

The classical “discrete” orthogonal polynomials can be defined as the polynomial solutions of the difference equation

+

+

CT(~)AVP,(Z)T ( x ) A P , ( z ) A,P,(x)

=0

(4)

where of(.) = f(x) - f(x - 1) denotes the backward difference operator. We get the same conclusions as before: 0

0 0

This leads-besides

the classification7:

+

=0

0

.(x) .(x)

=1 =z

0

deg(a(z),x) = 2

0

+ + +

translations-to

a(.)

0

+

implies ~ ( x=) dx el d # 0 implies a ( x ) = ax2 bx c implies A, = -n(a(n - 1) d )

n=l n=2 coefficient of xn

falling factorials xn = x(x - I) . . . (x- n I ) translated Charlier polynomials falling factorials, Charlier, Meixner and Krawtchouk polynomials Hahn polynomials

Again, the falling factorials are not orthogonal, whereas the other systems are. The discrete weight function p(x),x E Z, corresponding to the difference equation (4) satisfies Pearson’s difference equation

Hence it is given by the term ratio

3. Hypergeometric Functions The power series a17 . . . 1 ap Fq ( b l , . . . ,b,

whose summands

al,

f f k+l -

ffk

I ’)

03

= k=O Ak z k

’

= A k z k have rational term ratio Zk+’ A k Zk

- (k -

(k

+ U l ) . . . ( k + aP)

2

+ b l ) ... ( k + b,) ( k + 1 )

330

is called the generalized hypergeometric function. The summand ck!k = A k Z k of a hypergeometric series is called a hypergeometric term w. r. t. k. Eq. (5) therefore states that the weight functions p ( z ) of classical discrete orthogonal polynomials are hypergeometric terms w. r. t. the variable 2.

For the coefficients of the hypergeometric function one gets the formula

, in terms of the Pochhammer symbol (or shifted factorial)

(a)k

= a(a

+ 1)..

'

(a

+ k - 1) = r(a+ IC) r(a)

.

From the differential or difference equation of a classical orthogonal system, one can determine a hypergeometric representation. This computation can be easily done with Maple. Here are the computations in the generic continuous case. We define the polynomials > sigma:=a*x^2+b*x+c; > tau :=d*x+e ; 0

:= a Z'

+bz +c

~ : = d ~ + e with arbitrary coefficients a , b, c, d and e and consider the differential equation > DE:= > sigma*diff (F(x) ,x$2)+tau*diff (F(x) ,x)-n*(a*n+d-a)*F(x) ;

DE := ( a ~ ~ + b s + c ) ( & F ( z )+)( d z + e ) ( & F ( z ) ) - n ( a n + d - a ) F ( z ) To convert the differential equation to a recurrence equation for the series coefficients, we load the gf un (generating functions) package. > with(gfun): This package contains a function dif f eqtorec which computes the recurrence equation for the Taylor coefficients Aj of a function M ..

F ( s ) = CAj ~j j=O

satisfying a linear differential equation DE: >

RE: =dif f eqtorec (DE,F(x) , A ( j) ;

331

+ (d - u ) j - n2u + ( c j 2+ 3 c j + 2c) A ( j + 2 )

RE

:= ( a j 2

-

nd+an)A(j)

+ ( b j 2 + (e + b ) j + e ) A ( j + 1)

In certain instances, this three-term recurrence collapses towards a twoterm recurrence generating a hypergeometric representation for the family. This gives, for example, for the Laguerre polynomials

n+a! a!+l

k=O

n-k

In the Jacobi case, the point of development should be x = f l to get a hypergeometric power series. As a classical discrete example, the Hahn polynomials are given by

Q p q x , N)= 3F2

(

-71,

+

-x,n 1+ a ! + p a+l,-N

IN

*

Note that in a similar fashion the basic discrete case can be handled and leads to a classification of the so-called q-Hahn class of orthogonal polynomials, that have a basic hypergeometric representation, see e. g. Lesky6. 4. Computation of the Recurrence Coefficients

Moreover, by linear algebra one can determine the coefficients of the following identities (written for the discrete case) 0

(Recurrence Equation)

x P n ( x ) = an P n + l ( X ) 0

(6)

(Difference Rule)

~ ( xAPn(z) ) = 0

+ bn Pn(z) + cn Pn-l(S)

P n + ~ ( x )+ Pn Pn(x) + ~n Pn-l(z)

(Structure Relation)

Pn(z)=EnAPn+l(z)+ZnAPn(x)+&APn-i(x) in terms of the given numbers a , b,c,d and e. A recurrence equation of type (6) is valid for every orthogonal system, whereas difference rule (or differential rule) and structure relation are characteristics of the classical systems. With Maple, we will now determine the coefficients an,bn and c, of the recurrence equation (6) in terms of a, b, c, d and e. A s an example, we

332

deal with the classical continuous case. Note that the discrete case can be treated in a similar fashion. We consider the three highest coefficients of the orthogonal polynomial: > p: =k [nl *x-n+kprime [nl *x^ (n-1) +kprimeprime [n]*x- (n-2) ;

+

+

p := k, xn kprime, x(,-') kprimeprime, x ( n - 2 ) The polynomial satisfies the differential equation D E = 0 with >

DE:=sigma*diff (p,x$2)+tau*diff (p,x)+lambda[nl *p;

k, xn n kprime, x(,-l) + + c)( k , x"2 2 n2 -~ + 22 22

DE := ( a x 2 b x -

kprime, x(,-l) ( n - 1) 22

+ kprimeprime, x(,-')

X2

+ ( d x+e)

kprime, x ( ~ - ' () n - 1)

+ A,

( n - 212

22

kprimeprime, x(,p2) ( n- 2 ) )

(F +

( n-

X

+ kprimeprime,

( n - 2) ) X

(knxn + kprime, x(n-1) + kprimeprime, x ( ~ - ' ) )

After division by xn-', we collect coefficients: >

de:=collect(simplify(DE/x-(n-4)) ,x) ;

+ +

+

+

+

+ +

+

+

de := (-ak, n A, k, d k , n a k, n2)x4 (-3a kprime, n b k , n2 a kprime, n2 2 a kprime, A, kprime, - d kprime, e k , n - b k, n d kprime, n ) x 3 (-5 a kprimeprime, n - 2 d kprimeprime, - e kprime, - c k, n - 3 b kprime, n 2 b kprime, c k , n2 e kprime, n 6 a kprimeprime, + d kprimeprime, n A, kprimeprime, a kprimeprime, n2 + b kprime, n 2 ) x 2 ( ckprime, n2 - 5 b kprimeprime, n 2 c kprime, - 3 c kprime, n e kprimeprime, n b kprimeprime, n2 6 b kprimeprime, - 2 e kprimeprime,)x - 5 c kprimeprime, n 6 c kprimeprime, c kprimeprime, n2 Equating the highest coefficient gives the already mentioned identity for A, : > rule1 :=lambda [n]=solve (coeff (de ,x ,4),lambda [nl ) ;

+

+

+ +

+

+

+ +

+

+

rule1 :=A, This equation can be substituted:

+

+

+

+

=-n(an+d-a)

+

333 >

de:=expand(subs(rulel,de)) ;

+

+

de := 2 x3 a kprime, 6 x2 a kprimeprime, 2 x 2 b kprime, + 6 x b kprimeprime, + 2 x c kprime, + 6 c kprimeprime, - x3 d kprime, - 2 x2 d kprimeprime, - x2 e kprime, - 2 x e kprimeprime, x2 c k, n2 - x2 c k , n - 2 x3 a kprime, n - 4 x2 a kprimeprime, n x3 b k, n2 - x3 b k, n x2 b kprime, n2 - 3 x2 b kprime, n + x b kprimeprime, n2 - 5 x b kprimeprime, n

+

+

+

+ x c kprime, n2 - 3 x c kprime, n + c kprimeprime, n2 -

5 c kprimeprime, n + x3 e k, n + x2 e kprime, n

+ x e kprimeprime, n

Equating the second highest coefficient gives kk as rational multiple of k,: >

rule2:=kprime[n]=solve(coeff (de,x,3),kprime[n]);

+ + +

k, n ( e b n - b ) -2a d 2an Equating the third highest coefficient gives kC as rational multiple of Ic,: > rule3 :=kprimeprime [n] = > solve(coeff (subs(rule2,de) ,x,2),kprimeprime[n]) ;

rule2

:= kprime, =

1

rule3 := kprimeprime, = -kn n ( 3 b e + 5 b2 n - 2 b2 + e2 n + 2 e n 2 b 2 - 5 e n b - e2 - 4 c n a + c n d + 2 c n 2 a + 2 c a - c d + b2 n3 - 4 b2 n 2 ) / ( ( - 2 a + d + 2 a n ) ( - 3 a d +2an))

+

Without loss of generality we consider the monic case and set >

k[n] :=I; k, := 1

We would like to compute the coefficients a,, b, and c, in the recurrence equation RE = 0 with: >

RE:=x*P(n)-(a[n]*P(n+i)+b[nl*P(n)+c

RE := x P ( n ) - a, P ( n

[nl*P(n-I)) ;

+ 1)- b, P ( n ) - c,P ( n - 1)

We substitute P,(x): > RE:=subs( > { P (n)=p ,P (n+I) =subs (n=n+I ,p) ,P (n-1) =subs (n=n- 1 ,p) } , > RE);

334

RE := z (9 + kprime, z('+') + kprimeprime, - a, (dn+') + kprime,+, zn + kprimeprime,+l dn-')) - b, (zn+ kprime, z(,-') + kprimeprime, z ( n - 2 ) ) - c, (dn-')kprime,-, dnp2) + kprimeprime,-,

+

We substitute the already known formulas: > RE:=subs( > {rule2,subs(n=n+l,rule2),subs(n=n-l,rule2), > rule3,subs (n=n+l,rule3) ,subs (n=n-1,rule3)}, > RE);

+

n ( e b n - b) dn-') n ( 3 b e 5 b 2 n - 2 b2 e 2 n -2a+ d +2 a n 2 e n 2 b - 5 e n b - e2 - 4 c n a + c n d + 2 c n 2 a + 2 c a - c d b2n3- 4 b 2 n 2 ) z ( n - 2 ) / ( 2 ( - 2 a + d + 2 a n ) ( - 3 a + d + 2 a n ) ) ) ( n 1) ( e b ( n 1) - b) xn - a,(z(n+') + (n 1)(3be - 2 a + d + 2 a ( n + 1) 5 b2 ( n 1) - 2b2 e2 ( n 1) 2 e ( n 1)2 b - 5 e ( n+ 1 ) b - e2 - 4 c ( n + l ) a + c ( n + l ) d + 2 c ( n 1 ) 2 a +2 c a - c d + b2 ( n+ 1)3 - 4 b2 ( n+ 1 ) 2 ) x ( " - 1 ) / ( 2(-2 a d + 2 a ( n+ 1))

RE := x ( x n +

+ +

+

+

+

+

+

+

+

+

+

+

+ +

+ +

+ +

+

+

3 b e 5 b2 n - 2 b2 e2 n 2 e n 2 b - 5 e n b - e2 - 4 c n a 2 c n 2 a 2 c a - c d b2n3 - 4b2n2)z(n-2)/(2

+

+

+

+

+c n d

( - 2 a + d + 2 a n ) ( - 3 a + d + 2 a n ) ) ) -&(dn-') ( n - 1 ) ( e b ( n - 1)- b) dn-') +(n-1)(3be+5b2(n-1) -2a d 2 a ( n - 1) - 2 b2 + e2 ( n - 1 ) 2 e ( n- 1 ) 2b - 5 e ( n- 1)b - e2 - 4 c ( n - 1)a + c ( n - l ) d 2 c ( n - 1)2a+2 c a - c d 4 b2 ( n l ) 2 ) d n - 3 ) / ( 2 (-2 a d 2 a ( n - 1)) + b2 ( n-

+ + + +

+

+

+ +

+ +

(-3a d 2 a ( n - 1 ) ) ) ) After putting this rational expression in the form numerator/denominator with normal, the numerator must vanish: > re :=simplify(numer (normal(RE) 1/x- (11-31 ) : Equating the highest coefficient gives for monk polynomials >

rule4 :=a [n]=solve (coeff (re,x, 4) ,a [nl ) ;

335

rule4 := a, = 1 and equating the second highest coefficient yields >

rule5 :=b [n] =f actor (solve(subs (rule4,coef f (re ,x,3) ) ,b [nl

rule5 := b,

=

>

;

-2bn2a+2bna+ 2ea- 2bnd -ed ( d + 2 a n ) (-2 a d 2 a n )

+ +

Finally equating the third highest coefficient leads to > rule6 :=c [nl= > factor(solve(subs(rule5,subs~rule4,coeff(re,x,2>>>,c[nl 1);

+

+ + +

+ +

rule6 := c, = -n ( a n d - 2 a ) ( 4 a 2n2 c - 8 a 2 c n 4 a 2 c - a b2 n2 4 a c n d 2 a b2 n a e2 - a b2 - 4 a cd - b2 d n - b e d c d 2 b2 d ) / (( d - a 2 a n ) (-3a+ d + 2 a n ) (-2a+d+ 2an)Z) In a similar fashion the differential (difference) rule and structure relation can be handled.

+

+

5. Zeilberger’s Algorithm Doron Zeilberger (1990)lO developed an algorithm to detect a holonomic recurrence equation for hypergeometric sums

c M

s, =

F ( n , k ).

k=-cc

A recurrence equation is called holonomic, if it is homogeneous, linear and has polynomial coefficients. A similar algorithm detects a holonomic differential equation for sums of the form s(z) =

c

F(z,k).

k=-m

Holonomic functions form an algebra, i.e. sum and product of holonomic functions are holonomic, and there are linear algebra algorithms to compute the corresponding differential / recurrence equations. As an example, we apply Zeilberger’s algorithm to the Laguerre polynomials

336

For this purpose, we load the package hsum.mpl from my book Hypergee metric Summation3b >

read "hsum9.mpl"; Package LLHypergeometric SummationLL,Maple V - Maple 9 Copyright 1998 - 2004 , Wolfram Koepf, University of Kassel

We define the hypergeometric summand of the Laguerre polynomials >

laguerreterm:=(-l)-k/k!*binomial(n+alpha,n-k)*x-k;

laguerreterm :=

(-l)k binomial(n

+ a , n - k) xk

k!

and use Zeilberger's algorithm to detect a recurrence equation for the sum, hence for the Laguerre polynomials: >

LaguerreRE:=sumrecursion(laguerreterm,k,L(n));

LaguerreRE := (n a 1)L(n) (x - 2 n - a - 3) L(n 1) (n Similarly, a recurrence equation w.r.t. a is obtained

+ +

>

+

+ + + 2) L(n + 2) = O

sumrecursion(laguerreterm,k,L(alpha));

(n + a + 1 ) L ( a ) - (x

+ a + I ) L ( a + 1)+ z L ( a + 2) = O

Next, we compute the differential equation of the Laguerre polynomials from their hypergeometric representation: >

LaguerreDE: =sumdiffeq(laguerreterm,k,L(x)) ; LaguerreDE := x (&L(x)) - (x - a - 1)

(5L(x)) + L(x) n = 0

Using the holonomic algebra, it is furthermore easy to find recurrence and differential equations for the square L ? ) ( Z ) ~ and for the product L e ) ( x ) L g ) ( z ) or any other sum or product of holonomic functions. The following computation using the gfun package takes the recurrence equation for L e ) ( x ) and generates the recurrence equation valid for the square of the Laguerre polynomials (including some initial values) >

'rec*rec' (LaguerreRE,LaguerreRE,L(n)) ;

bThe software can be downloaded from my home page http: //w.mathematik. mi-kassel.de/-koepf/Publikationen.

337 - 5 2 a + 2 n 4 + 13n3 + 2 7 a + 9 a3 + a4 - 4 z n ' + 7 n 3 a + 9 n ' a' 35n2 a + 31 na' + 5 n a 3 - x a3- 4 x a' - 3 x n ' a - 3 2 n a ' - 8 x n a - 5 2 n + 5 5 n a ) L(n) + (-66 + 702 - 1 4 9 n - 124n' - 4 7 a 2 + 7 5 r a - 6 n 4 - 4 5 n 3 - 222' + 2 2 - 91- - 11a3 - a4 + z3 n - 6 2 ' n ' - 232' n + 112 n 3 + 6 2 2 n ' - 1 5 n 3 a - 14n' a' - 84n201 - 52 n a' - 6 n a3 + z3 a - 3 zz a' - 172' a + 3 2 a3 + 26 2 a' - 9 2 ' n a + 22 zn' a + 1 4 z n a' + 82 z n a + 1152n - 1 5 3 n a ) L(n + 1) + (110 - 1 0 2 2 + 2 1 9 n + 160n' 2 2 2 - 48 I a 6 n 4 + 5 1 n 3 + 262' - 2 2 + 8 2 a + 2 a 3 - z 3 n + 6 2 ' n' + 251' n - 11z n 3 - 70 zn' + 9 n 3 a + 5 n 201' + 57n' a + 21 n a2 n a 3 + 6 2 ' a - 6 2 a ' + 32' n a - l l z n ' a - 3 2 n a ' - 4 6 2 n a ((10

- 2 2 + 2 9 n + 30n' + 25 a'

-zn3

+

+

+

+

-

147xn

+ 1 1 9 n a ) L ( n + 2) + ( z n 3 + 8 zn' + 21 z n + 18 2

- 2 n 4 - 1 9 n 3 - 66n' - 9 9 n - n3 a - 8 n 2 a - 21 n a - 18 a - 54) 1 4

L(n+3),L(2) = - - c o - c z

+ -

3 4

9

f - - C i - c 3 - --Co-c3

3

- --ci-c2

4 1 1 x 2 a a_c32 -co a - c 3 z --Coa'-Cz 4 4 2 1 1 1 1 4-Coaz-C3- - - C l a ~ ~ C ' + - - C 1 Q ~ - C 3 + - - G 1 X ~ - C 3 4 4 4 1 1 1 - -c12 4

4

1

+

- -c1

+

+--Coa-Cz--Coa-C3+ --Co-C32--Cla-Cz 2 4 3 1 + 3 - C 1 a - C 3 - --c1-c32+--c1z-c2, L(0) = - c o - c z , 2 2 4 L ( l ) = -c1-C3}

Next, we compute the differential equation for the square of the Laguerre polynomials >

'diffeq*diffeq'(LaguerreDE,LaguerreDE,L(x)); (-42n+4~~~1+2n)L(~)

+ ( 4 z n + 2 z 2 + 3 a + 1 + 2 a 2- 4 2 a - 4 2 ) ( & L ( z ) )

+

+

+

(gz

+ (6

(-322 3 2 a 32) L(s)) L(2)) 2 2 and this is finally the (fourth order) differential equation for the product L e ) ( z )L g ) ( z )of the Laguerre polynomials. > 'diffeq*diffeq'(LaguerreDE, > subs (alpha=beta,n=m,LaguerreDE) ,L(x) ) ;

338

+ 4 x n ' a p + 2 on2x - 2 m2 a x - m a' - 2 m' o x + m a 2p2 + 2 m a 0' + a n 0' - 2 p n a' - 12 m x' n' - 5 m' a' x + m2p2 x + 12m' x' n - 1 2 m x ' n a + 6 m x n a ' + 1 2 m p x ' n - 6 mp' x n - 4m' x a p - a3 o n - a3 m p + 4 x 3 n a - 4 x 3 no + 4 x 3 m a - 4 x 3 m p - 41' a' n + x a3 n - 2 x' an' + 42' m p 2 - 41' a n o

(50 ' n' x

- x a n ~ ~ + 4 x ~ m p a + x r n p+ a 5' x p n a 2 - 5 x m a p '

+ 8 x 3 n 2 - 8 x 3 m'

- x m p 3 + 22' m2

+ 8x'p' n - 5 x p 3 n + 5 2 m a3 + 14 x' m' a + 12m' x' - 122' n2 - na' - n a 3 - 6 x 2 n a + 2 x n Z a + 6 x n a 2 - 61' a m + 6 p x ' m + 6/32' n + 6 a' m x - 60' m x - 6 0' x n + p3 a n + p3 a m - a2n 0' - a' n2x + o4 n + 4 x' n3 + n 0' - ma' + m 0' + 2 p 3 n - 4 m 3 x 2 - m a 4 - 2 m a 3 + m p 3 ) ~ ( x+) (-a2+ p' - 2 m x P + 2 x n P - 9 0 x 2 a' - 6 p3 x a - 2 x m a + 4 x 3 n a + 2 0 x 3 n p - 2 0 x 3 m a - 4 x 3 m p + 42' a 2n - 2 1 a3n + 82' an' - 42' mp' - 6 p a m x - 16 x' a n p + 6 x anp' + 1 6 x ' m p a - 6 x m p a ' + 2 x f l n a ' - 2 x m a p ' + 8 x 3 a' - 3 p a3 + 6 p a x n + 3 p3 a - 16 x 3 n' + 16 x 3 m' - 32 m x 3 + 8 m x 4 + 2 x m p - 81' m' p - 122' 8' n + 2 1 p3 n + 8 x 2 on' + 122' ma' - 2 x m a3 - 81' m' a + 21 0' x' - 9 p 3 x + p4 - 12m' x 2 - 2 a 3 - a4 + 92' a 0 ' + 6 x a3 p + 3 2 x 3 n + 122' n' - 2 0 x ' n + 1 6 x 3 a - 212' a' - 102' a + 9 x 2 + 1 0 x a 2 - 162' n a - 2 x n a ' + 2 x n a + l 0 p x ' - 10fl' x - 2 a ' p 141' On' - 82' ma'

-

+20mx2+2~3-5x'a3-8~2x3+5~3x2-9xa(1'

+ 282' a m + 9 p x a' + 1 6 0 x 2 m - 28 px' n - 8 a' m x + 20' m x + 80' x n - a4 - a3 0' - 8 x 4 n - 4 x 4 a + 4 p x 4 - f14 x + p3 a' + p4a - 1 6 P x 3 + Zap' + a4 x ) ( & L ( x ) ) + ( 9 o x 2 a' + 3 p 3 x a - 16 x 3 n a - 2 4 x 3 n p + 2 4 x 3 m a + 1 6 x 3 m p + 22'

a' n - 22'

+ 8 z 3 n'

-

mp'

8 x 3 m'

+ 121'

+ 6 p 3 x - 92'

- 250' x' 26x3 a

($3

a2

+ 82'

a p2 - 3 1 a3 p - 52 x 3 n

-

62' ma'

+ 162'

n

+ 1 8 x ' n a - 802' + 78' x - 1 6 m x ' + 72' a3 + 150' x3 - 7 p 3 x' + 6 1 a 0' - 18 I' a m - S o x a' - 18 o x 2 m + 18px' n + 20 x 4 n + 10 x 4 a - 10Bx4 + p 4 x + 26px3 - a 4 x ) ( $ L ( x ) )+ ( - 8 p 2 x 3 + 6 p Z x 2 + 8 x 3 a' + 2 0 x 3 n + 1 6 m x 4 - 1 6 x 4 n - 2 0 m x 3 + l o x 3 a - 1 0 p x 3 - 6 I' a' - 8 x 4 a + 8 p x 4 + 21' a 0 ' - 8x3ma + 8 x 3 n a - 2 px' a' - 8 x 3 m p + 8 x 3 n o - 21' a3 + Z O 3 x ' ) -

+ 252'

a n 0 - 12 x' m p a - 15 x 3 a 2

+ 5 2 m x 3 - 2 0 m x 4 + 6 x 2P ' n a -6xa3 - 7xa2

L(x))

+ ( 2 x d a - 2 0 x 4 - x3 a' + 0'

23

- 4 m 24

+ 4x412) ($

~ ( x ) )

339

6. Petkovsek-van Hoeij Algorithm

Mark0 Petkovseks developed an algorithm to find all hypergeometric term solutions of a holonomic recurrence equation. This algorithm is not very efficient, but finishes the problem to find hypergeometric term representan

c F ( n , k ) like c (i)’algorithmi00

tions of hypergeometric sums s,

=

k=O

k=--03

cally. Mark van Hoeij’ gave a very efficient version of such an algorithm, and implemented it in Maple. We would like t o find a simple representation for >

s:=Sum(binornial(n-2*k,k)*(-4/27)-k,k=O. .floor(n/3)) ; n floor(-) 3 s := binomial(n - 2 k, Ic) 27

c-

(z)k

k=O

Hence we define the summand >

summand:=binomial(n-2*k,k)*(-4/27)”k;

-4 summand := binomial(n - 2 Ic, k) ( - ) k 27 and compute the recurrence equation for the sum s, using Zeilberger’s algorithm: > RE: =sumrecursion(summand, k, S (n)) ;

RE

:=

2 (TI

+ 3) S ( n ) + 3 ( n+ 4) S(n + 1)

-

9 (TZ

+ 2) S(n + 2) = 0

Since the recurrence R E is second order, we try to find the hypergeometric term solutions of R E using van Hoeij’s algorithm: >

res:=‘LREtools/hsols‘ (RE,S(n)) ; -1 2 4 res := [(y)”, (5)” (3 + n ) ]

Since our second order recurrence has two linearly independent hypergeometric term solutions, every solution of R E must be a linear combination of them. Hence we can set for s, >

result:=alpha*op(l,res)+beta*op(2,res);

-1 result := a (-)” 3

2 + p (-)“ 3

4 (-

3

+n)

and we can find the coefficients a and ,B using two initial values by linear algebra:

340

2 1 sol := { p = - a = G ) 3' Therefore, we have finally found the simple formula for s,: >

result:=subs(sol,result);

7. Recurrence Operators Assume we consider the holonomic recurrence equation

R f ( x ):= f(x

+ 2 ) - + 1)f ( +~ 1) + x 2 f(x) = 0 . (X

In the general setting the coefficients could be rational functions w.r.t. x. Let 7 denote the shift operator r f ( x ) = f(x 1). Then the above recurrence equation can be rewritten as R f ( x ) = 0 with the operator polynomial

+

R

:= r 2 - ( X

+ 1) + x2 .

Such operators form a non-commutative algebra. The product rule for the shift operator

is equivalent to the commutator rule

rx-x7=r in this algebra. An operator polynomial has a first order right factor iff the recurrence has a hypergeometric term solution. Hence the Petkovsek-van Hoeij algorithm finds first order right factors of operator polynomials. Multiplying an operator polynomial from the left by a rational function in x is equivalent to multiply the recurrence equation by this rational function. Multiplying an operator polynomial from the left by r is equivalent to substitute x by x 1 in the recurrence equation.

+

341

Let us construct a fourth-order recurrence equation from R. To construct the equation S f(x) = 0 with operator

s := ( I C ( Z + 1)? + 2 3 7 + ( 2+.

.

- 1 ) ) .R

,

we just add the equations

( 2+

x3 (f(. z .(

- 1) (f(.

+ 2) - .( + 1)f ( . + 1)+ z2f(.)

=0

+ 3) - .( + 2) f(. + 2) + .( + q2f(. + 1)) = 0

+ 1) (f(z+ 4) - .( + 3) f ( . + 3) + .( + 2)2 f(. + 2)) = 0 .

This leads to

s := z .( + 1) (4. + 3) T4

--Z

+(a:+

T3

1) ( 3 z 2 + 6 z - 1)

+.( + 1) ( 2 4 + x3 + (2+ z 1) 2 .

-

x 2 - 5 + 1) 7

-

Given S , a factorization procedure by Mark van Hoeij can compute the factorization S = L R, again. For this purpose, we load his package: >

read “Fa~torOrder4-discrete’~ ;

Warning, the name delta has been redefined

-Env-LRE-x := x -Env-LRE-tau := T The following command factors S and returns the right factor found: >

fact:=FactorOrder4(RE) ;

fact := {

+ (-x - 1)T + x2}

T ~

which, of course, equals R.

8. Classical Orthogonal Polynomial Solutions of Recurrence Equations Previously we had shown how the recurrence equation can be explicitly expressed in terms of the coefficients of the differential / difference equation of the classical orthogonal systems.

342

If one uses this information in the opposite direction, then the corresponding differential / difference equation can be obtained from a given three-term recurrence. In this algorithm a polynomial equation system containing linear and quadratic polynomials must be solved5. Let the recurrence

be given. Our program computes that for a = 1/4 this corresponds to translated Laguerre polynomials, and for a < 1/4 translated Meixner and Krawtchouk polynomial solutions occur: >

read "retode.mpl" ;

Package "REtoDE", Maple V - Maple 8 Copyright 2000 - 2002, Wolfram Koepf, University of Kassel > RE :=P (n+2)- (x-n-I) *P (n+l)+alpha* (n+l)^2*P(n)=O ;

RE := P(n + 2) - (X - TI

- 1)P(n

+ 1)+

+

( n 1)2P(n) = 0

The following command gives the classical continuous solutions >

REtoDE(RE,P(n) ,x) ;

Warning : parameters have the values, { b = 2c, a = 0, a

=

1 4'

- c = c, d = -4c, e = 0}

1 [Z(2z+1)(&P(n, z))-2z(&P(n,z))+2nP(n1 z)=O,

and finally, the following command gives the classical discrete solutions:

>

REtodiscreteDE(RE,P(n) ,x>; f2

-1

Warning : parameters have the values, { a = -, f = f , d = d , 4f2 1 1 1 1 c = -- f 2 d - d - g d f - g d , a = 0 , g = g, e = - g d , 4 4 2 2 1 1

+

b=--fd--d} 2 2

+

+

343

f

[a(z) = -

2

+ z - -21 - g , a(.) + +)

=

(f -1)(f + 2 z + 1 - 2 g ) 2 (f + 1)

1 7

9. Final Remarks

I hope this survey article has shown some of t h e capabilities t h a t lie in t h e use of computer algebra systems when working with orthogonal polynomials and special functions. I wish you much success with their use! References 1. S. Bochner: Uber Sturm-Liouvillesche Polynomsysteme. Math. Z. 29, 730736 (1929). 2. K. 0. Geddes, S. R. Czapor and G. Labahn, Algorithms for Computer Algebra, Kluwer, Boston/Dordrecht/London, 1992. 3. W. Koepf, Hypergeometric Summation, Vieweg, Braunschweig/Wiesbaden, 1998. 4. W. Koepf and D. Schmersau, Representations of orthogonal polynomials, J . Comp. Appl. Math. 90, 57-94 (1998). 5. W. Koepf and D. Schmersau, Recurrence equations and their classical orthogonal polynomial solutions, Appl. Math. Comput. 128, 303-327 (2002). 6. P. Lesky, Eine Charakterisierung der klassischen kontinuierlichen, diskreten und q-Orthogonalpolynome, Shaker, Aachen, 2005. 7. A. F. Nikiforov, S. K. Suslov and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer, Berlin, 1991. 8. M. Petkovsek: Hypergeometric solutions of linear recurrences with polynomial coeffcients. J. Symbolic Computation 14, 243-264 (1992). 9. M. van Hoeij: Finite singularities and hypergeometric solutions of linear recurrence equations. J. Pure Appl. Algebra 139, 109-131 (1999). 10. D. Zeilberger: A fast algorithm for proving terminating hypergeometric identities. Discrete Math. 80, 207-211 (1990).

344

CONVERGENCE TO A PERIOD-TWO SOLUTION FOR A CLASS OF SECOND ORDER RATIONAL DIFFERENCE EQUATIONS

M. R.

s. KULENOVIC AND ORLANDO MERINO Department of Mathematics University of Rhode Island Kingston, Rhode Island 02881-0816

We investigate convergence to a period-two solution for the second order difference equation

where all parameters a ,p, 7, A , B, and C and initial conditions x-1,10 are nonnegative, and such that A Bxn Cxn-l > 0 for all n. Necessary and sufficient conditions for existence of prime period two solutions were given by Kulenovid and Ladas. Here the authors study the basins of attraction of period-two solutions of ( E ) for all cases where the right-hand-side has either two non-zero coefficients in both numerator and denominator, or one non-zero coefficient in numerator and three non-zero coefficients in the denominator. More precisely, it is shown that for these cases, a map associated with the equation has a unique saddle point in the positive quadrant, and it has two periodic orbits. The latter have large basins of attraction consisting of subsets of the positive quadrant of initial conditions in many cases separated by the stable manifold of the unique saddle point, which is the graph of a continuous monotonic function. Answers are given to three open problems posed by KulenoviC and Ladas.

+

+

1. Introduction and Preliminaries We investigate the convergence of solutions to a prime period-two solution of the second order difference equation

where all parameters a, ,B, y,A, B , and C and initial conditions 2 - 1 , zo are nonnegative and such that A Bz, Cz,-l > 0 for all n. A sequence 2 = {&} is a prime period two solution to (1) if it is a solution and there exist two distinct real numbers 4 and such that 2, = 4

+

+

$J

345

if n is even, and 2 , = I) if n is odd, n 2 0. The basin of attraction of a prime period two solution h is the set of all initial conditions ( 2 - 1 , z o ) for which the corresponding solution IC to (1) converges to 2 , that is, zn - 2n -+0 as n + 00. A prime period two solution h = (I),4, I),4,. . .) to equation (1) is said to be unique if there are no other prime period two solutions besides (4, I),4, I),. . .). We shall use the abbreviation (I),4) to refer to a prime period two solution ($,4, I),4,. . .). The problem of existence and uniqueness of period-two solutions of equation (1) was addressed and completely solved in [4],where necessary and sufficient conditions were given. However, the global behavior of prime period two solutions has been an open problem since the appearance of [4]. In this article, we state and prove a description of global behavior of prime period two solutions of (1) when they exist, for a large class parameters. Qualitatively different cases arise when one or more parameters of equation (1) is zero. In order to facilitate the study of the equation, a nomenclature based on the number of nonzero parameters was introduced in [4]. For example, the “(2,3)-type” refers to all equations (1) where there are two nonzero parameters in the numerator and three nonzero parameters in the denominator of the right-hand-side of (1). Clearly, “(2,3)-type” has three different subcases, depending on whether a = 0, ,f3 = 0 or y = 0. Regarding prime period two solutions to (l),the main result from [4] is the following.

Theorem 1.1. = 0 and B > 0 , equation (1) has a prime period-two solution i f and only i f y = P + A , and in this case, 4, I),4, I),. . . is a prime period-two solution of equation (1) i f and only if

i. If C

a + P(4 + I))= B4I) .with 4, I) E [O, 00) and 4 # I), or equivalently,

ii. If C > 0 and (1) possesses a prime period-two solution 4,+,q5,I), . . ., then in addition to y > P + A one of the following three conditions must be satisfied:

B # C , a=p=O, B=C, a=p=O, B>C, a+p>0,

and and

$I)=O

(3)

4I)>O

(4)

and

+I)>O

(5)

346

9,.

When (3) holds, the two cycle is 0 , ~, 0 , . . . When (4) holds, the two cycle is 4, with 0 5 4 < - 4, 4, cA - 4 , . . . Finally when (5) holds, the values 4 and $ of the two cycle are given by the two roots of the quadratic equation

9.

and we should also require that the discriminant of this quadratic equation be positive. That is,

a < (7 - P - A)[B(r- P - A) - C(Y 4c2

+ 3P - 41.

(6)

By performing a case-by-case analysis of the statement of Theorem 1.1 we may see that difference equations (1)with a unique positive equilibrium and unique (in the sense restricted above) period-two solution arise in the following cases: (a) (1,2)-type, (b) (2,2)-type (three subcases, two of which have unique prime period-two solution), (c) (1,3)-type, (d) (2,3)-type (three subcases, two of which have unique prime period-two solution), (e) (3,2)type (three subcases, one of which has a unique prime period-two solution), and (f) (3,3)-type. In this paper we study the basin of attraction of the unique prime periodtwo solution of (2,2)-type and of (1,3)-type, and show that in many cases it is bounded by the stable manifold of the positive equilibrium. The stable manifold is the graph of a monotonic function. We also give an answer to Open Problems 6.10.3, 6.10.4 and 7.5.1 from [4]. Our proof uses a recent general result for systems of difference equations of the form

where f , g are continuous functions and f ( x ,y) is non-decreasing in x and non-increasing in y and g ( x , y ) is non-increasing in x and non-decreasing in y in some domain B. Such systems are called competitive. Competitive systems of the form (7) have been studied by many authors such as Clark, KulenoviC, and Selgrade 111, Hirsch and Smith [2], KulenoviC and NurkanoviC [7],Smith [8]and others. We now present some basic notions about competitive maps in plane. Define a partial order 5 on R2 so that the positive cone is the fourth quadrant, that is, ( x l , y l ) 5 ( x 2 , y 2 )if and only if x 1 5 x2 and y1 2 y 2 . A map T on a set B c W2 is a continuous function T : B + B. The

347

map is smooth if it is continuously differentiable on B. A set A c B is invariant for the map T if T ( A )c A. A point z E B is a fixed point of T if T ( z )= z. The orbit of z E B is a sequence {Te(z)}e",o. A prime period ~ XO # z1 and z o = z2. The map T two orbit is an orbit { X ~ }forE which is competitive if T ( z l ,y') 3 T ( z 2 y2) , whenever (z1, y') 5 (z2,y2), and T strongly competitive if T ( z l , y l ) - T ( z 2 , y 2 )is in the interior of the fourth quadrant whenever ( z 1 y') , 3 (z2,y2). Consider a competitive system (7) where f , g : B -+ IR are continuous functions such that the range of (f,g ) is a subset of B. Then one may associate a competitive map T to (7) by setting T = (f,9 ) . If T is differentiable, a sufficient condition for T to be strongly competitive is that the Jacobian matrix of T at any (z, y) E B has the sign configuration

If (z,y) E B , we denote with Q e ( z ,y), l E {1,2,3,4}, the usual four quadrants relative to (z,y), e.g., Q l ( z , y ) = { ( u , v ) E B : u z, v 2 y }. For additional definitions and results see [6] and 8. A result from [6] we need is:

>

Theorem 1.2. Let T = (f ,g ) be a smooth competitive map on a rectangle R = [ a l ,a21 x [bl,bz] and strongly competitive on its interior R",where -co < a1 < a2 5 co and -03 < bl < b2 5 03. Suppose that the following

hypotheses are satisfied: 1. R" is an invariant set. 2. T has a unique fixed point E in R". 3. Either (a) T has no prime period two orbits in Ql(G) U Q3(E), 0r,

4.

8

(b) det & ( 8 ) > 0 and T(x) = 8 only for x is a saddle point.

= 8.

Let R(-, -) := {(z,y) E R : f ( z , y ) I z, g ( z , y ) 5 Y }, R(+,+) := {(z,y) E R : f ( z , y ) 2 2, g ( z , y ) 2 y }, and set A := { ( a l , y ) : bl < y < bz}, B := {(z, b l ) : a1 < z < az}. Let W s ( E )denote the global stable manifold of T at E , and let

w+:= {x E R \ W " ( 8 ): w- := {x E R \ W " ( E ) :

3x' E W ( E ) such that x' + x}

3x' E

Then the following statements are true.

ws(e)such that x + x'}

348

i. If T has no fixed points in B, then (xn,yn) := Tn(x) satisfies xn -+ a2 for every x in W - , ii. If T has a single fixed point x in B, and i f {x : k 5 x} c R(-,-), then the basin of attraction o f k in R is W- . iii. If T has no fixed points in A, then (xn,y,) := Tn(x) satisfies yn -+ b2 for every x in W+, iv. If T has a single fixed point 9 in A, and i f {x : 9 5 x} c R(+, +), then the basin of attraction of y in R is W+ 2. Main Results

It is shown in [4], pages 92, 113, and 133, that (2,2)-type subcases, and (1,3)-type may be reduced, through some simple algebraic transformations, to the following difference equations:

+ Yn-1 QYn + Yn-1 P + Yn-1 Yn+l = QYn + Yn-1

yn+1=

Yn+l =

PYn

Yn- 1

P+qYnfYn-~ We now proceed to give a characterization of the basins of attraction of prime period two solutions to equations (8) and (10). The basin of attraction of prime period two solution of Eq.(9) can be found in a similar fashion. 2.1. Basin of attraction of p r i m e period two solutions to

5.

The only equilibrium point of equation (8) is B = It is known from [4] and [5] that the equilibrium jj is locally asymptotically stable if p > q and q < pq + 1 3p, and it is unstable (and more precisely a saddle point equilibrium) when q > pq 1 3p. The latter inequality is equivalent to 1 3p O < p < l , and q > - . 1-P The following result gives the global behavior of solutions of equation (8) when (11) is satisfied.

+

+ +

+

Theorem 2.1. Consider equation (8) when (11) is satisfied. Let ($,$) be the unique prime period-two solution of equation (8) for which $ < $.

349

Then the global stable manifold Ws(jj,jj) is the graph of a smooth increasing function with endpoints o n the boundary of B = ( 0 , ~ x) (0, w ) ~ and such that every solution with initial point below W"(jj,ji)converges to ($,4), while every solution with initial point in B above W"(ji,jj)converges to (4,$). Consequently, except for solutions with initial point in WS(jj,ji), every solution converges to the period-two solution. Proof. Set un = yn-l and v, = yn for n = O , l , . (8) as the first order system

Pun vn+l = qv,

.., and

, n = O , l ,....

+ un

+ u,

s).

Let T be the map on B defined by T (u ,v) := (v, of T is given by

T 2 ( u , v )=

write equation

The second iterate

u + p v p u + p 2 v +uv +qv2 u+qv' q u + p q v +uv+qv2

( ~

)

One can see from a direct inspection of the Jacobian matrix of T 2 and the inequality q > p > 0 (which follows from assumption (11) ) that T 2 is a strongly competitive map on B = (0, co)x ( 0 , ~ ) :

The fixed points of T 2 in B satisfy T 2 ( u , v )= (u,v), that is, u=-

pv qv

+ +u 1 'L

and

pu+v qu+v'

v=-

which are exactly the equations satisfied by period-two solutions of the difference equation (8). Hence the fixed points of T 2 in B are (4,Q), ($,4) and ( j j , j i ) , where a direct calculation of 4, and j i shows that they can be chosen so that 4 < jj < Q. A consequence of this and of the fact that T 2 is strongly competitive is that B' = [$,$)I x [q5,$] is an invariant box, with a unique fixed point in its interior, namely ( j j , j j ) . This can be seen from the fact that points ( x ,y) in B' satisfy (4, $)I 5 (2,y ) 5 ($, 4 ) , hence ($,$) = T2(q5,$)4 T 2 ( z , y ) 4 T2(+,$)= ($,$). Furthermore, (B')" is invariant as well since T 2 is strongly competitive on B'.

+

350

A straightforward calculation gives that the Jacobian determinant of T 2 at

(p,y) satisfies

In addition, we have that the only point in B mapped by T 2 to the fixed point (g,g) is the fixed point itself. To see this, note that the equation T ( u ,v) = (y, g) may be written as

Pv+u qv+u=Y,

pu+v -qu+v - Y

Straightforward algebraic manipulations show that ( 1 2 ) implies u = w = j j . We now proceed to apply Theorem 1.2 to the restriction of T 2to the box B'. Thus far we have seen that conditions 1-4 of Theorem 1.2 are satisfied by T 2 on the box B'. Note that statements (i) and (iii) of Theorem 1.2 are true, hence the conclusion of the Theorem follows. That is, the portion of W " ( p , y )inside B' is the graph of an increasing function of the first coordinate, and every orbit with initial point ( ~ 0 , 2 1 0 ) = (y-1,yo) E B' below W"(y,y) converges to ($, 4), (in particular, in B' n Q2(ij,jj)), while every orbit with initial point (uo, vo) = (y-1, yo) E B' above the global stable manifold W"(jj,j j ) converges to (4,+). In particular, orbits with initial point in Q2(jj, p)" n Q4(4, $) (respectively, in Q4(jj,jj)' n Q2($, 4)) converge to (4,$) (respectively, to ($, 4)). By using the same argument found in the proof of Theorem 1.2 in [6]we conclude that W " ( yy) , extends outside of B' to the graph of an increasing function with endpoints on the boundary of B . The increasing character of W"(tj,g) guarantees that given any point (z,y) E B \ W " ( g , g ) ,there exists (z*,y * ) E W"(jj,jj)such that either (z, y ) 4 (z*,y * ) ("(z, y ) is above W"(y,g)"),or ( z * , y * ) 4 (z,y) ( " ( z , y ) is below W " ( g , g ) " ) .Then, either T 2 " ( z , y ) 4 T2"(z*,y*)for n > 0, or T 2 " ( z * , y * ) 4 T 2 " ( z , y ) for n > 0. Following the argument in the proof of Theorem 1.2 in [6] we conclude that, either T2"(z,y ) eventually enters Q2(jj, j j ) " , or T2"(z,y ) eventually enters Q4(jj,Y)'. Therefore, to finish the proof, it is sufficient to show that (a) every orbit with initial point in &2(p, 3)" converges to (4,+), and, (b) every orbit with initial point in Q4(jj,B)' converges to ($, 4). We need the following result from [3]: Let I, := (0, x (1,m) and I, := ( 1 , m) x (0, If p , q satisfy ( l l ) , then all solutions t o the difference equation (8) with initial conditions in I, U Ie converge t o the prime period two solution. Note that I, c Q2(4,$) and Ie c Q4($,4), since < 4 < ?I, < 1. Since Q2(4, $) and Q4($, 4) are invariant sets for T 2 ,it follows that

E).

E)

351

every orbit with initial point in Q2(4, $) converges to (4, $), and every orbit with initial point in Q4($, 4) converges to ($,4). To prove (a), note first that we have already shown that orbits with initial point in Q~(jj,g)" n Q4(4,$) converge to ( A $ ) . Clearly for every point (z,y) E Qz(jj,jj)" there exist points (z*,y*) E I, and (z*,y*) E Qz(jj,jj)" n Q 4 ( + , $ ) such that (zc*,y*)5 (z,y) I (z*,y*). Therefore, for all n 2 0 we have Tan(z*,y*)5 TZn(z,y)5 T2n(z*,y*).By taking limit as n .+ 00 we conclude that T 2 n ( xy) , .+ (4, q). The proof of (b) is similar to the proof of (a), now using the sets Q 4 ( j j , ~ j ) " n Q2($,4) and Ie = ( 1 , ~x )(0, and the fact proved in [3] that every orbit with initial point in It converges to to ($, 4). Th'is com0 pletes the proof of the theorem.

g),

Remark 2.1. Theorem 2.1 gives an answer to Open Problem 6.10.4 in [4].

Figure 1. The phase plane for equation (8), showing the stable manifold W"(B,rj) (increasing curve), the unstable manifold W " ( Q , p ) (decreasing curve), the sets s,, Se, and the initial conditions of period-two solutions.

2.2. Basin of attraction of p r i m e period two solutions to Iln--1

%+I

=

P+qYn+Yn-l

The equilibrium points of equation (10) are the zero equilibrium yo = 0 and a unique positive equilibrium ji = when p < 1. The following facts have been established in [4]: the zero equilibrium is globally asymptotically stable when p 2 1 and unstable (a repeller) when p < 1. If p < 1the positive

352

equilibrium is locally asymptotically stable when q < 1 and unstable (a saddle point) when q > 1. Equation (10) has prime period-two solutions ( 0 , l - p ) and (1 - p , O ) if and only if p < 1, and in that case the prime period-two solutions are locally asymptotically stable when q > 1. Our main result on prime period two solutions of equation (10) is the following.

Theorem 2.2. Consider equation [lo) with p < 1. Then the global stable manifold W"(jj,jj)is the graph of a smooth increasing function with endpoints o n the boundary of the closed positive quadrant of initial conditions, and such that every solution with initial point below W"(jj, jj) converges to (1 - p , 0 ) , while every solution with initial point in B above W"(jj,jj) converges to ( 0 , l - p ) . Consequently, except for solutions with initial point in W"(jj,jj), every solution converges to the period-two solution.

Proof. The proof of this result is similar to the proof of Theorem 2.1 and we present only the major steps and formulas. First write equation (10) as the system

=

p

+ qv,U n + u, , n = 0 , 1 , . .

Let T be the function on the closed positive quadrant [0,m) x [0, m) defined by T ( u ,v ) = ( v , *). The second iterate of T is given by

and the corresponding Jacobian matrix is

Clearly, T 2 is a competitive map on [O, m) x [0,m). The fixed points of T 2 are 0, ( j j , j j ) , (1- p , 0 ) , and ( 0 , l - p ) , of which only (jj,jj) is in the interior of the positive quadrant. A straightforward calculation gives the Jacobian matrix of T 2 at jj) satisfies

(a,

353

We now verify t h a t the only point mapped by T 2 t o the equilibrium @ , g ) is the equilibrium itself. The equation T 2 (uv) , = (jj,jj) may be written as the system U

= Y,

V

-

-

- Y.

p+qv+u p+qB+v Straightforward algebraic manipulations give u = v = p. Thus conditions 1-4 of Theorem 1.2 are satisfied by T 2 as a map on the closed positive quadrant. We now verify (i) of Theorem 1.2, t h a t is, we assume (1- p , 0) 5 ( I C , ~ and ) verify (z,O) E R(-,--). We have, T2(z,0)- (z,O)= (&,O) -

9

(z,O) = ( 9 , O ) . Since 1 - p < IC, we have < 0. Thus (i) of Theorem 1.2 applies, and we conclude t h a t (1 - p , 0) has as basin of attraction all orbits with initial point below the stable manifold W”(jj,jj). A similar reasoning may be used t o show t h a t (iii) of Theorem 1.2 applies and conclude t h a t ( 0 , l - p) has as basin of attraction orbits with initial point above the stable manifold. 0

Remark 2.2. Theorem 2.2 gives an answer t o the Open problem 7.5.1 posed by KulenoviC and Ladas [4]. References 1. D. Clark, M. R. S. KulenoviC, and J.F. Selgrade, Global Asymptotic Behavior

of a Two Dimensional Difference Equation Modelling Competition, Nonlinear Anal., T M A 52(2003), 1765-1776. 2. M. Hirsch and H. Smith, Monotone Dynamical Systems, Handbook of Differential Equations, Ordinary Differential Equations (second volume), Elsevier, (2005) (to appear). 3. S. KalabuSik and M. R. S. KulenoviC, Rate of Convergence of Solutions of Rational Difference Equation of Second Order, Advances in Difference Equations, 1 (2004), 121-140. 4. M. R. S. KulenoviC and G. Ladas, Dynamics of Second Order Rational

Difference Equations with Open Problems and Conjectures, Chapman and Hall/CRC, Boca Raton, London, 2001. 5. M. R. S. KulenoviC, G. Ladas, and W. Sizer, On the Recursive Sequence Xn+l - (axn ,8zn-l)/(yzn Sz,-l) , Math. Sci. Res. Hot-Line 2(1998),

+

+

1-16. 6. M. R. S. KulenoviC and 0. Merino, Competitive-Exclusion versus Competitive-

Coexistence for Systems in the Plane, Discrete Contin. Dyn. Syst. Ser. B (to appear) 7. M. R. S. KulenoviC and M. Nurkanovik, Asymptotic Behavior of a Linear Fractional System of Difference Equations, J. Ineq. Appl., (2005), 127-144 8. H. L. Smith, Planar Competitive and Cooperative Difference Equations,J. Dzffer. Equ. Appl. 3(1998), 335-357.

354

TWO NORMAL ORDERING PROBLEMS AND CERTAIN SHEFFER POLYNOMIALS WOLFDIETER LANG Institut fur Theoretische Physik, Universitat Karlsruhe, Physikhochhaus, 076128 Karlsruhe, Germany E-mail: [email protected], www-itp.physik.uni-karlsruhe.de/- wl The first normal ordering problem involves bosonic harmonic oscillator creation and annihilation operators (Heisenberg algebra). It is related t o the problem of finding the finite transformation generated by Lk-1 := -zk a,, k E Z,z E C (conformal algebra generators). It can be formulated in terms of a subclass of Sbeffer polynomials called Jabotinsky polynomials. The coefficients of these polynomials furnish generalized Stirling number triangles of the second kind, called S2(k;n,m) for k E 75. Generalized Stirling-numbers of the first kind, S l ( k ; n, m) are also defined. The second normal ordering problem appears in thermo-field dynamics for the harmonic Bose oscillator. Again Sbeffer polynomials appear. They relate t o Euler numbers and iterated sums of squares. In a different approach to this problem one solves the differential-difference equation fn+l

= f;

+ n2 f n - l

, n >= 1, with certain inputs fo and

f1 =

f;.

In this case the integer coefficients of the special Sheffer polynomials which emerge have an interpretation as sum over multinomials for some subset of partitions. Keywords: Sheffer polynomials, normal ordering, quantum physics

1. Introduction

Two exercises from physics lectures on quantum field theory will be discussed.

Problem 1: Normal ordering of harmonic Bose oscillator operators related to the exponential e q ( c z k&), z E C,k E Z.This will introduce a family of generalized Stirling numbers of both kinds, called S 2 ( k ; n , m )and S l ( k ; n, m ) . The problem is related to the Witt algebra (conformal Lie algebra for C).

355

Problem 2: Rightsided normal ordering in thermal quantum field theory of the harmonic Bose oscillator. In both problems Jabotinsky and Sheffer number triangles, resp. polynomials will show up. 2. Problem 1 The Heisenberg algebra [a,a+] = 1 is considered in the (infinite dimensional) holomorphic representation [ l ,] [2], [3], [4]: aAa,=-

a az

a+ A ~ E C .

These operators act on the space of holomorphic functions C functions) endowed with a scalar product.

(1) -+

@. (entire

The problem is to solve the following equation for g :

and the linear normal ordering symbol : 1 : = 1 and : ( z 8,). : = for p E N.

zp

a:

This is useful because if g is known, one has, via Taylor’s expansion for holomorphic functions,

This rephrases the problem of finding the finite conformal transformations on C generated by L k - 1 := - z k d Z , with k E Z. These generators obey the conformal Lie algebra

[Lm,Ln]= (m--)Lm+n7

m7n E Z .

Together with the complex conjugated generators, this is the Witt Lie algebra. Some special cases:

k k

+

= O : g ( O;c ;z)= c / z ; z’ = z c; L-1 generates translations. = 1 : g ( 1 ; c ; z ) = exp(c) - 1 ; z’ = e x p ( c ) z ; Lo generates

scalings (dilations) and rotations. k = 2 : g ( 2 ; c ; z )= c z / ( l - c z ) ;z’ = z / ( l - c z ) ; L+I generates special conformal transformations. {L-1 , Lo , L+1} generate the globally defined SL(2,C) Mobius transformations.

356

There are at least 4 different ways to solve for g ( k ; c; z ) : To derive from the Lie algebra the Lie group one solves the differential Eq. ~ . ~ ( a ! ) with

dZ(a) - -

da

z' = ~ ( a = ! 1 ) and z

and finds the solution z' = ( 1

+

1 g ( k ;c; z ) For all k sheets.

=

=~

(=a0),

(4)

+ g(k;c; z ) )z with

( 1 - (k - 1) c z k - - l ) - A

1.

(5)

# 0 , 1 , 2 there appear Ik - 1 I-th roots defined on Riemann

One may also use a transformation of variables, viz. 1 , k #1. This y k-l := lreduces the problem to a translation in the y-variable e.g. [5] . The physicist's solution: Use the multiple commutator formula for ezp(B) 2' exp(-B) with B = - C L k - l and [Lk-l,z'] = -lz'+"'. A resummation of the ensuing series in powers of c ( k - 1 ) zk-' with coefficients ( l / ( k - l ) ) K / nleads ! to the above given result for 1 g ( k ; c ; z ) .zKstands for therisingfactorialz(z+l)...(z+(n-l)) . Direct solution generalizing Stirling numbers of the second kind (the case k = 1 is reached as a limit). This approach has been used in [6]. Case k = 1:

+

cI 00

,.az

-

Cn

n=O n.

$ c S 2 ( n , m )zmdzm ( 6 ) n=l n. m=l

= 1

+

00

= 1

+

a,),

n

00

E,"

G2,(c) zm

aZm

(7)

azm

with E," := ( z = C:=, S 2 ( n , m) zm , n E N. G2,(c) = (G2(c))" with G2(c) = exp(c) - 1 the e.g.f. (exponential generating function) of the first ( m = 1) column of the S 2 ( n , m ) number triangle. For this triangle see [7], nr. A008277. eczaz = : e z p ( G 2 ( c ) z d z ): = : ezp[(ezp(c)- l ) z a z ]: , i.e. 1 g(1;c; z ) = e x p ( c ) .

5

+

357

This fourth way signals that S2(n,m) is a special instance of a Sheffer triangle, called by D. E. Knuth [8] Jabotinsky matrix [9]. The row polynomials S2,(x) := C",=, S2(n,m)xm are therefore exponential (also called binomial) convolution polynomials, satisfying, with S2o(x) := 1 ,

c (i)

c (">

n

s2n(x

+ 9)

=

n

~ 2 p ( xs2n-p(y) )

=

p=o

p=o

P

~ 2 p ( ys2n-p(x) ) .

(8) General k-case: One can write everywhere S2(k; n, m) with k E Z.With n

E

~= (;2aZln ~ = ~

C ~ 2 ( kn,; m)

azm ,

n EW,

(9)

m=l

and the triangle convention: S2(k;n,m) = 0 for n bn,O. The recurrence relation for each k is: F k ; n , m ) = ((k-l)(n-l)+m) S2(k;n-l,m)

< m, and

S2(k;n,O) =

+ S2(k;n-l,m-l) I.

(10)

Number triangles of this type have been investigated by Carlitz [lo]. Special cases: The k = 0 triangle is the lower part of the unit matrix. The k = 2 triangle was known as the (unsigned) Lah number triangle. [7], nr. A008297. The k = -1 triangle is related to a Bessel triangle. [7],nr. A001497. The e.g.f.s for the first columns ( k all what is needed):

# 1): (for Jabotinsky triangles this is X

G2(k; x) = ( k - 1)g2(k; -) k-1

, with g2(k; y)

= (1 - (1 - (1 - k)Zy)+k)

/(1- k ) .

(11)

g2(k;y) is the 0.g.f. (ordinary generating function) of the first column of the associated triangles s2(k; n, m) := ( k - l),-,

m! n!

- S2(k; n, m)

,

(12)

with recurrence

k-1 s2(k;n,m) = - [ ( k- l ) ( n - I) m] s2(k; n - 1,rn) n m - s2(k; n - 1,m - 1) , n

+

+

(13)

358

where s2(lc;n,m)

=

0, n

<m ,

s2(k;n,0)

= &,o,

s2(k; 1,1)= 1

.

which appears here as g2(k;y) = yc2(1 - k ; y ) is, for I # 0 the 0.g.f. for generalized Catalan-numbers (1 = 2 corresponds to the usual case).

3. Sheffer group and Jabotinsky subgroup

Before commenting on generalized Stirling numbers of the first kind, S l ( k ; n, m ) ,an interlude on the Sheffer group and its Jabotinsky subgroup. This is similar t o the case of ordinary convolution polynomials where the corresponding group has been called in [ll]Riordan group with its associated subgroup.

+

Elements of the Sheffer group are ( g , f ) , with e.g.f.s g(y) := 1 gk yk/k! and f(y) = y fn yn/n!, standing for the infinite dimensional, lower triangular matrix S

+ c,"==,

cyr1

for n 2 m 2 0, and 0 if n < m . Multiplication is defined as matrix multiplication S(l) . S(') = S(3)which produces the law (g(l),f(')) . (g('), f'')) = (g(3),f ( 3 ) ) ,with g(3) = g(l) (g(') 0 f ( l )) and f ( 3 ) = f(') 0 0

The unit element is matrix 1,

, corresponding to

0 The inverse element to ( g , f ) is ( g , f ) - l compositional inverse 3 f - l of f. The Sheffer polynomials [12] s,(z) =

7

pZZ0-l.

(17)

(1,y).

:= (l/(g o

C:=,

(16) f(1)

7), 7) with

the

S ( n , m ) z m have e.g.f.

(1,f) are the elements of the Jabotinsky subgroup Z: (1,f(')) . (1,f'')) = (l,f(')0 f'l). (1,f)-1 = (1J) . 0 The { s n ( z ) } ,together with the associated Jabotinsky polynomials {pn(x)} with (1,f ) are exponential convoution polynomials: 0

359 0

A multinomial M 3 expression for Jabotinsky matrix elements is Ref. 8

&Pa(,,,)

k=l

where Pa(n,m) denotes the partitions of n with m parts written in the exponential language as (lQ1 ...nan) with aj E No and fk := [yk]f ( y ) .

+

Example: (l,ezp(y) - 1) for S2(n,m) and its inverse ( l , l n ( l y)) for Sl(n,m ) . s 2 . s 1 = 1, = S l ' S 2 . Also for the general k E Z case: S 2 ( k ) . Sl(k) = 1, = Sl(k) . S2(k) . Neuwirth [13] (also private communication) observed that for k # 1 one has S2(k) = kS1 . S 2 , as well as Sl(k) = S1 . kS2, with kSl(n, m) := (1 - k),-, Sl(n,m) , and kS2(n, m ) := (1 - k),-, S2(n, m ) . 0

The e.g.f. for S row sums T , := C;=, S(n,m ) is g(z) ezp(f(z)) .

A recurrence relation is given by sTX(z) = .[ (In(g(J(t))))']Itk=dksTl-l(z) p.50.

+

7

12

2 1, so(z)

=

1 . See [14]

0 Orthogonal polynomial systems of the Sheffer type have been classified by Meixner [ 151.

4. Problem 2 The second problem involves rightsided normal ordering in thermal quantum field theory for harmonic Bose oscillator operators [16]. The following identity will be proved.

This is the thermo-vacuum l 0 ; P ) with the inv. temperature P = l/(kT), and tanh(8) = ezp(-Phw/2) where w the frequency of the oscillator. The operators A+ and A act on a direct product space according t o

A+ ._ .- a + @ 5 + ,A := a B 5 , 10) :=

IO>@l6>

.

(21) The tilde-system 5,ii+ with Id > is a twin version of the harmonic Bose oscillator. One has a Lie algebra su(1,l) with J- := A , J+ := A+ = (J-)+, 5 3 := (ll N)/2 = (J3)+.

+

[ A , A + ] = ll

+ N ,with N

:=

N@i

+

l@N,

(22)

ll := l @ i .

(23)

where N := a+ a and N := ii+ 5 ,

",A]

=

-2A,

[ N A,+ ] = + 2 A + ,

360

A holomorphic representation is, cf. Ref. 5 eq. (1.3.43)

A+

A

(1/2) 8," , A

+ 1/2) . (24)

(.az

(1/2) z2 , (1+ N ) / 2 A -(1/2)

Compute the above given 1.h.s. of the thermo-vacuum with N 10) = 0 and A I 0) = 0, keeping ll I 0) and A+ 10) . Rightsided normal ordering means to write for every monomial all A and N to the righthand side. Polynomial functions of A and A+ are first rewritten, using the commutation relation, in such a form that all A+'s are moved to the left of the A : O(A,A+) = U ( 0 ) .This expression U ( 0 ) is then decomposed according to U ( O ) = N r ( O ) R ( O ) with N r ( O ) I O ) = 0. Here N r is the rightsided normal ordering symbol and R stands for the remainder.

+

+ +

+

Example: U ((A+ - A)2) = A + 2 - A + A - ( A + A ll N) A2 , ie. N r ( (A+ - A ) 2 ) = - 2 A + A A2 - N and R((A+ - A ) 2 ) = A + 2 - ll. The interest is in R ( O ). If x is used instead of A+, and 1 instead of ll then R((A+ - A)") becomes a polynomials R , ( x ) . E.g.Rz(z) = x 2 - 1 . One finds an integer coefficient triangle for R(n,m) := [xm]R , ( z ) .See Tab.1 and Ref. 7, nr. A060081.

+

a ( n - (2 k - l),k) xnP2 , where a ( n , k ) = 1)j2, with input a(n,O) = 1 . is a rectangular array satisfying the following recurrence. R,(x)

=

C k L"' =2O J

Cy=,a ( j + 1,k

-

I a(n,k)

= a(n-1,k)

+ n%(n+l,k-l)

1,

with input a ( n ,-1) = 0 , a(0, k) = bo,rc. This array is shown in Tab.2 and appears as triangle in Ref. 7, nr. A060074. Example: &(x) = z6 - a(5,l) x4 a(3,2) x 2 - a ( l , 3 ) 1 = x6 - 55x4 331x2 - 61 .

+

+

The Rn(z)polynomials are Sheffer for (l/coshy, tanhy), i.e. 03

C ~ , ( z )y n / n !

= (l/coshy) exp(x tanh y)

.

n=O

For x A+ ,y .+ 0 this then proves the thermo-vacuum identity. The associated Sheffertriangle corresponding to the polynomials (1,tanh y) is shown in Tab.3 and it appears in Ref. 7, nr. A111593. ---f

Euler numbers En (signed, aerated) appear in the first (and second) column of the a(n,m) array. In symbolic notation (exponents instead of indices) they are defined by

36 1

( E + 1)k

+ (E - l)k

= 0, Ic E

N,Eo

= 1.

(27)

To the author's knowledge one obtains here a new representation for Euler numbers En = (-l)nEzn,n E No and their generalizations, via iterated sums of squares:

x*"

=

%(x)

=

c",=, R(n,m)P.

(30)

5 . Alternative Approach t o Problem 2

Define with Umezawa et al. Ref. 16 fn(e) := (0 I An exp(-Q (A+ - A)) 10) Consider

fA(6) = -(0 I (A+

- A)U(-e)lO)

(0 I A" U(-e) 10). = -(0

I U(-B)(A+

and derive, using Bogoliubov transformations like

U(e)aU(-e) the differential-difference eq.

=

coshea - sinhog+, etc.

with inputs fo(0) = l/coshe and f1(e) = fh(0). For general input fo(e) one uses fo(0) = s n ( g j )fo(O), with fn(6) = C",O f ( n , m )

-

A)\0)

362

m'

Thus {sn(e --t z)} become Sheffer polynomials for (2 Artanhy). The f(n,m ) triangle is the inverse of the R(n,m) triangle. See Tab.4 and Ref. 7, nr. A060524 (Author: V. Jovovic) The associated Sheffer triangle coresponding to the polynomials (1,Artanhy) is shown in Tab.5 and it appears in Ref. 7, nr. A111594. There is the following combinatorial interpretation for the f(n, m) numbers:

f(n,m) = Cs;EPao(n,m)M 2 ( 4 7 where Pao(n,m) stands for partitions of n with m odd parts (and possibly even ones). Again, partitions are written in the exponential form with exponents 6 := (a1,..., an). M 2 are the multinomial numbers [18], pp. 823, 831: n!/ j a 3 aj! . Example: 5 = f ( 3 , l ) = M2([3]) M2([1,2]) = 2 3. This is a reformulation of exercise 3.3.13. on p.189 of [19] (V. Jovovic, priv. comm.)

+

+

n:==,

For the considered physical problem the input is fo(0) = l/cosh6' leads to

f,(e) = n!(i/coshe) (-tanhe)"

. This

,

(33)

which coincides with the matrix elements (0 I An U ( - e ) 10) with u(-e)lo) = (l/coshO) ezp(-tanh(Q) A+) 10) due to (0 )A"(A+)" 10) = ( n ! ) z6n9rn. 6. Conclusion 0 Two simple harmonic quantum oscillator problems feature some nice elements of the Sheffer group. 0

Problem 1: Sometimes it is rewarding not to take the diretissirna.

0 Problem 2: Sometimes it is rewarding to take different routes to the same summit.

Acknowledgements The author would like to thanks the DESFA (ISDESIDEOPSFA) organizing committee for the invitation, the interesting program and the stimulating atmosphere. Thanks go also to Dr. Stefan Gieseke (ITP Karlsruhe) for sharing his foil program.

363

References 1. V. A. Fock, ,,Verallgemeinerung und Loesung der Diracschen statistischen Gleichung“, 2. Phys. 49, 339-57 (1928) 2. V. Bargmann, “On a Hilbert Space of Analytic Functions and an Associated Integral Transform, Part I”, Commun. Pure tY Applied Math. 14,3, 187-214 (1961) (K. 0. Friedrichs anniversary issue) 3. I. E. Segal, “Mathematical characterization of the physical vacuum for a linear Bose-Einstein field”, (Foundations of the dynamics of infinite systems. 111), Illinois J . Math. 6, 500-23 (1962) 4. I. E. Segal, “The complex wave representation of the free Boson field”, in

Topics in functional analysis: essays dedicated to M. G. Krein on the occasion of his 70th birthday, Advances in Mathematics: Supplementary studies, Vol. 3 (I. Gohberg and M. Kac, Eds.), pp. 321-344 (Academic Press, New York, 1978) 5. G. Dattoli, P.L. Ottaviani, A. Torre and L. VBzquez, “Evolution operator equations: integration with algebraic and finite difference methods [...I”, Riu. del Nuouo Cimento, 20,2, 1-133 (1997) 6. W. Lang, “On Generalizations of the Stirling Number Triangles”, on-line Journal of Integer Sequences, 3, Article 00.2.4 (2000) 7. N. J.A. Sloane’s On-Line Encyclopedia of Integer Sequences, http//:www.research.att.com/ -njas/sequences/index.html 8. D. E. Knuth, “Convolution polynomials”, The Mathernatica J., 2.1, 67-78 (1992) 9. E. Jabotinsky, “Analytic Iteration”, Trans. Amer. Math. SOC.108, 457-77 (1963) 10. L. Carlitz, “On Arrays of Numbers”, A m . J. of Mathematics 54,4, 739-52 (1932) 11. L. W. Shapiro, Seyoum Getu, Wen-Jin Woan and L. C. Woodson, “The Riordan Group”, Discrete Appl. Maths. 34, 229-39 (1991) 12. I. M. Sheffer, “Some Properties of Polynomial Sets of Type Zero”,Duke J . of Maths. 5, 590-622 (1939) 590-622 13. E. Neuwirth, “Recursively defined combinatorial functions: extending Galton’s board”, Discrete Maths. 239,33-51 (2001) 14. S. Roman, The Umbra1 Calculus (Academic Press, New York, 1984) 15. J. Meixner, ,,Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion“, J. London Math. SOC.,9, 6-13 (1934) 16. H. Umezawa, H. Matsumoto and M. Tachiki, Thenno Field Dynamics and Condensed States (North-Holland, Amsterdam, 1982) 17. Th. Spernat, “Clifford-Algebren in der Quantenmechanik” , Diplomarbeit Dortmund, (2004), http://doom.physik.uni-dortmund.de/pub/theses/TSpernat.pdf 18. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, tenth printing, National Bureau of Standards Applied Math. Series 55, reprinted, (Dover publication, New York, 1972) 19. I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, (Wiley, New York, 1983)

364

l+

0

m

r-

co

m

*

M

hl

d

0

0

0:

rl

r l o * o

1

0

0

%

d O O O C . 1

2

Tab. 2: a(n,m) array (as triangle Ref. 7, nr. A060074) 1

0 1 2 3 4 5 6 7 8

1 1 1 1 1 1 1 1 1

0 0 1 5 5 61 14 331 1211 30 55 3486 8526 91 140 18522 204 36762

5

6

7

0 0 50521 1385 2702765 50521 49164554 663061 510964090 5162421 3706931865 28862471 127838711 20829905733 96508175400 475638163 1544454483 384154309032

0 2702765 199360981 4798037791 64108947631 584856590956 4059150905356 22882712047924 109415187933364

0 199360981 19391512145 596372040824 9954077496120 111432850130020 935210483855284 6296554692590120 35575114290521256

4

2

31 0 61 1385 12284 68060 281210 948002 2749340 7097948

0

c n ua

Tab. 3: associated Sheffer triangle (1,tanh y), Ref. 7, nr. A111593 -

-

n/m

0

1

2

6

3

- -

0

1

1 0

8 9 10

0 0 0 0 0 0 0 0 0

-

-

2

3 4 5 6

7

1 0 -2 0 16 0 -272 0 7936 0

1 1 0 1 -8 0 0 -20 0 0 -40 136 0 0 616 2016 -3968 0 0 0 -28160 176896 0 -135680

1 0 -70

0 5376 0

1 1 0 -112 0 1 0 -168 0 12432 0 -240

1 0

Tab. 4: f(n,m) Sheffer triangle Ref. 7, nr. A060524 ~

n/m

0

1 2

3 4 5 6

7 8 9

10

0

1

2

1 1 0 1 0 1 0 5 0 9 0 14 0 89 0 225 0 439 0 3429 0 11025 0 24940 0 230481 0 893025 0 2250621

3

4

1 0 1 30 0 0 55 1519 0 0 4214 122156 0 0 463490

5

6

1 0 1 91 0 0 140 10038 0 0 21378

368

m 00

b

m

* M

m

4

0

4 0

co

do000

0

CD

M

d O O O * O b M

0

m M

0

*

l - l o m o * o o o o o m m m

m

m

b

M

4 o o o * o m o m

b

d 0 0 0 0 0 0 0 0 0 0

369

EXISTENCE OF SOLUTIONS FOR EVEN ORDERED BOUNDARY VALUE PROBLEMS ON A TIME SCALE

JOHNNY HENDERSON Department of Mathematics Baylor University Waco, Texas, USA

BONITA A. LAWRENCE Department of Mathematics Marshall University Huntington, West Virginia, USA E-mail: [email protected], lawrenceOmarshall.edu

Utilizing a result due to Krasnosel'skii and Zabreiko, we establish the existence of non-trivial solutions for the family of even ordered dynamic equations

(-i)"uAzm(t) = f ( U u ( t ) ) ,t E

[o, 11n T

with the boundary conditions A2i

W+lU A2i

'Yi+lU

(0)- P i + l U

(41)) - &+lU

A2i+l

(0)= 0 A2i+1

(4)) =0

Ojijm-1

We will determine sufficient conditions on the function f to insure the existence of a solution.

1. Introduction Our study takes place within the structure of time scale calculus, first introduced in the Ph.D. dissertation of Stefan Hilger in 1988 12. The differential operator in this structure, known as the delta derivative, as well as solutions of the dynamic equation that utilizes this operator are defined on a closed subset of the reals known as a time scale (usually denoted by T). The opportunity for gaps in the domain set is one of the features that gives the studies in this structure a new and interesting twist and the mechanics are handled using jump operators, for the forward jump and p

370

for the backward jump (defined from the time scale to the time scale). These operators essentially "jump" over gaps in the time scale if the next point in the set can be determined. In the event that the point is dense in either direction, the appropriate operator maps the dense point to itself. (Note that u2 denotes the (T operator composed with itself.) The recent books of Bohner and Peterson 4*5 and the original book on the topic by Kaymakqalan, Lakshmikantham, and Sivasundaram l 3 offer the reader an introduction t o this unified calculus and current foundational results. For completeness, we will give characterizations of both the delta and nabla derivatives, the fundamental differential operators in time scale calculus. Theorem 1.1. Assume f : T

+ R is a function and let t E TK (or TK).

Then the delta and nabla derivatives can be characterized as follows: (I) I f f is continuous at t and t is right-scattered (or left scattered), then f is delta differentiable ( or nabla differentiable) at t with

where (T and p are the forward and backward jump operators, respectively. (11) If t is right dense (or left dense), then f is delta (or nabla differentiable) at t i f

exists and is finite. I n this case the delta and nabla derivatives are defined in the standard way

In this work, we are interested in the 2m-th order dynamic equation

( - l ) m u A 2 " ( t )= f (u"(t>),t E [O, 1 1n T

(1)

or t E [0,1]n T, with Sturm-Liouville boundary conditions, A2i W+lU

(0) - Pi+l.

a"'+'

(0) = 0

(2)

371 A2i

Yi+lU

(41))- &+1U

A2‘+1

( 4 ) ) =0

(3)

for 0 5 i 5 m - 1. We give f the freedom to be nonlinear, making no restrictions on its sign, and determine the conditions that will insure solutions. A fairly standard approach for establishing the existence of solutions for an even ordered dynamic equation is to use a fixed point theorem and define the operators required by the fixed point theorem using an appropriate Green’s function. A search of the literature will reveal a variety of existence results for second order nonlinear boundary value problems on a time scale using a variety of fixed point theorems. For example, applications of the the Guo-Krasnosel’skiXxed point theorem ’, the Avery-Henderson fixed point theorem 3 , and the Leggett-Williams fixed point theorem 15, have been used to establish the existence of positive solutions and multiple positive solutions when f is nonnegative for conjugate and right focal boundary value problems. With the goal of positive solutions and the same conditions on f , others have successfully found conditions to insure existence of solutions of even ordered delta-nabla dynamic equations as well as with Sturm-Liouville boundary conditions In the particular case when certain growth conditions are assumed for f Henderson verified the existence of solutions for a 2m-th order problem like the one of interest to us with Sturm-Liouville boundary conditions lo. Sun and Li l 6 weakened the assumptions of previous results for second order right focal boundary points by allowing the function f to change signs and verified that solutions exist using the Krasnosel’sk’i-Zabreikofixed point theorem. In ”, Henderson expanded their work to the more general Strum-Liouville boundary conditions and we will extend his results to 2m-th order problems described in (1),(2), (3). In Section 2 we will introduce the Krasnosel’ski’i-Zabreiko fixed point theorem and construct the operators required by the theorem that will be used to develop our own theorem. In Section 3 we will verify that these operators satisfy the conditions of the theorem and determine any additional assumptions we will need on our function f . In Section 3 we state our main result. 17216,7.

2. The Krasnosel’skiY-ZabreikoFixed Point Theorem

The main result of this work is an application of the Krasnosel’skiY-Zabreiko fixed point theorem so we begin with a statement of the theorem as it was originally presented.

Theorem 2.1. Let E be a Banach space and F a completely continuous

372

operator defined o n all of E which is asymptotically linear. If 1 is n o t a n eigenvalue of the operator F'(oo) then x = Fx has at least one solution. Note that f is said to be asymptotically linear if and only if there exists a continuous linear operator A such that

The operator A is called the derivative of F at infinity and is denoted by F'(co).The theorem has three main requirements: (i) a Banach space to work in, (ii) an appropriate linear operator that does not have 1 as an eigenvalue,and (iii) appropriate operators that satisfy (4). Our Banach space of choice is ( X , 11. 1 1) where X = C[O,a2(1)]and

llull = { lu(t)l I t E

[0JJ2(1)1

1.

The operators of choice, denoted by F and A , will be defined in terms of the Green's functions for the associated homogeneous problem (-i)muA2" = 0, t E

[o, 11n T,

(5)

with boundary conditions (2),(3). For this 2m-th order problem, the Green's function is iterated as follows: We define a sequence of second order homogeneous boundary value problems -uAA = 0, t E

[o, 11n T

(6)

indexed by boundary conditions aiu(0)- p i u y o ) = 0

(7)

yiu(a(1)) - SiUA(C7(1)) = 0

(8)

and

< <

for 1 i m and denote the Green's function for each problem in the sequence by Gi(t,s). Note that the constants ai, ai,yi,and Si are precisely those from our original boundary conditions (2),(3). The Green's function for each BVP in the sequence (6),(7), (8) is defined

G i ( t , s )= -

+

+ +

(ait Pi)[-yi(a(l)- a ( s ) ) Si] if t 5 s (aia(s) Pi)[yi(a(l) - t ) Si] if a ( s ) 5 t

+

(9)

373

with the following assumptions on the constants in the initial conditions,

and

With these restrictions Gi(t,s) 2 0 for ( t , s ) E [O,a(l)] x [O,l]. (See for a reference on the sign of the Green's function.) Using this sequence of functions we recursively define Hj ( t ,s), 1 5 j 5 m by

Hj(t,S)

=

I"'"

Hj-1

( t ,r)Gj(r,s ) A ~

As it turns out, the end result of our construction, H,(t, s) is the Green's function for the homogeneous problem (5), (2), (3), and H,(s, t ) > 0 for ( t ,s) E (0, ~ ( 1 )x) (0, ~ ( 1 ) )We . are making the additional assumption that ~ ( 1 is) right dense. With this we define our completely continuous operator F : X + X, in terms of a continuous function f , (10) We know that we have a solution u of (l),(2), (3), if and only if u is a fixed point of F . For the second operator, we consider the linear homogeneous dynamic equation (-l)"uAZrn(t) - p ( u U )= 0

(11)

satisfying boundary condition (2), (3). Using the Green's function for ( l l ) , (2), (3), we construct the bounded linear operator, A : X + X ,

( A ( u ) ) ( t= ) p/'(l) H,(t,s)uU(s)As 0

(12)

As with our operator F , solutions of ( 5 ) , (2), (3) are fixed points of (12). The converse is also true. Next we verify that these operators will fit the bill.

374

3. Development of the Main Result

Our first step is to verify that 1 is not an eigenvalue for our operator A . Note that if m = 0, the only solution of (5), (2), (3), is the trivial solution. If m # 0 we obtain the following estimate:

Now if we have

b=

1

J:")

~ ~ ~ t g [ o , i ] n T~

m ( tS)AS ,

> 1Pl

then the above inequality reduces to

IlAull < bllull;

1 =

llull

Therefore, 1 is not a eigenvalue of A. Our next task is to verify that the following limit exists:

From the definitions of F and A , and the properties of absolute value, we obtain

If we can show that this expression is less than or equal to ~ l l u l l(some . constant, perhaps)

375

for any

E

> 0 or, equivalently, that If(u"(s)) - PU"(S)l < E I I 4 I , st E [O, 4111

then we are done. To this end, assume that our function is such that f(r) lim - P. I l T l l b C 7-

This is the source of p in the linear equation (12). Now choose any E > 0. The limit above insures that we can find C1 > 0 such that when Irl > C1,

If(r) - PI < &Id Using this C1, we denote by C ,

c=

SUP

IITlllC1

If(.)I.

Then, for s E [O,a(l)]n T , if it is the case that Iu"(s)I

I C1,

I If(u"(s))I + IPllu"(s>I I c + IPIG We can also find a constant M > C1 such that If('lL"(S)) - P.ll"(S)I

C

+ (p(C1< M .

E,

(13)

(14)

and so using (13) and (14) and choosing u E X such that 1)u11 > M,we obtain If(."(s)) - PU"(S)I

Alternatively, if lu"(s)l

> C1 we have a similar bound,

If(u"(s)) - PU"(S)I

So, for all u such that llull is

IIJYu) - A(u)II =

I EM I E l l 4

I EIU"(S)I i E l l 4

> M , the estimate for the norm of this difference

SUP

IlU(l)

t~[o,i]n~

IElbll

~m(t,s)[f(W)) Pu"(s)lAs

1

"(1)

SUP

t ~ [ o , i ] no~

Hm(t,s)As

1 Ell.11,

Therefore our limit exists and we have satisfied all the conditions on the operators. In addition, we have determined the conditions necessary on our function f t o insure the existence of a solution.

376

4. Existence Theorem

From the our construction in the previous section we now formulate the main result.

Theorem 4.1. Let f : R 4 R be a continuous function such that

f (u) = p. lim -

u-iw

21

If 1

t, ~ ~ ~ t c : [ o ,J:i")] n T~ m ( S)AS

=b

> IPI,

(16)

then the nonlinear 2m-th order boundary value problem (1),(2),(3), where the constants found in the bounda y conditions satisfy the conditions

and

, has a solution. Proof. Define operators F and A as in (10) and (12), respectively. It has been verified that 1 is not an eigenvalue of A. In addition, if the limit condition (15) holds and satisfies (16) we have verified that

is satisfied. All assumptions of the reported fixed point theorem of Krasnosel'skG and Zabreiko, Theorem 2.1 are satisfied and therefore F has a fixed point which is a solution of (l),( 2 ) , (3) characterized by 0

The scope of this lovely theorem is not lost on the authors. In a point work to be published soon with Dr. Basant Karna (Marshall University) , we consider second order multi-point boundary value problems.

377

References 1. D .R. Anderson, Eigenvalue intervals for a second-order mixed-conditions problem on a time scale, Znt. J. Nonlin. Differential Equations 7 (2002), 97-104. 2. D.R. Anderson, Eigenvalue intervals for a second-order Sturm-Liouvlee dynamic equations, Int. J . Nonlin. Differential Equations in press. 3. R. I. Avery and J. Henderson Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. Appl. Nonlinear Anal. 8, (2001), 27-36. 4. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: A n Introduction with Applications, Birkhauser, Boston, 2001. 5. M. Bohner and A. Peterson, eds., Advances i n Dynamic Equations on T i m e Scales, Birkhauser, Boston, 2003. 6. L.H. Erbe and A. C. Peterson, Positive solutions of a nonlinear differential equation on a measure chain, Mat. Comput. Modelling 32 (2000), 571-585. 7. L.H. Erbe and A. C. Peterson, Eigenvalue conditions and positive solutions, J . Differ. Equations and Appl. 6 (2000), 165-191. 8. L.H. Erbe and A. C. Peterson, Green’s functions and comparison theormes for differential equations on measure chains, Dynam. Continuous, Discrete & Impulsive Systems 6 (1999), 121-137. 9. D. Guo and V. Lakshmikantham, Nonlinear Problems i n Abstract Cones, Academic Press, San Diego, 1988. 10. J . Henderson, Multiple solutions for dmth order Sturm-Liouville boundary value problems on a measure chain, Journal of Difference Equations and Applications, 6 (2000), 417-429. 11. J. Henderson, Nontrivial solutions to a nonlinear boundary value problem on a time scale, Comm. on App. Nonlinear Analysis, 11 (2004), 65-72. 12. S. Hilger, Ein Masskettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. Thesis, Universitat Wurzburg, 1988. 13. B. KaymakCalan, V. Lakshmikantham, and S. Sivasundaram, Dynamic Systems o n Measure Chains, Kluwer Academic Publishers, Dordrecht, 1996. 14. M. A. Krasnosel’sk& and P. P. Zabreiko, Geometrical Methods of Non-Linear Analysis, Springer-Verlag, New York, 1984. 15. R. W. Leggett and L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. 28 (1979), 673688. 16. J. P. Sun, A new existence theorem for right focal boundary value problems on a measure chain, Appl. Math. Letters, 18,No. 1 (2005), 41-47.

378

ASYMPTOTICS AND ZEROS OF SYMMETRICALLY COHERENT PAIRS OF HERMITE TYPE

M. G. DE BRUIN Delft Institute for Applied Mathematics Technische Uniuersiteit Delft, The Netherlands

W. G. M. GROENEVELT Korteweg-De Vries Instituut uoor Wiskunde Uniuersiteit uan Amsterdam, The Netherlands

F. MARCELLAN* Departamento de Matemdticas Universidad Carlos III de Madrid, Spain H. G. MEIJER Delft Institute for Applied Mathematics Technische Uniuersiteit Delft, The Netherlands

J. J. MORENO-BALCAZAR~ Departamento d e Estadistica y Matema'tica Aplicada Uniuersidad de Almen'a, Spain and Instituto Carlos I de Fisica Teo'rica y Computacional Uniuersidad d e Granada, Spain

*Research partially supported by Direcci6n General de Investigacih (Ministerio de Educaci6n y Ciencia) of Spain under grant BFM2003-06335-C03-02 and I N T A S Research Network NECCA I N T A S 03-51-6637. +Researchpartially supported by Ministerio de Educaci6n y Ciencia of Spain under grant MTM2005-08648-C02-01 and Junta de Andalucia (G.I. FQM 0229).

379 We consider the Sobolev inner product

where ( p 0 , p i ) is a symmetrically coherent pair with one of the two measures the Hermite measure. We give a survey of the analytical properties of the corresponding Sobolev orthogonal polynomials and establish a new result about the asymptotic behaviour of these Hermite-Sobolev orthogonal polynomials inside the support of the measures po and p i . MSC: Primary 42C05;Secondary 33C45 Key words: Sobolev orthogonal polynomials, Hermite polynomials, Asymptotics, Symmetrically coherent pairs, Zeros.

1. Introduction Consider the Sobolev inner product

(f,g ) s =

1:

f(z)g(z)dPo(z)+

f'(Z)d(Z)dP1(4,

/a -a

(1)

where po and p1 are symmetric measures on ( - a , a ) and X > 0. Let {P,}F=o and {Q,}F=odenote the sequences of polynomials orthogonal with respect to po and p1 respectively. In Iserles et al. introduced the concept of (symmetrically) coherent pairs, that is, the pair ( p 0 , p l ) is said to be symmetrically coherent if there exist non-zero constants C, (see Theorem 4 in 5 , such that

Q n ( z )= Cn+l p;+,(z)- Cn-1 p;-l(z),

n

2 2.

In all (symmetrically) coherent pairs of measures have been determined. Especially, it has been proved that, for symmetrically coherent pairs, one of the two measures po and p1 must be either a Hermite or a Gegenbauer measure. In this paper we consider the inner product

s_, 00

(P14)S =

P(z)q(z)dPo

+

/

cn

P'(44'(4@1

(2)

-W

where (pol p1) is a symmetrically coherent pair of measures corresponding to Hermite weight, that is, the measures are either Type I : Type I1 :

+ u2)e-x2dz,dp1 = e-"'dz, u E R, or dpo = e - x 2dx, dpl = C Z 2 / ( z 2 + u2)dz, a E R \ {0}, dpo = (z2

according to the classification given by Meijer in

6.

(3)

(4)

380

A more general framework about the study of polynomials orthogonal with respect to Sobolev inner product like Eq. (2) is presented in ‘. There, asymptotic properties for sequences of monic orthogonal polynomials with respect to a vector of measures with dpo = $(x) exp (-Q(x)) d x , d p l = exp ( - Q ( x ) ) d x where Q is an even and continuous function on the real line satisfying some extra conditions and $ is a measurable and positive function on a set of positive measure on R are analyzed. Nevertheless, according to P. Halmos “. . . the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in essence the same as a small and concrete special sense”. Our present work, focussed on a vector of measures associated with the Hermite weight function, fits the above sentence. Furthermore, the results appearing in this work are not recovered for those obtained in (see Example 1.6 in 4). The aim of this paper is to give a survey on asymptotic properties and zero distribution of the so called Hermite-Sobolev polynomials orthogonal with respect to the inner product Eq. (2). Furthermore, a new result concerning the inner strong asymptotics of these Hermite-Sobolev orthogonal polynomials is introduced (Theorem 2.3). The organization of this paper is as follows. In Section 2 we give the relative asymptotics for the HermiteSobolev and scaled Hermite-Sobolev orthogonal polynomials in terms of the classical Hermite polynomials and the inner strong asymptotics of these polynomials (Theorem 2.3). In Section 3 we deduce the properties of localization of their zeros and its asymptotics including Mehler-Heine type formulas. Next, we use the notation: Hn(x) = 2nxn . . . for the classical Hermite = 2nxn . .. , SiH2’(x) = 2“xn . . . for the polynomials and SiH1’(x) Sobolev polynomials orthogonal with respect to Eq. (2) where the measures are given by Eqs. (3) and (4), respectively.

+

+

+

2 . Asymptotics

In the framework of symmetrically coherent pairs, we have the algebraic relation between the classical Hermite polynomials and the Sobolev orthogonal polynomials (with another normalization see Theorem 4(c) in 5),

for i = 1 , 2 , where

giH1’ = 0 for all n E N.

381

The key to obtain the asymptotic behaviour of the Sobolev orthogonal polynomials is related to compute the limits of the sequences and aiHa), i = 1,2, when n 4 00. It is important to note that the sequences aiH1)and aiH2)involve the ratios

uiH2’

respectively, where Q n ( z )= 2nxn+. respect to the weight function

. . are the polynomials orthogonal with

e-xa x2

a

+ a2 ’

# 0.

From a technical point of view the analysis of type I is more simple than type 11. In fact, in Lemmas 2.2, 2.4, and 2.6 in it is obtained:

where

cp(x) = z

+ @T

(8)

with d n > 0 if x > 1. This is enough to deduce the relative asymptotics between Sobolev orthogonal polynomials and Hermite polynomials in type I. However, for type I1 more information about the asymptotics of 0 L H 2 ) / ( 2 n )is needed to get a useful result. Thus, in Proposition 2 in it was deduced

From this we get (see Theorem 2.3 in

and Theorem 2 in

Theorem 2.1. Denote

e(x) = (p(2X ++1 )1) -1. (a) Outer relative asymptotics.

uniformly on compact subsets of C \ IR

7,

382

(b) Outer relative asymptotics.

+ ip E

uniformly o n compact subsets of half planes C + := {x = Q C , p > 0 ) and C - := {x = Q + i p E C , p < O}, respectively. Again, cp is given by Eq. (8).

Sketch of the proof. (a) If we divide Eq. (5) by Hn+z we get

From Perron’s formula (page 199 in ’) we deduce lim 71-00

nHn(z)

= -1/2.

Hn+2(z)

Taking into account this and Eq. (7) we get

Now, we write

and Eq. (12) is rewritten like

Therefore, using Eq. (13) and 0 < l/cp(2X+1) < 1 we obtain that ZiH1’(z) is convergent and its limit is e(A). (b) To prove Eq. (11) we take the following steps: 0

We consider the limit polynomials

383

’

(H2,m)

In particular, these polynomials satisfy (see and 3): Sn (z) = limx-tm SAH2)(z). Then, we get using the same technique as in (a) (see formula (16) in ’):

0

Using Eq. (9), we deduce (see Proposition 5 in

7,

(16)

+

uniformly on compact subsets of half planes C + := {z = Q i p E C , p > 0) and C - := {z = Q i ,B E C p < 0}, respectively.

+

Eq. (11) follows from Eqs. (15) and (16).

0

On the other hand, the asymptotics of scaled Hermite polynomials was obtained in Theorem 3.1 in

’:

Theorem 2.2. (a) Outer relative asymptotics for scaled polynomials. lim n-+m

SiH”(fiz) Hn(Jn.)

-

(3) (P(2X + l)p2 (3) +1 (P(2X + 1 ) v 2

[-a, a].

uniformly o n compact subsets of C \ (b) Outer relative asymptotics for scaled polynomials.

uniformly on compact subsets of C \

[--al a].

Again, cp is given by Eq. (8). Sketch of the proof. We need the ratio asymptotics for the scaled Hermite polynomials (see page 126 in ’): for every nonnegative integer k

384

uniformly on compact subsets of C \ [-&, &I. (a) If we make the change of variable x ++ fix in Eq. (12) and apply this formula in a recursive way, we get

k=O

where gn,o(fix) = 0 and [u] denotes the integer part of a. Using Eqs. (7) and (17) we deduce that for every Ic fixed limn+wgn,k(fix) exists uniformly on compact subsets of C \ &] and, furthermore we can find a majorant convergent sequence for & & ( f i x ) on compact subsets of C \ [-&, . Thus, we apply Lebesgue's dominated convergence theorem in order to obtain (a). (b) We proceed like in (a), but now using the limit polynomials SiH2'm'(x) defined in Eq. (14) and taking into account

[-a,

a]

It is important t o note that for the scaled asymptotics of type I1 it is enough t o know Eq. (7). However, for the non-scaled asymptotics obtained later on in some extra information about the sequence aiH2)given by Eq. (9) is needed. This follows from the fact that

'

+

y

SLH2'")(x) Hn+2 (x) a ;:;) H, = lim n+m H,(z) ,+a H n (x) S i H 2 qf i x ) = lim %+2(fix) a : ; ) ? lim n+m H,(fiX) n+m Hrl ( f i x ) lim

+

= 0,

Hn(fix)

a],

# 0,

uniformly on compact subsets of C \ R and C \ [-&, respectively. Next we give the asymptotic behaviour of the Sobolev polynomials SLHi'(x) (i = 1 , 2 ) for z inside the support of the measures po and p1, i.e. the so-called inner strong asymptotics. We give the asymptotic behaviour with respect to the Hermite polynomial H,.

385

Theorem 2.3. Let (PO,11.1) denote a symmetrically coherent pair of measures of Hermite type. Denote

+

A*n = r ( n / 2 1) ' r(n) and O(X) is given by Eq. (10).

(a) For n + 00, the polynomial SLH" satisfies

X*e-iz2SLH1)(z) n = O(A) Fn(z)+ o ( l ) ,

z E R,

where Fn(z) = X;e-fz2H,(z). The convergence is uniform o n compact intervals of the real line. (b) For n .+ 00, the polynomial SiH2)satisfies

'. where Gn(z) = {X;-lzHn-l(z)- ( a ~ X ~ - 2 H , , - ~ ( z ) } e - ~ z The convergence is uniform o n compact intervals of the real line. To prove this Theorem we need two technical results:

Lemma 2.1.

(a) Let X; be defined in Theorem 2.3. Then %+l

1

n + 00. A,: &' (b) The asymptotic behaviour of F,(x) defined in Theorem 2.3. - N -

uniformly o n compact intervals of the real line.

(18)

386

(b) See page 200 in

0

8.

Proof of Theorem 2.3. (a) F’rom Eq. ( 5 ) we have A;+2 e - i x 2 H n+2 (X) =

If A* , - + z 2 S ( H 1 ) n

n

(z) = e(x) Fn(z)

then we only need to prove limndm &(x) (20) as

=

+ Rn(z),

0. Now, we can rewrite Eq.

and so

with 0 have

< l/(p(2X

+ 1) < 1. On the other hand, from Eqs.

(7) and (19) we

From Eqs. (21) and (22) we can use standard arguments to prove limn+m &(z) = 0. Therefore, (a) of this Theorem is proved. (b) The proof of case (b) is similar to case (a). First, notice that from the three-term recurrence relation for Hermite polynomials, as well as from Eqs. (9) and (18), we can express the left hand side of Eq. (5) as ~ n + 2 ( 5+a;:;) )

F H n ( z )=

(X:,+

2 z ~ n + 1 ( -z 2) l a l ~ n ( z )

Xn+l

.

387

Now, write

XLl

+

-+.z

SiH2)(z) = O(X) G,(z) W,(z). 2 Taking into account the expression obtained for the left hand side of Eq. (5) we obtain -e

-*

Wn+z(= ~)

An-

+

+

akHZ) Wn(z) (1 - e(A))(Gn+2(z) G,(z))

1

Notice that

G,(z) = zF,-i(z)

- IaIFn-z(z)

(23)

and so 1

Gn+z(z)+ Gn(z)= O(-).

(24)

6

Therefore, from Eqs. (7) and (18) we get

with 0 < l/p(2X

+ 1) < 1, and from Eqs.

+

(7), (18), and (24) we deduce

+

lim [(I - e(A))(Gn+2(z) G n ( z ) ) G n ( z )x

n-+m

We deduce limn..+00W, = 0 using the same steps as in (a) and therefore (b) follows. 0

Remark. From Eqs. (19) and (23) we have for n -+ 00

3. Zeros and its asymptotics

’.

The zeros of Hermite-Sobolev polynomials were studied in Applying Gaussian quadrature formulas and other technical tools, the location of the zeros of Hermite-Sobolev polynomials for both types I and I1 is deduced. We summarize the results as (see Theorems 4.7, 4.8, 4.10, 5.7, 5.8, 5.10, 6.6 in ’):

388

Theorem 3.1. Let hn,1 < . . . < hn,m denote the positive real zeros of Hermite polynomials Hn . Then

(a) The zeros of SLH” and S2n+l ( H a are real and simple. S i F ) has at least 2n - 2 real and simple zeros. (b) Let pn,l < . . . < pn9m denote the positive real zeros of polynomials orthogonal with respect to the weight function ( x 2+ a 2 ) e-”’ and let < . . . < siHh) denote the positive real zeros of the polynomials SiH”. Then, for n 2 3,

SLY)

SAY)< . . < hn>m< s ~ , SAY)< pn,1 < . . . < sAHA) < pn,m. hn,1 <

(HI)

,

,

~ (25)

(c) For n 2 3, the zeros of SLY:) interlace with the zeros of SiH” (d) Let < . . . < siH2) denote the m - 1 largest positive real zeros of polynomials SiH2’. Then,

SAY) hn,1

<

hn-2,1

SAY)< < ’ . < SLY’ < ... < hn,2

< hn,m-l < sn,m ( ~ <~hn,m, ) hn-2,m-l < s~H:). ’

(26)

Furthermore, if SLH2)has n real and simple zeros and denoting the smallest positive real zero, then f o r n 2 3 we have by < hn,l.

SLY)

SAY)

Sketch of the proof. The proof uses information about the sign of certain integrals and Gauss-quadrature. For both types we introduce

1

00

Il :=

$m(x)7r(x)e-”2dx,

J-00

J-00

where n ( x ) is an even monic polynomial of degree 2 k , with all its zeros on the real line, and ( S n )are the monic HermiteSobolev polynomials. For SLH1’(x)= xn . . . we then have 11,I2 # 0 and:

+

type I:

sgnIl= ( - - I ) ~ + ~

( m 2 2, 1 F k F m, n ( z )# x 2 ) ,

sgnI2 = (-l)m+k

( m 2 1, 0

5 k 5 m),

(28)

389

while for SiH2'(x) = xn type 11:

I

+ . . . again I1,I 2 # 0 and the signs satisfy now

s g n I I = (--I)"+"' sgn12

= (-~),+~+l

( m 2 2, 1 I k I m - I), ( m 2 1, o 5 k 5 m - 1).

(29)

The proof of these facts uses recurrence relations for the moments of the measures and is left out here. Let now, for instance, hl < . . . < h, be the positive real zeros of the monic Hermite polynomials fin (x)and consider

then the integrals in Eq. (27), with ~ ( x = ) TI(X) for n = 2m and ~ ( x = ) T ~ ( x for ) n = 2m 1, lead to

+

Application of Gaussian quadrature on the zeros of Hn shows

or sgnSiH1)(hi)= -sgnHA(hi), as hi > 0. Now HA has opposite signs at consecutive zeros, and thus the same holds for SAH1). Then there is a zero of this Sobolev polynomial on the interval (hi, hi+l) and on (-hi+l, -hi) (because the odd-index polynomial is odd and the even-index polynomial is even, real zeros appear in pairs h i , together with x = 0). This accounts for 2(m - 1) real zeros. Moreover, fi;(h,) > 0 implies SiH1)(h,) < 0: because is monic, there is another zero for the Sobolev polynomial for x > h,, and then also one with x < -hm. This accounts for 2 more real zeros. Finally S ~ ~ ~ l=( 0, 0 leading ) to the final real zero in the odd-index case. This proves the statement in (a) for type I.

,!?AH1)

Type I1 can be treated in the same way, starting from Eq. (29), for the even-index case. For the odd-index case one needs an interlacing property for the points where the Sobolev polynomial and the monic polynomials orthogonal with respect to dpl have their extremata i.e. points where SA(x) = 0 and QL(x) = 0.

390

The other statements also use Gaussian quadrature and the integrals in Eqs. (28) and (29); there is only one case where the Gaussian quadrature with remainder term is needed: statement (d) about the interlacing property with the hn-2,i. For

with hl < . .. < h,-l the positive zeros of f i n - 2 ( 5 ) , application of Gaussian quadrature on the zeros of H n - 2 has a remainder term I?:-2(~)dp0 > 0 because the upper bound for exactness is 2(n-2) - 1 = 2n - 5 and the degree of the integrand is n ( n - 2 - 2) = 2n - 4,leading to

sw

+

and we can proceed as before.

0

Remarks.

0

In Theorem 5.15 in it is also shown that the zeros of the polynomials Q 2 n orthogonal with respect to the weight Eq. (6) interlace with those of S ’,:i and the zeros of Qzn+l interlace with those real ( H 2 ) zeros of S2n+2. It is clear that for large n, Sif2) does not have complex zeros. But, for small n , complex zeros can appear. In a study of the zeros of has been done when X -+ 00. In this paper the authors state that the limit polynomials: Sif2’m’(~),n 2 2, have complex zeros when la1 < 0.21 and have no complex zeros when -1.1 > 2.5. An open question about the complex zeros of these polynomials is posed in 3: what happens if 0.21 < )a1 < 2.5?

Sif2)

The study of the asymptotics of the zeros of Hermite-Sobolev polynomials has been done in for type I and in for type 11. They follow from some generalizations of the Mehler-Heine type formulas for Hermite polynomials. We have (see Theorem 1 in and Theorem 1 in 7,

Theorem 3.2.

391

(a) Strong asymptotics. Let SLH" be the Hermite-Sobolev orthogonal polynomials of type I. Then,

hold both uniformly o n compact subsets of C . (b) Strong asymptotics. Let SiH2)be the Hermite-Sobolev orthogonal polynomials of type II. Then,

hold both uniformly o n compact subsets of C . Again,

'p

and e(A) are given by Eqs. (8) and (lo), respectively.

Sketch of the proof. Using the relation between Laguerre and Hermite polynomials (see page 106 in ') we can rewrite the Mehler-Heine type formula for Laguerre polynomials given in page 193 in as

'

both uniformly on compact subsets of C and uniformly on j E N U 0. (a) For the even case, we make the change of variable z H z / ( 2 & ) in Eq. (5) and applying this formula in a recursive way, we get

(- 1 )

fiSE1)(/. 22n n!

n

k=O

( 2fi) )

392

Now, using Eqs. (7) and (30) we can assure that limn,,jn,k(z/(2fi)) exists and holds uniformly on compact subsets of the complex plane and, furthermore, we can find a majorant convergent sequence for j n , k ( f i Z ) on compact subsets of the complex plane. As in Theorem 2.2, the result follows from the application of Lebesgue’s dominated convergence theorem. We can proceed in a similar way when n is odd but now using Eq. (31). (b) As in (a), now we can obtain for the even case

In Proposition 4 in

it is proved that

uniformly on compact subsets of the complex plane. Finally, in order to obtain the result we can proceed as in (a). To prove the result for the odd case it is necessary to take into account (see Proposition 4 in 7,

uniformly on compact subsets of the complex plane. It is important to highlight that the key to prove Proposition 4 in

Eq. (9).

is

0

Therefore, using Hurwitz’s theorem in the results of this theorem we obtain the same limit relation between the zeros of Hermite-Sobolev polynomials for types I and I1 and the zeros of the elementary trigonometric functions cos(z) and sin(z). That is (see Corollary 1 in and Corollary 2 in 7),

Corollary 3.1. With the notation of Theorem 3.1, we have

Remark. In the Mehler-Heine type formulas as well as the zero asymptotics for the standard polynomials Qn are deduced. In fact, the asymptotic behaviour of the smallest zeros of the Hermite polynomials H,, the polynomials Qn , and the Hermite-Sobolev orthogonal polynomials (types I and 11) is the same.

393

References 1. M. Alfaro, J.J. Moreno-Balcfizar, T.E. PBrez, M.A. Piiiar and M.L. Rezola, J. Comput. Appl. Math. 133, 141 (2001). 2. L. Castaiio-Garcia and J.J. Moreno-Balcbar, J . Comput. Appl. Math. 150, 25 (2003). 3. M.G. de Bruin, W.G.M. Groenevelt and H.G. Meijer, Appl. Math. Comput. 132, 135 (2002). 4. J.S. Geronimo, D.S. Lubinsky and F. Marcellfin, Constr. Approx., 22, 309 (2005). 5. A. Iserles, P.E. Koch, S.P. Norsett and J.M. Sanz-Serna, J. Approx. Theory 65, 151 (1991) 6. H.G. Meijer, J. Approx. Theory 89, 321 (1997). 7. J.J. Moreno-Balcbzar, J. Comput. Appl. Math. 179, 289 (2005). 8. G. SzegB, Amer. Math. Soc. Colloq. Publ., 23, (1975). 9. W.Van Assche, Lecture Notes in Mathematics, 1265, (1987).

394

ASYMPTOTIC BOUNDARY VALUE PROBLEMS FOR DISCRETE SYSTEMS

M. MAFUNI AND S . MATUCCI Department of Electronics and Telecommunications University of Florence, I-501 39 Florence, Italy E-mail: [email protected], [email protected]

P. REHAK* Mathematical Institute Academy of Sciences of the Czech Republic ZiZkova 22, CZ-61662 Brno, Czech Republic E-mail: [email protected]

The existence of nonoscillatory solutions with prescribed asymptotic behavior is studied for a nonlinear system of two coupled second order nonlinear equations. A general existence theorem for functional boundary value problems, stated by the authors in [l],is employed and the main advantages of this approach are illustrated.

1. Introduction The existence of a class of positive solutions of the difference system A(@.,(AZk)) =

-f@,

Ykfl),

A(@'P(AYk))= g ( k , Zk+l),

(1)

is examined, where A is the forward difference operator defined by Ax, = ~ k + 1- xk, @~(u) = Iz1IA-'sgnu with > 1, and f,g : N x IW -+ IW are continuous functions, nondecreasing with respect to the second variable, such that uf(lc,u)> 0, ug(lc,u)> 0 for every u # 0 and k E N. System (1)can be viewed as the discrete counterpart of a coupled nonlinear differential system with plaplacian operators which appears in studying spherically symmetric solutions of certain nonlinear elliptic systems (see, *Supported by the Grant No. KJB1019407 of the Grant Agency of ASCR and by the Grant No. 201/04/0580 the Czech Grant Agency.

395

e.g., [2] and the references therein). System (1) also covers a wide class of fourth order nonlinear difference equations (see, e.g., [3, 41). The oscillation and nonoscillation of (1) have been considered in [5] and in [l,61, respectively. This paper is a continuation of [l]: here we deal with certain new nonoscillatory boundary value problems (BVP’s) associated to (1). Our approach is based on a general existence result stated in [l]. In Section 2 we recall such a result, completing the discussion with some remarks and comments about the advantages of this approach with respect to the use of well-known fixed point theorems as Schauder or Tychonoff theorems. In Section 3 we classify solutions of (1) with respect to their asymptotic behavior and we apply the previous method to solve some BVP’s associated to ( l ) ,which complete the results in [l]. 2. The topological approach

Boundary value problems in infinite intervals can be tackled by means of several approaches. Besides the most classical ones, as the use of the Schauder [Schauder-Tychonoff] fixed point theorem in a suitable Banach [Frechkt] space, new methods have been developed recently. We quote, for instance, the nonlinear alternative of Leray-Schauder type in Frkchet spaces, or the so-called cone compression and expansion theorem in Frkchet spaces, which has a close connection with the well-known Krasnoselskii fixed point theorem on cones in Banach spaces. We refer to the monographs [7, 81 for a survey on this topic. Let

N

P be the Frkchet space of all real sequences defined on N,

=

{k E

2 m}, m E N, endowed with the topology of uniform convergence on [m,n]= {m,m+ 1 , . . . ,n} for every n E W,. Further, let IFN be the : k

Frkchet space of all N-vector sequences endowed with the topology induced by the Cartesian product. An element of FN will be denoted by z. In the solvability of BVP’s the use of the Frkchet space I F N , instead of a suitable Banach space, is advantageous especially for the compactness test. Even if this is true also in the continuous case, in the discrete case the situation is even more simple, since any bounded set in IF is relatively compact (see, for instance, [8, Theorem 5.3.1]), and the same holds also in I F N . We recall that a set R c P is bounded if and only if it consists of sequences which are equibounded on every compact subset of N,, i.e., if and only if there exists z E IF such that lzkl 5 zk for each k E N, and z E R. The compactness test is therefore very simple just owing to the topology

396

of IF. On the other hand, the compactness in discrete Banach spaces can require some checks which are not always immediate. For instance, in the Banach space em of all real bounded sequences defined on N,, according t o [9, Theorem 3.31, R C em is relatively compact if it is bounded and uniformly Cauchy in em, i.e., for every E > 0 there is n E N, such that lxi - z j l < E whenever i , j 2 n and z E R. In particular the above conditions are necessary and sufficient in the Banach space of bounded sequences having finite limit a t infinity, as the following result shows.

Proposition 2.1. Let infinity. A subset R c and uniformly Cauchy.

er be the space of sequences having finite limit at tr is relatively compact if and only if it is bounded

Proof. The assertion can be obtained as the discrete counterpart of a result in [lo]; we give a sketch of the proof for reader’s convenience. The sufficient part follows from [9, Theorem 3.31 (see also [8, Remark 5.3.11). Let us show that, if R is relatively compact in then it is bounded and uniformly Cauchy. For every E > 0 there exists a finite +net for R: {Q’,... ,QT}. Let M = maxj=1.., llQT1lm. Hence for every q E R there exists j E { l , . . ., r } such that 114 - Qjllm < E , which implies that R is bounded and llqllm 5 M for every q E R. For every j E { l , . . ,r}’ . let nj = n j ( ~such ) that IQi-QiI < E for k,s > nj. Putting fi = maxj,l, ... ,T nj we obtain IQi - Qil < E for k , s > and for every j = l , . . .,r. Let w E R, and let j E (1,.. . , r } such that IIw - Qjllm < E . Then, for k , s > f i , we have IWk - w,I I IWk - Qi1 Iw, IQi - Qil < 36, which gives the assertion.

lr,

+

+

If the boundary conditions involve the asymptotic behavior of unbounded sequences, to work in a Banach space it is necessary t o choose a weighted space as, for instance, e: = {z E P : SUpkEN, W k 1 z k l < 00)’ with w : N, ( 0 , ~ and ) IIxllw = SUpkEN, Wklzkl. The check of compactness in this case is even less immediate. Theorem 3.3 in [9], with a clear modification due to the different norm, works also in this space, otherwise it is possible t o use the following criterion, whose proof can be easily derived, with minor changes, from the continuous counterpart [ll]. .--)

Proposition 2.2. Let {hk} be a positive sequence defined o n N, such that limk,,wk/hk = 0. Then R c t: is relatively compact i f it is bounded with respect to the norm 11 . llh, i.e., there exists M > 0 such that SUPkEN, hklXkl < M for every 2 E a.

397

To summarize, the choice of the Frechet space IF (or I F N ) , allows to treat in the same manner problems having bounded and unbounded solutions, and reduces substantially the complexity of calculus. In a recent paper [l,Theorem 11 we proposed a different approach, which is a generalization of a previous result [12, Theorem 1.21 stated in the continuous case. Theorem 1 in [l]works for functional difference equations and therefore allows to treat in a unifying way, for instance, equations with deviating arguments or the discrete version of integral-differential equations. We recall here our abstract existence result. Consider the functional boundary value problem (FBVP)

where F : N, x RN x IFN I F N . The following holds.

-+

RN is a continuous map, and t? is a subset of

Theorem 2.1. ([l]).Let G : N, x such that, for ( k , g ) E N, x IFN

G(k,u,,

x IF2N

.+

RN be a continuous map

u,, u, u) = F ( k ,u,, u).

If there exists a nonempty, closed, convex and bounded set R c IF" such that: a) for any q E R, the (Yinearized") problem

has a unique solution y = T(q); b) T(R) c R; c ) T(R) c t?; then ( 2 ) has at least one solution. Notice that if the boundary set t? is closed, as it often happens for boundary value problems on finite discrete intervals, then condition c) is trivially satisfied. For BVP's on infinite discrete intervals, if the boundary conditions involve the behavior of the solution at infinity, i.e. t? is, roughly speaking, of asymptotic type, then 23 may be not closed in I F N , as, for instance, it happens if t? contains sequences approaching zero, or sequences for which the first quasidifference tends to a constant in R".

398

Observe that condition b) implies the relative compactness of the set

T(R), while condition c) (assuming b) satisfied) implies the continuity of the operator T . Condition b) can be usually verified by applying the Lebesgue dominated convergence theorem and therefore, roughly speaking, both condition a) and b) are often a simple consequence of good a-priori bounds. As pointed out in [l,Remark 31, condition c) can be replaced by the weaker condition

c l ) if {qi} is a sequence in R converging in R and T (-q i )4 9" (in the qw E B. topology of FN), then In particular, if conditions a) and b) are satisfied, then it is easy to verify that condition c1) is also necessary for the continuity of T in R. Finally, Theorem 2.1 enables to treat problems in which the dependence of the map F on the unknown is of functional type, while this possibility is not considered in the continuous case [12, Theorem 1.21. Such a functional dependence allows to apply our result t o a wide class of BVP's, involving e.g. the difference equations with deviating arguments or sum-difference equations. Another advantages in the use of Theorem 2.1 lies in the possibility of applying suitable changes of variables which lead to FBVP but reduce the complexity of the computations. An application of this approach is given in the next Section: even if our original problem is not of functional type, we use a change of variables which reduces the order of the problem, leading to a functional difference system. 3. Two asymptotic boundary value problems

In this Section we study the existence of some nonoscillatory solutions of (1) with prescribed asymptotic properties. By solution of (1) we mean a vector sequence (2, y ) = ({xk}, {yk}) satisfying (1) for k E W. The component x [y] of a solution is called nonoscillatory if it is positive or negative for large values of the index. Otherwise it is called oscillatory. In view of the sign assumptions on f,g , both the sequences x,y have the same behavior with respect to oscillation, i.e. either x,y are both nonoscillatory or x,y are both oscillatory. Thus a solution (x,y) is said to be oscillatory or nonoscillatory according to x,y are both oscillatory or nonoscillatory. System (1) is said to be oscillatory if all its solutions are oscillatory. In [5], where the conditions guaranteeing oscillation were derived, we also established basic properties of nonoscillatory solutions of (1). In particular, any nonoscillatory solution (x,y ) of (1) belongs

399

to one of the following four classes (for large k):

(c1)xkyk > 0, zkzF1> 0, ykyF1 < 0; (c2) Z k Y k (c3)xkyk (c4) xkyk

> 0, < 0, < 0,

zkzL1 > 0, 9kYF1 > 0; xkxF' < 0, YkYF1 > 0; 2kxf1 > 0, yky;' > 0.

The complete characterization of these four classes follows from the study of positive elements of the classes ((21) and (C2), thanks to the symmetry properties of (1). Each of these classes can be divided according to the limit behavior z, y, and y['] denote the quasidifference and y[l], where of z and y, respectively, i.e.,

In particular, every solution (z,y) in ( ~ 1satisfies ) lirnk,, $1 = 0, while in (c2)we have lirnk,,yk = 00, see [5]. In [I], we studied solutions of class (C2) for which both components are unbounded. Our aim here is to complete the examination of class (C2), studying the remaining two subclasses of (C2):

where Mx,MLE (0,co). In addition to the conditions posed on the nonlinearities f and g in Section 1, here we assume that both these functions are regularly varying at infinity (RVI). We say that a continuous function h : N x R --+ R is RVI if limsup,,, h(k,X t ) / h ( k ,t ) < oo for any X > 1 and k E W. Taking into account that f and g are also nondecreasing with respect to the second variable, then for every a,X > 0 there exists C = C(a,X) > 0 such that f (k,A t ) 5 C f ( k ,t ) for every t E [a,oo) and k E N. Analogously for g. The next result gives necessary and sufficient conditions for the existence of solutions in class (I). In what follows, a* and ,B* denote the conjugate numbers of a and p, respectively, i.e., 1/a l/a* = 1 and 1/,B 1/p* = 1. We also make use of a usual convention, namely CiIAak = 0 for any sequence a and any n E N.

+

+

400

Theorem 3.1. System ( 1 ) has a solution in class (I) if and only if 00

(4) k=l

(5) Proof. Necessity. Let ( z , y ) be a solution of (1) in class (I). By summing from k to 00 the second equation in (1) we obtain

c 00

ML

=

-

00

g ( j ,x j + l )

j=k

2

Cdj,

zk),

j=k

and thus condition (4) follows. Condition (5) follows from summing the first equation in (1) twice, i.e.,

\ i=j

j=k

and taking into account that limk,, y k / k E (0,oo). Suficiency. Let A > 0 and B > 0 be two positive constants, and m E be a sufficiently large integer such that 00

c 00

g ( k A ) S B and k=m

("

k=m

f(j,@ p * ( B ) ( j - m

1

+ 1))

N

I A. ( 6 )

j=k Let Si : IF -+ IF, z = 1,2, be the operators defined by S l [ w ]= {(S1[w])k}, S 2 [ 4 = { ( S 2 [ 4 ) k } ,where

c 00

(Sl[W])k= A -

@a*(wj), (s2[z])k =

j=k

c

k-1

@p*(zj).

(7)

j=m

Consider the FBVP

Awk = -f (k,( S z [ z ] ) k + l ) , Azk = g(k,(S1[W])k+l), lim w k = 0, lim .z'k = B . k-00

(8)

k+cn

Observe that (8) has the form (2) and therefore we can apply Theorem 2.1 to solve it. Let R c IF2 be defined by

40 1

and for every (u, w) E R consider the linearized BVP

Awk = -f(k, (SZ[v])k+l), Azk = g ( k , (Sl[u])k+l), lim W k = 0, lim Z k = B. k-co

(9)

k-wx

It is immediate to check that (9) has a unique solution ( w , z ) = T ( u , v ) , where T : R H IF2 is given by T ( u ,w) = (Tlv,Tzu),with 00

00

j=k

j=k

k 2 m. In view of condition (6), the estimates

(10) imply that the map T is well defined in R and T(R) & R. In order to apply Theorem 2.1, it remains to show that for every sequence (ui,wi)in R such that (ui, wz) (uo,wo) in R and (T1vi,T2ui) + (wo, zo) in IF2, we have --f

( w o , z o )E B

=

{(w,z)

E

IF'

: lim W k = 0, lim Z k = B k+m

k-+m

1.

This is an easy consequence of the discrete Lebesgue dominated convergence theorem, whose applicability is guaranteed by estimates (10). Theorem 2.1 gives the existence of at least a solution ( w , z ) of (8). Then (5, y) = ( S ~ [ W S z][, z ] )is a solution of (1) in class (I) with M , = A and MIY = B. 0

Remark 3.1. Theorem 3.1 implies in particular that for any A, B > 0 , there exist infinitely many solutions (5, y) of (1) in (I) satisfying M , = A and Mh = B . This can be proved by using the same arguments, with minor changes, modifying the definition of the operator S2 in (7) as (&[z])k= a [email protected]*(zj), where a is an arbitrary nonnegative constant. Notice that a plays, roughly speaking, the role of the initial value for the second component of a solution, i.e., ym = a.

+

The following result deals with the existence of solutions in the class (11).

402

Theorem 3.2. System (1) has a solution in class (11) if and only i f /i-1

\\\

Proof. The assertion can be obtained by using similar arguments t o that given in the proof of Theorem 3.1, with minor changes. We omit the details; note that, in this case, the FBVP t o be solved is (8) with the boundary conditions: limk wk = 0, zm = 0. The set 0 becomes

where A is a positive constant. The fixed point operator T is T ( u , v ) = (Tlv, Tzu),with

cf(j,

(T1V)k

=

c

k-1

oo

(S2bJl)j+l),

(T2U)k =

j=k

9 ( j , (SlbI)j+l).

j=m

A slight modification yields also a multiplicity result similar t o that given in Remark 3.1. Theorems 3.1 and 3.2 can be easily stated for more general systems with deviating arguments of the form A(rk@cx(Axk)) = -f ( k i Yo(k)) A(qk@P(AYk))

=

(ki

>

x:e(k))7

(11)

where r, q are real positive sequences such that

and c,e : W

-+

Z,limk ~ ( k=) 00, limk a ( k ) = m.

References 1. M. Marini, S. Matucci, P. Rehcik, Boundary value problems for functional

difference equations on infinite intervals, Adv. Differ. Equ., to appear.

403 2. P. ClBmen, J. Fleckinger, E. Mitidieri, F. de ThBlin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Diff. Equat. 166 (2000), 455-477. 3. M. Migda, A. Musielak, E. Schmeidel, On a class of fourth-order nonlinear difference equations, Adv. Differ. Equ. 1 (2004), 23-36. 4. J. Yan, J., B. Liu, Oscillatory and asymptotic behaviour of fourth order nonlinear difference equations, Acta Math. Sinica 13 (1997), 105-115. 5. M. Marini, S. Matucci, P. Rehtik, Oscillation of coupled nonlinear discrete systems, J. Math. Anal. Appl. 295 (2004), 459-472. 6. M. Marini, S. Matucci, P. Rehtik, Strongly decaying solutions of nonlinear forced discrete systems, in: Proceedings of the Sixth International Conference on Difference Equations, Taylor and Francis, 2004, 493-500. 7. R. P. Agarwal, M. Meehan, D. O’Regan, Fixed Point Theory and Applications, Cambridge Tracts in Mathematics 141, Cambridge University Press, 2001. 8. R. P. Agarwal, D. O’Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, 2001. 9. S. S. Cheng, W. T. Patula, An existence theorem for a nonlinear difference equation, Nonlinear Analysis 20 (1993), 193-203. 10. C. Avramescu, Sur 1’Bxistence des solutions convergentes d’kquations diffBrentielles non linbaires, Ann. Mat. Pura Appl. 4 (1969), 147-168. 11. K. Zima, Sur l’existence des solutions d’une Bquation int6gro-diffbrentielle, Ann. Polon. Math. 27 (1973), 181-187. 12. M. Cecchi, M. Furi, M. Marini, On continuity and compactness of some nonlinear operators associated with differential equations on noncompact intervals, Nonlinear Anal. TMA 9 (1985), 171-180.

404

THE LIMITS OF THE SOLUTIONS OF A LINEAR DELAY DIFFERENCE SYSTEM*

H. MATSUNAGA Department of Mathematical Sciences, Osaka Prefecture University Sakai 599-8531, Japan E-mail: [email protected]. ac.jp

In this paper we will classify the limits of the solutions of a linear delay difference system completely. In paticular, if the solution tends to an equiliblium point or a periodic orbit, we will give the explicit expressions in terms of the initial conditions and the delay parameters.

1. Introduction

We consider a linear delay difference system ~ , + 1 - 2 , = A ( x , - ~- z,-e),

71. = O,1,2,.

.. ,

(1)

where A is a 2 x 2 real constant matrix, the delays k and l are nonnegative integers with k > l , and the initial conditions are as follows:

x,=&

for n = - k , - I c + l ,

. . . ,0.

(2)

Notice that for the system (1) each point on R2 is an equilibrium point. Driver et al. [2] first gave the limit of the solution of (1). That is, if

(k - Wll < 1, then the solution

2,

(3)

of (1) with (2) satisfies k

where IlAll is the operator norm of A and I is the identity matrix. Also, several authors ([l, 2,4]) have studied the limits of the solutions of more general linear and nonautonomous delay difference systems. "This work was partially supported by a grant for the Basic Science Research Projects of the Sumitomo Foundation.

405

The purpose of this paper is to classify the limits of the solutions of (1) completely. In paticular, if the solution of (1) tends to an equiliblium point or a periodic orbit, we will give the explicit expressions in terms of the initial conditions and the delay parameters. We note that in case e = 0, the author et al. [3] have already solved this problem. By the transformation x, = Pun where P is a nonsingular matrix, the system (1) can be written as

un+l-

U, = P-lAP(u,-k

- un-e),

n = 0 , 1 , 2 , .. . .

(: a) ,

Thus it suffices to consider the matrix A in the three cases of Jordan form:

(

),

( ),

(ii) A = a1 0 0 a2

(iii) A

where a , al, a2 and 8 are real numbers with 0

< 181 5 7r/2.

(i) A = a cos 8 - sin 8 sin8 cost9

=

2. Preliminaries In this section we summarize some auxiliary results which are useful to study the asymptotic behavior of the solutions of (1). Let

where d = 2(k

+ 1). Then (1) becomes an ordinary difference system Yn+l

= BY,,

(5)

where B is the d x d matrix defined by

1

k--L+1

k + l

Hence, the solution y, of (5) with the initial value yo is given by Yn = Bnyo.

The characteristic equation for (5) is the form

F(X) 3 det(X1- B ) = 0.

406

We will decompose the solution of (5) to observe its behavior. Let F ( X ) = (A - A,)vl(X

- X2)VZ

.*.(A

- X,)Va,

where XI, X 2 , . . . ,A, denote distinct eigenvalues of B and v1+v2+. .+v, = d , and let V1,V2,. . . ,V, denote generalized eigenspaces of B belonging to XI, X 2 , . . . ,A,, respectively. Then, by a well-known theorem from linear algebra, the solution space Rd is expressed as a direct sum

R d = & @ V 2 @... @V,. Moreover, for each j , 1 5 j 5 s, the projection of Rd onto V, is represented by a d x d matrix Pj which satisfies the relations

+ P2 + + P, = I , ~ j =” ~ j ,P,P~= o (i # j ) . PI

(7) (8)

* * *

Then by (7) we rewrite the solution yn of (5) as Y n = PI ~n

+ P2 + .. . + P . yn. ~n

Here, for the projected solution Pj yn, the following result is known. Proposition 2.1. If l X j l < 1, then Pj yn tends to the zero vector as

TI

4

00.

We will now give the projection Pj explicitly. For brief, let A j be simple, $j denote a basis of V, (i.e. B& = Xj&), and $j denote an eigenvector of BT belonging t o X j (i.e. $jB = Xj$j). Then it is easy to see that

From the spectral decomposition of B , we have

B = A,P1

+ X2P2 + . . + ASP,, ‘

which, together with (8), implies that PjB = XjPj. Therefore we can obtain

Pj yn = Pj Bnyo = XnP. 3 3 yo.

(10)

Remark 2.1. In case X j is a multiple eigenvalue, the explicit expression of Pj yn corresponding to (10) is also given, provided that X j is semisimple, that is, the multiplicity of X j = dimKer ( A j I - B ) .

407

3. Characteristic equation

In this section we investigate the roots of the characteristic equation for (1) det(Xk+lI - XkI

+ (Ak-'

- l ) A ) = 0.

(11)

Notice that by ( 6 ) det(X1- B ) = det(X"l1-

Xkl

+ (Ak-'

- 1)A).

Hereafter we only consider the case where the matrix A is given by

)'

A = , ( cos 0 - sin 0 sine cose

where a and 0 are real numbers with 0

XkI

det(X"'1-

= -

I

xk+l

-

+ (Ak-'

< 181 5 1 ~ 1 2Then . we have

- l)A)

xk + a ( ~ k - --e 1) cos 8

a(Xk-e - 1) sine

P+l-xk +

{ x ~ + '- xk + u ( x ~ - ' - 1)cos e}2 + {a(Xk-'

-{ ~ ~ + X~ l

+ a ( ~ ~ - ' 1) cos e}2 - {ia(~"' -

- { ~ k + ' - ~k + aeie(~k-' - 1)){,,jk+' -

=

- 1) sin 8

-a(X"'

(A

-

- ~k

- 1)

e)2 1) sin e}2

- 1) sin -

+ ae-ie(X"'

-

111

1)2f(X)f(x)'

where

f(x) = ~k + ae"(Xk-'-' + A'-'-'

+ . . . + 1).

This, together with 0 < 101 5 n/2, implies that X = 1 is a double root of (11). Thus it is sufficient to discuss the roots of f(X)f(x) = 0. There are two possible cases t o consider.

+

Case (I): k = L 1. Let X be a root of f(X) = 0. Then, from l X k l = la[, we can immediately obtain the following lemma.

Lemma 3.1. Suppose that the mat*

A is given by (12) and k

If la/ < 1, then (11) has a double root X

=C

+ 1.

= 1, and the remaining roots are inside the unit disk. (ii) I f a = -1, then (11) has a double root X = 1 and 2k-simple roots X = exp(fi(0 2 m n ) / k ) f o r m = 0 , 1 , . . . ,k - 1. (iii) If a = 1, then (11) has a double root X = 1 and 2k-simple roots X = exp(fi(t9 ( 2 m 1 ) 1 ~ ) / kfor ) m = 0 , 1 , . . . ,k - 1.

(i)

+ +

+

408

+

Case (11): k 2 k' 2. By an argument similar to the proof of Lemma 3.1 in [3],we can get the following lemma. For simplicity, we put sin(w,/2) sin(( k - l?)wa/2)'

a=-

'

Clearly, a!

w, =

28 k+l?+l'

(13) sin(wp/2) 2 ( r - 14) -sgn(8) k + 1 . wp = sin((k - t)wp/2) 7 < 0 and /3 > 0. We also note that a! = -1 and /3 = 1 if k = k'+ 1.

e+

Lemma 3.2. Suppose that the matrix A is given by (12) and k 2 k' + 2.

(i) If a < a < /3, then (11) has a double root X = 1 and the remaining roots are inside the unit disk. (ii) If a = a, then (11) has a double root X = 1 and two simple roots X = e f i w a , and the remaining roots are inside the unit disk. (iii) If a = /3, then (11) has a double root X = 1 and two simple roots X = e f i w g , and the remaining roots are inside the unit disk. 4. Main Result

In this section we give the explicit expressions of the asymptotic equilibrium point and the periodic orbit.

Theorem 4.1. Suppose that the matrix A is given by (12). Let x, be a solution of (1) with (2). (i) If

a!

< a < /3, then the solution x, tends to b as n -+ 00, where

(ii) Let Ic = k'

+ 1. If la1 = 1, then the solution x,

is expressed as

k-1

xn = b

+ C Xm(n)cm, m=O

where

k

1 c, = -(I - xm(-l))-l C X , ( j - l)(t-j+l- < - j ) . k j=1

(15)

409

(iii) Let k 2 e

+ 2.

If a

=a

or a = p, then the solution x, tends to the

invariant orbit

xt = b + X(n)c,

(16)

as n -+00, where X(n) =

cos nw - sin nw sinnw cosnw

b=(I

w, ( a = a ) w = { w/3 ( a = P ) ’

),

+ (k - l)A)-l ( t o + A

k t-j)

,

j=e+i c=

( I + kATX(l) - lATX(k))-I k

x (Eo+ATEX(l+j)&

e -ATEX(k+j)Cj).

j=l

j=1

Remark 4.1. In Theorem 4.1 (i), the limit of the solution of (1) with (2) is the same as (4). Remark 4.2. In Theorem 4.1 (iii), from the definition of w, we find that (a) if 8/n is rational, then the orbit x: is periodic; (b) if 8/n is irrational and the initial conditions are not the same values, then the orbit x: densely fills out a circle with its center b and its radius Ic(.

Remark 4.3. If a < a or a > P, then (11) has a root X with 1x1 > 1 and therefore, there exist solutions of (1)which are unbounded as n -+ m. The proof is carried out in a similar way to Theorem 4.1 in [3] and will be omitted. Finally, to illustrate our main result, we consider the case where k = 3, e = 1 and 8 = n/3. Then it follows from (13) that sin(n/ 15) 2 a=M -0.51117, w - -T, sin(2n/15) - 15 ff

Figures 1-3 show some solutions of (1)with the initial values 6, = co1(7/2+ n/3,1+ (-l),), and Figure 4 shows a solution of (1) with the initial values = col(1 - n/3,1 (-l),). On the other hand, Figure 5 and Figure 6 show the periodic orbits x: calculated by (16), which correspond to Figure 3 and Figure 4, respectively.

<,

+

410 Y

Y

4~

3.5 3. 2.5 2 1.5 1

0.5 1 1.5 2 2.5 3 3.5 q X

Fig. 1

(Y

< a = -0.49 < ,B

Fig. 2 a = -0.52

Y


Y 31

41

3.5

2.5

3 2

2.5

1.5

2 1.5

1 1 0.5

0.5 0.5 1 1.5 2 2.5 3 3.5 q X

0.5

1

1.5

2

2.5

3x

2.5

ix

Fig. 4 a = ,B

Fig. 3 a = a

Y 41

2.5 2

2.5 2

1.5

1.5

1.

1 0.5

0.5

’

0 1 5 1 1.5 2 2.5 3 315 q X

Fig. 5 a = a

0.5

i

115

2

Fig. 6 a = /3

411

References 1. R. P. Agarwal and M. Pituk, Convergence to equilibria in recurrence equations, Advances in difference equations 11, Comput. Math. Appl. 36 (1998), no. 10-12, 357-368. 2. R. D. Driver, G. Ladas and P. N. Vlahos, Asymptotic behavior of a linear delay difference equation, Proc. A m e r . Math. SOC.115 (1992), 105-112. 3. H. Matsunaga, R. Ogita and K. Murakami, Asymptotic behavior of a system of higher order linear difference equations, Nonlinear Anal. 47 (2001), 46674677. 4. M. Pituk, The limits of the solutions of a nonautonomous linear delay difference equation, Advances in difference equations 111, Comput. Math. Appl. 42 (2001), no. 3-5, 543-550.

412

RIEMANN-HILBERT PROBLEM FOR A GENERALIZED NIKISHIN SYSTEM

ANA FOULQUIEMORENO* Departamento de Matemcitica, Universidade de Aveiro, Campus de Santiago 3810, Aveiro, Portugal E-mail: foulquie@@mat.ua.pt

Recently it has been shown that multiple orthogonal polynomials are solutions of a Riemann-Hilbert problem. For the class of Generalized Nikishin systems we give the normalization at infinity for this Riemann-Hilbert problem.

1. Introduction Recently, using the approach given by Fokas, Its and Kitaev, it has been observed that there is a natural Riemann-Hilbert problem associated with a system of orthogonal polynomials with weight function W(Z) on the real line. There have been attempts to extend this approach to other kind of orthogonality. In [12] multiple orthogonal polynomials, related to Hermite-Pad6 approximation of type I or type 11, corresponding to a system of Markov functions have been shown to be solutions of a Riemann-Hilbert problem. In [la] the normalization of this Riemann-Hilbert problem for two particular well known systems of Markov functions, the Angelesco and the Nikishin systems have been given explicitly.. In this paper we consider the class of Generalized Nikishin systems, that combine Nikishin and Angelesco systems and were introduced by Gonchar, Rakhmanov, and Sorokin [7]. We give the normalization for this RiemannHilbert problem using the solution of an equilibrium problem. *This work is supported by UID MatemAtica e Aplicqces, by FFRH/BSAB/448/2004 grant and by Intas Research Network NeCCA 03-51-6637

413

2. Generalized Nikishin systems

Let G be a tree graph (or merely a graph in what follows) with m vertices ( m 2 2). We can regard the set of vertices of G as a partially ordered set (G, <) having the inductive property, that is, each non-empty segment { p : /3 < a } has a maximal element a-. We also consider the extended tree graph = G U { w } , where w 6 G is the minimal element of (the root vertex of this graph). We shall write ,O -+ a if p = a-. Let a+ be the set of all minimal elements of the set { p : ,O > a } ; it is clear that a+ is the set of vertices "immediately succeeding" a : a+ = { p : a t p}. We define a "neighborhood relation" between the elements of a+. We shall write p H y if p and y belong to a+ for some a or, equivalently, "immediately succeed" a. Assume that with each a E G we associate a closed interval F, in R and a (finite positive Borel) measure (T, with support in Fa. Thoroughout, we assume that

c,

0

0

two intervals in the set { F , : a E G} are disjoint if there is a relation of "immediate succession" -+ or a "neighborhood relation" between their indices; C J> ~ 0,a.e.in Fa, for all 01 E G.

To each a E G there corresponds a unique chain y-+...+p+a,

YEW+

of elements of G. We call it the defining chain for a E G. Now, for each a E G, and using this defining chain sequence, we are going to define a measure s,, recursively as SLY

= (07,. . ., UP,(Ta) = (07,(.

and d(op,(T,)(x) =

/

. . a g , 0,))

d a " ( t ) d q ( x ) = B,(x)dap(z). x-t

Definition 2.1. By a generalized Nikishin system (GN system) corresponding t o a graph G and a fixed collection of measures {a,},& E G we mean a system of Markov functions

where y -+ . . . -+

p -+ a is the defining chain for the element a.

414

For a GN-system f = { f a : a E G} and a multi-index n = {n,,a E G} (for the sake of simplicity we use n instead of 8)we now consider the type I1 Hermite-Pad6 problem. Find a polynomial Qn such that

Qn(z)$0,

deg Qn

i In1= C LXEG

and

where we have a series in increasing powers of 1/z on the right-hand side and P,,, is the polynomial part of the power expansion for the Qn f a a t the point a t infinity. We will normalize the polynomial Qn with unit leading coefficient.

2.1. Orthogonality relations It follows from (2) that Qn satisfies the following system of orthogonality relations

s

Q n ( z ) z k d s , ( x ) = 0,

k = 0 , 1 , . . . ,n, - 1, a E G.

(3)

Throughout we shall assume that the components of the multi-index n satisfy the following condition

a < P,

np

5 n,

(4)

Starting from the polynomial Qn we consider the functions by induction as follows:

We fix a E

and set

G,

-

=

{ P : /3 > a } , G,

Qn,,

defined

= G, U { a }

c,).

( a is the root vertex of the graph For P E G, we now define the measures s; in terms of the graph G, in the same way as the s, were defined in terms of G = G, (for each P E G, there exists an unambiguously defined chain y + . . . --$ P with y E a+, and we set s; = (ay,. . . , go)). It follows from [7, proposition 11 that for each a E ??the function Qn,, satisfies the following system of orthogonality relations

s

Q n , , ( ~ ) x k d s $ ( x= ) 0,

k

=O,1,.

. . , n p - 1,

,l E ? G,

(5)

415 Now, we introduce the following notation. For an arbitrary

Q

E G we set

kEE,

Let qn,, be the polynomial with unit leading coefficient such that its zero set (with multiplicities taken into account) is the zero set of Qn,,-in the interval F,. Let us denote

If a+ = 0, then the last expression is equal to one by definition. It follows from [7, proposition 21 that for indices p E such that we have the following system of orthogonality relations:

1

Q n , p ( ~ ) ~dua(Z) k = 0,

qn,a+

(XI

k

= 0,1,. . . ,N , - 1,

Q:

p+ # 0

E p+.

(6)

It is also proven in [7, proposition] that qn,, has N , simple zeros in F, and degq,,, = N,. In particular for the Hermite-Pad6 polynomial Q, we have that

and the degree of Qn is In( 2.2. The equilibrium problem

The zero distribution of orthogonal polynomials is usually given by an extremal problem in logarithmic potential theory. The following results were obtained by Gonchar and Rakhmanov [5]. For multiple orthogonality one needs to study an extremal problem given by vector potentials [4],[9]. Suppose that v is a probability measure on the real line and define the logarithmic potential of v by V ( Z )

= /log

1

-dv(x). IZ - 21

Let us assume that

Denote by p(q) the discrete measure associated with the polynomial q. The following theorem appears in [7, Theorem 3).

416

Theorem 2.1. For each a E G we have the limit relations

where p = { p a : a E G } is the equilibrium measure for the interaction matrix A = (a,,p) with elements a,,p such that i ) a,,p = 2e: i f a = p, ii) a,,p = -8,Qp i f a! + ,6 or ,6 3 a (there is an immediate relation between the indices), iii) a,,p = 8,Op if a H p (there is a neighborhood relation between the indices) iv) a,,p = 0 in other cases. The vector equilibrium problem for pa is written explicitly as:

W,.(~C> = e,i,, W,.(X) 2 &Ax,

5 E 5

F,*

E F,

\ F:

(8)

where

p is equal t o a-

and F: is the support set for the measure pa.

3. The Riemann-Hilbert problem

Let us use the following notation: for a set of indices K we can consider M = {Ma,@a,P , E } a MIKlx~K~ matrix complex function. This matrix can be defined by rows as PEK

where a E K (we are expressing these rows as vectors in the canonical basis {e,},a E K ) , i.e., e, is the canonical vector with the a-entry equal to 1 and the others entries equal to 0. In [12, Theorem 3.11 the following theorem for multiple orthogonal polynomials is given. We have rewritten it for the particular notation used for the Generalized Nikishin systems.

Theorem 3.1. Consider the following Riemann-Hilbert problem: deter~ a complex function Z such that mine a M i ~ matrix

417

(1) 2 is analytic in C \ R

(2) O n the real line there is the j u m p condition 2+(2)= 2- (.)W(2)

(9)

where W is defined by rows as:

4. Normalization of the Riemann-Hilbert problem at infinity Let us consider a E G and y 4 . . . ,8 4 a the defining chain for a. Let us denote by l(a) the length of this chain. So for a E w+ we have that l(a)= 1, and so on. We also define s = sup{l(a),

E

G}.

(11)

Let us define for z E C Y1

= ZVI

where Vl is a M c x matrix ~ complex function defined by rows as

{

(K)u(.)

= eu,

(vl)ol(z)

= ea - C P g G , 2 F ( z ) e p ,

( V 1 M z ) = ea,

Clearly Yl is analytic in C \ Ul(,)=l,zFa.

z ( a )= l(a)> 1.

418

J1

Let us calculate the growth condition of Yl(z)as z + 00. Let us consider = J-lVl J that can be defined by rows as

(JlM.) = e w ,

i

( J ~ ) , ( Z= ) e, - CPEG, 2;(z)z-(na-nfl)ep, Z(a)= 1 , ( J d , = e,, l(a)> 1.

using i ; ( z ) z - ( n a - n ~ )= z-(na-na)-lf or z 4 00 and taking into account the condition (4) we get that Yl(z)has the same growth condition as Z ( z ) asz--,0O.

Now, let us calculate the jump condition on the real line for the matrix E R we have

Y1. For x

Yl+(Z) = Y,-(z)W1(x)

where W1(x) = {v,-}-l(z)W(x)vl+(x).

(12) Let us now calculate the jump matrix Wl(x).For doing this we are going to take into account that the matrix product can be seen as the composition of the linear applications associated to these matrices, and that the k-th row is the image by this linear application of the vector e k . Then, describing W1 by rows we get

In general we can define

419

Now using the same kind of arguments as before we get that Y has the same growth condition as Z as z 4 00. For x E R we have

Y f ( 2 )= Y - ( x ) V ( z ) where the jump matrix V ( z )is given by rows as

(V),(x) = e, ( V M x )= e,

+ C&+ 2nia,(x)e,, + &+, 27riop(z)ep

we recall that for indices a such that a+ = 0 the sum is equal to zero. Let p = { p a ,a E G } be the equilibrium measure given by (8) The normalization will be of the form

M

= LYGL-I

where the matrix L is given by rows as ( L ) w ( z ) = ew

{ ( ~ ) , ( z ) e,p-lnl(l5+."+ll;+l~) =

e,.

where y + . . . ,B 4 a is the defining chain for a. We also define the matrix G by rows as

where ga(z) =

/

log(z - Y ) & u ( Y ) ,

o! E G ,

F;=[aE,b;]

and the numbers 1; are obtained by rewriting the equilibrium problem (8) as

-0,

[g,'(x)

+ 9, ().I

ekg,

-

kEP+,k#a

where 1;

-

(x)+

c

ekgkf(x)

+ ePgp

kEa+

1, is an integer multiple of ?,for x E F,* and nI

= 1:

420

and that

Then

[

lirn M ( z ) = L lim Y ( z ) J ( z ) J - ’ ( r ) G ( z ) ]L-l Z W ’

Z-+W

and taking into account (13) and (14) we get lim M ( z ) = I Z+W

Let us now calculate the jump on the real line

M (x)’ = LY + (z)G+(z)L-’ = LY - (X)V (z)G+(x)L= LY - @)G- (Z)L- L{ G- (.)}

’

- v ( ~ ) G).(+ L= M(~)-L{G-(~)}-~v(z)G+(z)L-~.

This matrix can be described by rows as:

Taking into account that

we see that the normalized Riemann-Hilbert problem M ( z ) has a rather simple jump matrix , because on the supports of the measures pa, that is on F,* we see that we have oscillatory terms exp2pa(z) on the diagonal (recall that pa(.) is purely imaginary) and on Fa \ F,* the entries are equal to one.. The other terms that appear decrease exponentially fast provided that 1, > 0. Such a Riemann-Hilbert problem with oscillatory and exponentially decreasing jumps can be analyzed by using the steepest descent method introduced by Deift and Zhou in [1]-[2].

2

42 1

References 1. P. Deift, X. Zhou, A steepest descent method f o r oscillatory Riemann-Hilbert problems, Bull. Amer.Math. SOC.26,119-123 (1992). 2. P. Deift, X. Zhou, A steepest descent method f o r oscillatory Riemann-Hilbert problems, Asymptotic f o r the M K d V equation, Ann. of Math. (2) 137,295-368 (1993). 3. A. A. Gonchar, On the convergence of generalized Pad6 approximants f o r meromorphic functions, Mat. Sb. 98 (140) (1975), 564-577; Math. USSR Sb. 27,503-514 (1975). 4. A.A. Gonchar, E.A. Rakhmanov, On the equilibrium problem for vector potentials, Uspekhi Mat. Nauk 40 (1985), 155-156; English transl. in Russian Math. Surveys 40:4 (1985) 5. A.A. Gonchar, E.A. Rakhmanov, T h e equilibrium measure and distribution of zeros of extremal polynomials, Mat. Sb. 125 (167)117-127 (1984). 6. A.A. Gonchar, E.A. Rakhmanov, On the convergence of simultaneous Pad6 approximants f o r systems of functions of Markov type, Trudy Mat. Inst. Steklov. 157 31-48 (1981). English transl. in Proc. Steklov Inst. Math. 64,(1989) 7. A.A. Gonchar, E.A. Rakhmanov, and V.N. Sorokin, Hermite-Pad6 Approximants f o r systems of Markov-type functions, Mat. Sb. 188 (1997), no. 5, 33-58; Sb. Math. 188,(1997), no. 5, 671-696. 8. E. M. Nikishin, On simultaneous Pad6 Approximants, Math. USSR Sb. Vol. 41, 409-425, (1982). 9. E. M. Nikishin, V. N. Sorokin, Rational Aproximations and Orthogonality, Translations of Mathematical Monographs 92 Amer. Math. SOC.,Providence RI, (1991). 10. E.B. Saff, V. Totik, Logarithmic Potentials with external Fields, Grundlehren der Mathematischen Wissenschaften 316,Springer-Verlag, Berlin, (1997). 11. H. Stahl, V. Totik, General Orthogonal Polynomials, Enciclopedia of Mathematics and its Applications, Cambridge University Press, (1972). 12. W. Van Assche, J. Geronimo, and A. Kuijlaars, Riemann-Hilbert problems f o r multiple orthogonal polynomials, in Special Functions 2000: Current Perspectives and Future Directions (J. Bustoz et al., eds.), NATO Science Series 11. Mathematics, Physics and Chemistry 30, Kluwer, Dordrecht, pp. 2359,(2001) ,.

422

INEQUALITIES AND TURANIANS FOR SOME SPECIAL FUNCTIONS

A. LAFORGIA AND P. NATALINI Department of Mathematics R o m a Tre University Largo S a n Leonard0 Murialdo, 1 00146, Rome, ITALY E-mail: [email protected] E-mail: [email protected] We use a generalization of the Schwarz inequality t o prove T u r h - t y p e inequalities for some special functions, such as the polygamma and the Riemann C functions. Finally we extend t o z > 0 the range of validity of a lower bound for the func-

e-tPdt, proved by Elbert and Laforgia only for 0 < I < p

>

1. This function is related t o gamma and incomplete gamma functions by

Key words and phrases: Turhians, polygamma functions, Riemann zeta function, gamma and incomplete gamma functions, Exponential integral 2000 Mathematics subject classzjkation: primary 26D07; secondary 33B15; 33B20.

1. Introduction and backgrounds The first part of this paper is devoted to the study of new inequalities of Turh-type for some special functions. TurBn' proved that the Legendre polynomials P,(z) satisfy the determinantal inequality

where n = 0 , 1 , 2 , . . . and equality occurs only if z = f l . This classical result has been extended in several directions: ultraspherical polynomials, Laguerre and Hermite polynomials, Bessel functions of first kind, modified Bessel functions, etc. In view of the interest of inequalities of the type (l),

423

Karlin and Szego' named determinants as (1) Turcinians. The proof given by Turfin is based on the recurrence relation, see p. 81 of Ref. 3, (n

+ l)P,+l(Z)

{ P_l(Z)

= 0,

+

(2n l)zP,(z) Po(.) = 1.

=

-

12 = 0,1,.

nPn-1(z),

..

(2)

and on the differential relation, see p. 83 of Ref. 3, (1 - x2) PA(.)

= nP,-1(.)

- n Ic

P,(z).

(3)

In section 2 of this paper we prove new Turfin-type inequalities for the polygamma and Riemann zeta functions by using a different approach by those ones used by Lorch4, Elbert and L a f ~ r g i a ~based, - ~ prevalently, on the Sturm theory. Here our main tool is the following generalization of the Schwarz inequality

where f and g are two nonnegative functions of a real variable and m and n belong to a set S of real numbers, such that the integrals in (4)exist. In the second part of the paper we extend a result previously established by Elbert and Laforgia8 related to a lower bound for the integral function

1"

e-tpdt with p > 1. This function can be expressed by the gamma and

incomplete gamma functions defined, respectively, by

lm e-t

t a - 1d t , a > 0,

%z

> 0. (5)

Indeed we have

If p = 2 it reduces, by means of a multiplicative constant, to the well-known error function e r f ( x )

erf(.) =

2

"

e-t2dt

or to the complementary error function e r f c ( z )

ePt2dt= 1 -

-$l"e-t2dt.

Many authors established inequalities for the function

l"

e-tPdt.

424

Gautschig proved the following lower and upper bounds

1 2

- [(xp + 2)'Ip where p

- x]

lrn

< ex'

e-tPdt 5 up

> 1, x 2 0 and up =

[r

(1

+

31

P/(P-1)

The integral in (7) can be expressed in the following way

Alzer'O found the following inequalities

where p

> 1, x > 0 and QI

=

[.

(1

+

Qi and Guo" established, among others, the following lower bounds for P>l 1

- x (1

2

if 0

< x < (1 - l/p)'/',

Ix

+ e-I')

5

Iz

e-tpdt,

0

(9)

while

Elbert and Laforgia established in Ref. 8 the following estimations for the functions

etpdt and

I" e-tPdt

425

and

v(z”)

1-- p + l

<;b 1

e-tpdt,

forO
)

9 ( 3 p + 1 ) llP , p > l (12) 4(2P + 1 )

where

In section 3 we prove the following result. Theorem 1.1. For p

> 1, the inequality (12) holds for any z > 0.

We conclude this paper, section 4,showing some numerical results related to theorem 1.1. 2. TurAn-type inequalities Theorem 2.1. For n = 1 , 2 , . . . we denote b y $ ~ ~ ( = z ) $ ~ ( ~ ) (the z) polygamma functions defined as the n-th derivative of psi function

with the usual notation for the gamma function. T h e n +m

where

+n(.)

2 $2*

).(

7

m + n . as a n integer. 2

Proof - The polygamma functions have the following integral representation

tn

$hn(z)= ( - l ) n + l

eWxtd t ,

> 0 , n = 1 , 2 , . ..

z

(15)

We choose the integers m and n both even or odd, in such a way that e-xt

( m + n ) / 2 is an integer. By (4)with g ( t ) = 1 - e-t b = +m, we get ~

1

e-xt

03

tn d t .

~

[I

e-xt

00

1 - ect

tm dt 2

f ( t ) = t and a = 0,

~

1 - e-t

2

*t

dt] , ( 1 6 )

that is

$m(z)$n(z)2$&(z),

m , n = l , 3 , 5 , ... o r m , n = 2 , 4 , 6

,... ( 1 7 )

426

The proof is complete. 0

Remark 2.1. When m = n

+ 2 we find

I-‘ n + ~ ( ~ )

‘n(z)

‘n+lk)

n

= 1,2,. .. , z

> 0.

‘n+2(4’

Theorem 2.2. W e denote by <(s) the R i e m a n n zeta function. T h e n

Proof tion

-

For s

> 1 the Ftiemann zeta function satisfies the integral rela-

1 By (4) with g ( t ) = - f ( t ) = t and a = 0, b = +m, we get et-1’

-!? [I“ I n lrn 00

ts-1

dt .

et - 1

tS -d t I 2 .

dt 2

et

-

1

By (20) this inequality becomes

r(s)C(S + 2) r(s+ 2) L c2(s + 1) r2(s+ 1) or, by the functional relation r(z+ 1) = z r(z), <(s)

(s

+ 1) C(s)

C(s

+ 2)

L

s C2(s

+ 1)

which is equivalent to the conclusion of Theorem 2.2.

Remark 2.2. Many other TurGn-type inequalities can be obtained for the functions which admit integral representations of the type (15). For example, starting from the integral representation for the exponential integral function, see formula 5.1.4 p. 228 of Ref. 12,

1

00

E n ( z )=

e-xt t-n d t ,

and using inequality (4)we find

n = O , l , ..., z > 0 ,

427

3. Proof of Theorem 1.1 By means the series expansion of the exponential function e-tP, we have

and 00

w(x") =

X(-l)n-l-nn!' Xn*

n=1

consequently the inequality (12) is equivalent to the following

i.e. XP

1--+ p

+1

X2P

(p

x3p

+ 1)2.2!-(p+

+. . . < I--

1)3.3!

+ 1+ (2p + 1)2!-(3p + 1)3!+'

XP

p

X2P

x3p

' '

Since for every integer n 1

+ l)n!

(V

1

-

n(p

+ i)n!

--

(p

+

n-1 i ) n . n!(np

+ 1)'

by putting z = xp the inequality (12) is equivalent to s(z) =

1

p

*

n-1 zn ( n p + 1 ) n . n!

+ 1 D1)" n=2

> 0;

it is clear that the series to the right-hand side of (26) is convergent for any z E I%. We can observe that, for p > 1,

when 0


m.

As a consequence of a well known property of

Leibniz type series we have 0 < sg(z) < s ( z ) for 0 < z < just like was proved by Elbert and Laforgia in Ref. 8. It is easy to observe that z = 0 represents a relative minimum point for the function s ( z ) defined in (26). In fact we have s ( z ) > 0 for z < 0 and

o < z < 9 0 .

4(2~+1)

Now we can prove the theorem 1.1 Proof - First of all, we prove that the function s ( z ) , defined in (26), have not any relative maximum point in the interval (0,+m).

428

For any n 2 1 consider the partial sum of series (26)

and multiply this expression by p zl/P;we have

Deriving and dividing by

zl/P-'

we obtain

A new derivation give us the following expression (P+ 1 ) ( ( p

+

2n l)Shn(Z)

+ p z s&&(z)) = C

( - l ) k k=2

Ic-l k! 2 - 1

Dividing by z and re-writing] in equivalent way, the indexes into the sum t o the right-hand side, the last expression yields

Now consider the following series

we have for every z E 00

R

k+l

00

1

--

2

where f(z)

=

1- ( z

+ 1) e-'

-z + - z 2 - -z3+ . . . 3!

4!

5!

429

> 0 for z > 0, it follows that f(z) > 0

Since f ( 0 ) = 0 and f ' ( z ) = ze-z 'dz E (0,+m).

From eq. (27), by n -+ +m, we obtain

for every z E R. If we assume that Z > 0 is a relative maximum point of s ( z ) then s'(Z) = 0 and s"(Z) < 0, but this produces an evident contradiction when we substitute z = .?? in (28). As a consequence, we have that the function s ( z ) , defined in (26), have not any relative maximum point in the interval ( 0 ,+m). Now, since s ( z ) a point Z

CE

>

>0

'dz E

(0, 4o , if we assume the existence of 9(3p+1))

-such that s ( 2 )

( 9(3p+l) m , z - ) such that s(C)

=

< 0 then there exists at least a point 0. Let C, eventually, be the smallest

positive zero of s ( z ) , hence we have s(0) = s(<) = 0 and s ( z ) > 0 C). It follows therefore, that there exists a relative maximum point zo E (0,C) for the function s ( z ) , but this is in contradiction whit the above proved statement. 0

'dz E (0,

4. Numerical results

In this concluding section we use the computer algebra system Mathematica @ to compare numerically the known bounds established by Alzer, Gautschi, Qi-Guo and Elbert-Laforgia. The results show that the ElbertLaforgia bound is very stringent. We briefly put

I ( z )=

lx

e-tPdt;

( +3

A ( z )= r 1 the Alzer lower bound;

-

(1 - e-"')'/P

430

the Gautschi lower bound;

the Qi-Guo lower bound;

-)

E ( z )= z ( 1 - 4 X P > P+1 the Elbert-Laforgia lower bound. The numerical results we have obtained are shown in the following table:

Table 1. Approximation errors of the lower bounds E ( z ) ,A ( x ) , G(z) and Q ( x ) with respect to I(r). Here we put 19 =

(w) ”’.

p = 50

p = 100

p = 200

p = 300

x = 1.026

z = 1.013

x = 1.0065

x = 1.005

I9 = 1.01222...

I9 = 1.0060...

19 = 1.0040...

I9 = 1.02455...

I(x) - E ( x ) Z(X) - A(z) I(x) - G ( x )

0.000272222

0.0000690398

0.0000173853

9.95056 x lop6

0.000417332

0.000205222

0.000101731

0.0000312521

0.0108717

0.0107205

0.0106414

0.00474017

I ( x )- Q(x)

0.301341

0.308547

0.312265

0.313533

In these four numerical examples we can note that there exist values of z > 4(2p+l)) 9 ( 3 P + l ) ‘ I p such that E ( z )is the most stringent lower bound for I ( z ) .

(

Moreover we state that this is always true in general. Precisely we state the following conjecture: f o r any p > 1, there exists a right neighbourhood

(w)

of l l P such that E ( z ) represents the best lower bound of I ( z ) with respect to A ( z ) , Q(z) and G(s). References 1. P. TurBn, O n the zeros of the polynomials of Legendre, Casopis pro Pestovani Mat. a Fys, 75, (1950), 113 - 122.

431 2. S. Karlin, G. Szego, On certain determinants whose elements are orthogonal polynomials, J. d'Analyse Math., 8, (1960), 1 - 157. 3. G. Szego, Orthogonal polynomials, 4-th ed. Amer. Math. SOC., Colloquium Publications, 23,Amer. Math. SOC.Providence, RI, 1975. 4. L. Lorch, Tura'nians and Wronskians f o r the zeros of Bessel functions, SIAM J. MAth. Anal., 11,(1980), 223 - 227. 5. A. Elbert, A. Laforgia, Monotonicity results on the zeros of generalized Laguerre polynomials, J. Approx. Theory, 51, (2), 1987, 168 - 174. 6. A. Elbert, A. Laforgia, Some monotonicity properties f o r the zeros of ultraspherical polynomials, Acta Math. Hung., 48, (1986), 155 - 159. 7. A. Laforgia, Stumn theory f o r certain class of S t u n - L i o u u i l l e equations and Tura'nians and Wronskians for the zeros of derivative of Bessel functions, Indag. Math., 3,(1982), 295 - 301. 8. A. Elbert, A. Laforgia, An inequality for the product of two integrals relating t o the incomplete gamma function, J. Inequal. & Appl., 5 (ZOOO), 39-51. 9. W. Gautschi, Some elementary inequalities relating to the gamma function and incomplete gamma function, J. Inequal. & Appl., 5, (2000), 39-51. 10. H. Alzer, On some inequalities f o r the incomplete gamma function, Math. Comp., 66 (1997), 771-778. 11. F. Qi, S. Guo, Inequalities f o r the incomplete gamma functionand related functions, Math. Inequal. Appl., 2, (1999), 47-53. 12. M. Abramowitz, LA. Stegun (Eds.), Handbook of Mathematical Functions with Applications, Graphs and Mathematical Tables, Dover, New York, 1965.

432

ASYMPTOTICS IN THE COMPLEX PLANE OF THE THIRD PAINLEVE TRANSCENDENT

V. YU.NOVOKSHENOV * Institute of Mathematics Russian Academy of Sciences Chernys hevskii street 112 Ufa 450077 Russia E-mail: [email protected]

A uniform asymptotics in the complex plane for the third Painlev6 transcendent is constructed and proved. The leading term of asymptotics as IzI t 00 is given by the Boutroux ansatz, i.e. by an elliptic function with its modulus depending on arg 2. A functional equation for the modulus is universal for PI11 equation and does not depend on initial conditions. It can be solved as an Abel problem of inversion of elliptic integrals. Another component of the Boutroux ansatz is the phase shift in the elliptic function. It depends on initial data, and we calculate it wit.h the help of Isomonodromic Deformation Method (IDM). By solving a direct monodromy problem for a relevant Lax pair of operators, we fit given monodromy data with their approximations, coming from the leading term of asymptotics. This leads to explicit formula both for the modulus and the phase shift. Since a monodromy data for PI11 transcendent can be expressed explicitly through the initial conditions at z = 0 (see ’), we come to the connection formulas linking the PI11 transcendent asymptotics at infinity and at the origin. Finally, the IDM technique provides the proof of the above constructions, giving an analog of the Bolibrukh-Its-Kapaev theorem proved earlier for a similar asymptotic description of PI1 transcendent in the complex plane g .

1. Introduction

This paper continues the studies - of connection formulae for asymptotics of the Painlev6 transcendents. Here we study a particular case of the PI11 transcendent arising from the sine-Gordon equation utt - uyy sinu = 0, which will be called SG-PIII equation 1 u,, -u, sinu = 0, x,u E @. (1.1) X

+

+

+

*Work partially supported by grant 04-01-00190 of the Russian Foundation for Basic Research.

433

The initial condition for (1.1)is described by the asymptotics

(

u(x)= r l n x + s + 0 ~ a : ~ 2 - ~ I ,m ~z~-+ ) 0,

lImrl

< 2,

(1.2)

which parametrizes a non-degenerate two-parameter manifold of solutions 3 . Another parametrization of U ( X ) comes from the Isomonodromic Deformation Method (IDM), which is used for integration of the SG-PI11 equation (1.1). It provides two integrals of motion p and q, which appear to be a monodromy data of the Lax operator, as described below in $2. However, there is a direct map between this two ways of assignment, found in A e - 7 -Be? B e - 4 - Ae?f P= (1.3) A+B ’ A+B ’ where

We are going t o find the asymptotics of

U(X)

at infinity lr

1x1 + co, 0 < arga: < 2 and express it in terms of monodromy data (1.3). Thus we will find connection formulae similar to those found in s, for the real-valued x -++cm. Note also, that asymptotics in the complex plane of the first and second Painlev6 transcendents has been found in and It is based on the IDM, which will be slightly modified here for the particular PI11 equation (1.1). We assume the non-degenerate case of the monodromy data, which is equivalent to the condition p q 1 # 0. First let us explain a naive way to how to get the leading term of asymptotics directly from the SG-PI11 equation (1.1) itself. For large 1x1 the second term U,/X seems to be small, so the leading term is just mathematical pendulum equation

‘.

+

u,,

+ sinu = --.X

U X

Fix any initial point xo and a Cauchy data 210

= u(zrJ),

211

= U’(X0)

434

and define the total energy of the pendulum

F(z)= u3x) - cosu(x). 2 Now the Cauchy problem

has explicit solution U = U ( z , F ( z o ) expressed in terms of Jacobi sine function sin(U(t)-uO) 2 = sn ( ( z - - z o ) & F Z J F l ~+) l.

(1.5)

It is clear that

for any x running within a finite-radius circle and avoiding the lattice of poles of U . Making the standard uniformization

v = cosu, U," = F + V, v," = 2 ( F + V)(1 - v"),

(1.6)

choose a cycle L on a surface W 2 = 2 ( F + V)(1 - V2) and construct a sequence {zn}, n = 0 , 1 , .. . by recurrent formula

wn

=

f

13

J2(F

+

dV V)(1- V2)

435

In the case of non-zero homology cycle C this sequence, called by Boutroux a ”line of periods”, tends to a finite limit: w, = w,

+ O(z,l), dV

I:

d2(F,

+ V)(1-

V z )’

where F, satisfies transcendental equation

f /%dV

= 0.

13

This result expands the estimate (1.5) to a large domain with ranges of order O(lzl), t >> 1. Now the elliptic function (1.4) becomes ’modulated’ as zn+1-2, = O(1): with respect t o #J = a r g z which varies by o(zn+l -z,) V(X) = v(z - ~m(#J)IW#J)).

(1.9)

Here for the modulus F, of the elliptic function the equation (1.8) takes the form IT

arg Ga(F) = argGb(F) = -4 - -, 2

(1.10)

where #J

Ga(F) def =

= arg z,

Jdv,

f 1F -+vV

f

d VV . Gb(F) dkf / yF + 1- v2

(1.11)

b

a

and a, b are basic cycles of the elliptic integrals. Equations (1.10) are called the Boutrouz equations. They can be considered as modulation equations on the modulus K = ~ ( 4 of ) Jacobi sine function (1.9): dF d#J

- = -2i(F

+ 1)

with initial condition F(0) = 1,

(1.12)

436

which comes from the equations (1.10) as 4 = 0. Here E ( x ) and K ( x ) are standard complete elliptic integrals: 1

K = K ( x )=

dz

J(1-

22)(1-

K' = K ( d ) , H2z2)'

(1.13)

One can show that system (1.10) has a unique smooth solution in the first quadrant 0 < 4 < .; The graphics of real and imaginary parts of the function F = F ( 4 ) is shown in Figure 1.

Figure 1. Energy function F in the first quadrant 0

< argz < ~ / 2 .

Being applied to the modulus of elliptic function (1.5), this yields a deformation of period lattice in the first quadrant as shown in Figure 2. The Boutroux ansatz will be proved below in a rigorous way with the help of IDM. Moreover, besides of equations (1.12) this will give the for phase shift of the elliptic function (1.5). The main result of the paper is the following Theorem 1. A n y solution u(xIp, q ) of SG-PIII equation (1.1) parametrized by the monodromy data (1.3) with p, q # 0 and pq # -1 has the following asymptotics as 1x1 -+00, O < argx < lrf2

437

Figure 2. The deformation of period lattice for elliptic function V in the first quadrant 0 < argx < s/2.

where modulus

K

=~

( 4 of ) the Jacobi sine satisfies Whitham equation (1.15)

which is equivalent t o the modulation equation (1.10) for the full energy F = F ( 4 ) in (1.4) as 2 = 2. F+1 The phase shift xo is given by

2 . Direct monodromy problem

As it was shown in mutation relation

2,

the third Painlev6 equation (1.1) comes as the com-

438

where U = U(u, u', 2,A), A = A(u, u', z,A), are 2 x 2-matrices A =i

izu' cos u ( 7 - %)0 3 - -01 4A

f

i sin u -0 2 A2

7

(2.2)

u and

01,2,3

ixx =-Po3 8

iu' - -01, 2

(2.3)

are Pauli matrices 01

=

(; ;) ,

02

=

(a

,

03

=

(; "1).

The matrix equations (2.1) comes as compatibility condition of the Lax pair

9~= A 9 ,

{ 9,

=

(2.4)

u9.

Given values of x,u,u,, introduce two pairs of canonical solutions of the

Figure 3.

Domains for the canonical @-functions

A-equation (2.4), defined in the domains shown on Figure 3.

*?

= 11

SF = [I

+ O ( X - ~ >exp ] (-$.xo3) ,

A

+ 00,

AE

rl n R,,

+ O ( A - ~ >exp ] --&AO~ , ( 1 6 )

A

-+ 00,

AE

rzn R-,

(2.5)

439

and S$ = PO[I

+ O(X)] exp (-$03)

, X + 0 , ImX > 0 , (2.6)

SO_ = ~ o [ l + 0 ( ~ ) ] e x p ( - ~ a 3 )X, + O ,

ImX
where

Po=

(icosf sin

isin: cos f

The monodromy data for the canonical S-functions (2.5), (2.6) consist of the Stokes matrices S r , S r SF = q p s j ” ,

x E rj,

j = 1,2,

(2.7)

and connection matrix Q S i = SYQ,

X E f&.

The matrices 5’1, S2 and Q are parametrized by two scalar parameters p and q

Here p and q are the same as those in (1.3) parametrizing initial data (1.2), because we took the same definitions of canonical solutions as in 8. For simplicity, we will restrict ourselves to the generic case of non-degenerate monodromy matrices, i.e. assume 1+ Pq

# 0.

3. WKB approximation of *-function

Now we begin t o solve the direct monodromy problem (2.7) with the approximate ”potential” u = u, u, = ux, 1x1 >> 1 , where

Ux= .\/2cosU

+ 2F.

Introduce new variables scaled with respect to r = 1x1 >> 1

x = x / r = e@, 4 = i a r g x ,

< = rX

440

and rescale equation in X (2.4)

i X 2 +-i c o s ~ i X U , 16 E2 4E

is in^

--

A=rA=r

i X U , +-isinU

--

45

E2

iX2 16

icosU

The large parameter r allows to construct the asymptotics of the *-function explicitly by means of WKB-approximation. To do this first calculate the determinant of A

X4 X 2 F pU2(E) = det A = - - 256 8t2

+

1

4'

Stokes lines

Figure 4.

Stokes lines of equation (3.4)

Thus fip(<) are eigenvalues of A. They will form the phase integrals of the WKB-solutions we are going to construct. Let &
441

The Stokes lines for the phase integrals exp { f i r p(C)dC} are defined as solutions of the equation (3.4) These lines form the Stokes graph shown on Figure 4. Note that asymptotic directions at infinity coinside with the straight line arg A = T - 24 shown on Figure 3, thus the functions !€'4 and !€'I stand for !€'?and 9" respectively. Two small domains in the center of Figure 4 accomodate the functions and S t from the small upper and lower half-circles on on Figure 3. The normalizations of canonical solutions read irX2 Q j = [1+0(5-')]exp (-165n3) , ( + 00, 5 E rj, j = 1,4. (3.5)

!€'I"+

Since x and U are parameters in A-equation (3.2), we can specify these values in a suitable way. This means the monodromy data calculated via x and U will provide sufficient information to recover the elliptic function U ( x - xo(q5)lx(4)). The simplest way seems to choose the zeroes of the elliptic function U . If the modulus N is known, the zeroes of U uniquely determine the phase shift ZO. Namely, take x, U and U, in A-equation (3.2) in the form

z = T X , , ~ , cosU(x) = 1, and find

N

=

U, = ~ ~ C O+S2FU= d

m

(3.6)

,/& and Xn,m in terms of monodromy data p and q.

The matrix of coefficients A in (3.2) now simplifies

(3.7)

The WKB approximation for equation (3.7) follows from diagonalization of matrix A by the left-hand multiplier 9 e TQ, A I-+ T - l A T :

where

442

where C is some constant in (' matrix. Equation (3.9) defines a Riemann surface ( p , €Jwith canonical cycles defined as shown on Figure 4. The asymptotics of the phase integrals are given by the following lemma.

/

1

I

I \ \

Figure 5.

Canonical cycles on the Riemann surface (3.3)

Lemma 1. The phase integrals in (3.8) satisfy the estimates

at infinity by definition (3.3), and transform Proof. Note that p ( ( ) -+ the phase integral (3.10) for large such that E r2 (see Figure 4)

<

<

16

Since p ( 5 ) is even function, the last integral can be written as

(3.12)

443

The asymptotic formula (3.11) is proved similarly. 0 Theorem 2. Canonical solutions 9j (3.5) have the asymptotics as r

-+ co:

Proof. follows immediately from definition of canonical solutions (3.5) and asymptotics of phase integrals (3.10), (3.11) stated in Lemma 1. 0 We will apply the asymptotics (3.13) - (3.15) for the equation

(3.16) which follows from equation (2.7) (remind that 9 F is now 91 and 97 is 9 4 ) . The conjugation equation (3.16) takes place at a common domain of definition of both @-functions. Thus we need to make a extension of 91 and 9 4 into a common domain within the lines r2 and r4. The best way to do this is analysis of WKB-solutions at the turning points at the ends of the Stokes lines. 4. Asymptotics at the turning points

WKB-approximation (3.13) - (3.15) fails in a small neighborhood of the turning points E j . However, we can apply an Airy function approximation, since all the turning points are simple ones (remind that we study non-degenerate case of the fourth-order polynomial p2 (6)). This approximation enables an etalon extension of the *-function from one into another neighboring ?-sector around simple turning point. Similarly to procedures in this procedure is equivalent to a right-hand multiplication by etalon triangular Stokes matrices with f l off-diagonal entries (see lo for generic theory).

444

Lemma 2. A n y solution of equation !€j€ = A!€j with matrix (3.7) have the following asymptotics at the turning points Cj, j = 1 , 2

(4.1) .) where E r j , T = r2/3(<- &), yj”= X4(ca-c324&,oscj3 ’ , j = 1,2, D is constant in T matrix, and functions w , 55 are linear independent solutions of the Airy equation w,,

+ 3 6 7 ’ ~=~0.

Proof. It is easy to check that near the point asymptotics p = 67jr-1/342

(4.2) <j

the phase functionp has

+ O((S -

Ej)3/”.

5dx2

(4.3) Let $1, $2 be the components of the first column of !€j. The matrix equation (3.7) is equivalent t o the system of scalar equations

where

xl,2 = $l

X1,€

= -$(t

- 51)(1-

X2,€

= -$(E

+
(4.4)

f i$2. This system yields one second-order equation

At the turning point

<j

take a solution of (4.5) with a leading term x l , 2= w1,2

w1,2:

+ 0(~-1/3).

This yields a model equation for the leading order:

+ 36yjTW2 = 0,

where T = r2/3(5 - <j). obvious estimates

2

= 0,

(4.6) The remainder terms of the Taylor series lead to

W~,TT

W1

which prove the Lemma. 0 Now we use the well-known asymptotics of the Airy functions (l). Let us fix normalizations of basis functions w, of the Airy equation (4.2) by their asymptotics arg T = - arg y : W(T)

= T - l I 4 [1

+ o(T-3/2)] eXp (-4iyT3l2 - E! ’ 7 4 00. + o(T-3/2)] eXp (4iYT3/’ + 2),

W ( T ) = T-1/4 [1

(4.7)

445

Theorem 3. Let $1 and $4 be canonical solutions !PI and q4, continued to the domain domain 0 3 (see Figure 6). Then they have the following asymptotics as r -+ 00:

4

(4.10) r

1

(4.11)

f 2a--26

p(()d<]

, D4 = exp

[-?f

p(C)d(]

2a-b

Proof. First let us demonstrate how to continue q1 around the point (see Figure 6 ) . Consider an auxiliary WKB-solution

(1

Since (3.13) and (4.12) are solutions of the same equation 9~= A*,they should coinside up to the right constant multiplier in a common domain where 6 - 5 1 is small and T is large. All we need is to compare both !PI and $1 with Airy-type solution (4.1). First calculate the multiplier D in (4.1)

446

from asymptotics (3.13) of !PI as 5 + 6 1 :

{

x exp -ia3

kT1<3/2

+ 0 ( ~ - 2 / 3 ~ 5 / 2 ) ] D~ }

(4.13)

Apply the asymptotics (4.7) of the Airy function to (4.1) and compare the

Figure 6. Extension of

QJ1and QJ4

into a common domain

leading terms with (4.13): D =fiexp

Next do the same with \ir1 as integration from 5 2 to 51:

because

(y)

5 + 51.

D2.

(4.14)

Change first the lower limit of

447

Airy-function representation (4.1) and using

+r-l/6R(C,

(4.15)

T),

1-

where R(C,T ) = 0 (C3I4exp (f4iylC3I2)+ T - ~ C / 5/4 ~ Comparing the leading terms of (4.13) and (4.15), we get the constant C in the form

Finally we need to pass from $1 to $1 by going around the point 52. This is done similarly t o the case of but without change of lower limit of integration. Thus, the connection here reads

which proves formula (4.9). The asymptotic formula (4.10) is proved similarly, passing over

C counter-clockwise in formula (4.1). This gives the etalon matrix

-51

in

(; ;)

with left multiplier by D3, which comes from the change of lower limit of integration in (3.14) from -51 to 5 2 . 0

448

5. Zeros of the elliptic function ansatz The @-function representations (4.9)44.10) allow to solve the monodromy equation (3.16)

in terms of asymptotic parameters x = rei4 and F = F ( 4 ) . This will prove Boutroux equation (1.10) and determine the phase shift X O ( + ) of the Jacobi sine function sn ( x - X O ( + ) ~ N ( ~ ) )(1.5). Actually, we need an asymptotic distribution of zeroes of the Jacobi sine for r >> 1, 0 < 4 < 7r/2. The latter distribution will be found by inversion of asymptotic formulae for p and q , since the monodromy parameters are constant in x and assumed to be given (first integrals of SG-PI11 equation (1.1). Changing variable of integration 6 = $ u reduces phase integrals to the form

ir

J p(<)dJ = -%J u-2 J

m

u = -ixm n

d

4

J cL(E)dE, (5.2)

where

uf

=

-F

+ JF2-1,

Calculate S1 = \kT1\kl in the domain (4.9) and (4.10).

Theorem 4. The Stokes matrix

5’1

JFZ-1.

U;

= -F

03

in terms of WKB-approximation

-

has the f o r m (5.3)

where

(Y

= exp [ - g J p ( € ) d [ ] a

= (01)11,

0 = exp

[-:.,ib]

P(€)d€ =

(04)ll.

Proof. is a direct multiplication of matrices (4.9) and (4.10) and substitution of explicit expressions (4.11)

449

Lemma 3. The phase integrals (5.2) over the basic cycles are expressed as 2 f , G ( < ) d <= i h G b ( F ) = N [ K ’ ( N) E’(x)],

(5.4)

a

f

i ,G(<)d( = -Ga(F)

Jz

b

42

= --N E(N),

(5.5)

where G a ( F ) and Gb(F) are basic periods of the integral (1.11)

a,b

and E ( x ) , K ( x ) are standard complete elliptic integrals (1.13):

1 J( 1

K

= K ( x )=

dz 1 - 2 2 ) (1- N 2 2 2 ) ’

K’ = K ( d ) ,

F+1‘

Proof. Integrals (5.2) and G ( F ) have the standard uniformization

The latter modular transformation together with integration over basic cycles a and b gives formulae (5.4) and (5.5). C!

+

Corollary 1. Suppose F = FO rv1FI

+ O ( r - 2 ) , then

2 dZNK’( N )

2r

F1

+ O(r-2),

(5.7)

450

Proof. follows from Taylor series expansion of

f

J(F

+

dV = JZHK(H), V ) ( 1- Vz)

Proof of Theorem 1.

log(-p) = -2 ir

f

# b

J(F

@and notion that

+

dV = 2JZXK'(H). V)(1- V 2 )

Comparing matrices (5.1) and (5.3) we have ixm,n p(C)d('+ 2 ~ i N= 8

2a-b

2a-b

(5.9) log(-l- p q ) = ir

f

p(<)dC

+

ixm n 2 ~ i N= -4

f

ji(C)dC + 2 ~ i N .

Since Ixm,nl 1r = O ( N ) , argxm,n = 4, equations (5.9) yield Whitham equations on modulus

H

=~

(4)

#P(C)dC = #P(C)dC = -4. b

a

Due to Lemma 3 they are equivalent t o equations (1.10) 7r

G a ( F )= Gb(F) = -4 - -.

2

The zeroes xm,n can be found by their asymptotics in r-l : xm,n = m w

+ nw' + + ~ ( r - ' ) , [mi+In1 = ~ ( r ) , XO

(5.10)

where the periods of Jacobi sine function (1.14) are given as

w = 2 x K ( x ) , w' = ixK'(H). Calculate now terms of order O(1) in equations (5.9). They are formed by and r-lF1, which provide the following system

20

ixo log(-p) = - (K' - E' 2H

log(-1 - p q ) =

+ E ) + 2-\/z~F1( K + iK'),

3 (K' - E') + 2 J Z F 4 K ) . 2%

45 1

The determinant of this system is 7r/2 due to Legendre equality EK' E'K - K K ' = n/2. Thus the solution for 20 is straightforward:

xo(4) =

i 4 4 ) [ K ( 4 )W - P )

-

+

( K ( 4 )- i K ' ( 4 ) )log(-1 - P d l .

References 1. H. Bateman, A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, NY, 1953. 2. H. Flaschka and A.C. Newell, Monodromy- and spectrum-preserving deformations I, Comm. Math. Phys. 76 (1980) 65-116. 3. A.R. Its and V.Yu. Novokshenov, The Isomonodromy Deformation Method in the Theory of Painlev6 Equations, Lect. Notes in Math., v.1191, SpringerVerlag, 1986. 4. N. Joshi and M.D. Kruskal, An asymptotic approach t o the connection problem for the first and the second Painlev6 equations, Phys. Lett. A , 130 (1988) 129-137. 5. A.V. Kitaev, Elliptic asymptotics of the first and second Painlev6 transcendents, Uspekhi Mat. Nauk 49 (1994) No. 1,77-140; (English Transl.: Russian Math. Surveys 49 (1994) No. 1, 81-150). 6. V.Yu. Novokshenov, Boutroux ansatz for the second Painlev6 equation in the complex domain, Izv. Akad. Nauk SSSR ser. math., v.54 (1990) p.1229-1251. 7. V.Yu.Novokshenov, A.G.Shagalov, Bound states of the elliptic sine-Gordon equation, Physica D, v.106 (1997) 81-94. 8. V.Yu. Novokshenov, On the asymptotics of the general real solution of the Painlev6 equation of the third kind, Sov. Phys. Dokl.,30 (1985) 666-668. 9. A.A. Bolibruch, A.R. Its, A.A. Kapaev, On the Riemann-Hilbert-Birkhoff inverse monodromy problem and the Painlevk equations, Algebra i Analiz, v.16, N 1 (2004) 105-142. 10. W. Wasow, Linear Turning Point Theory, Applied Mathematical Sciences, Vol. 54, Springer-Verlag, New York, 1985.

452

SOLUTIONS OF DYNAMIC EQUATIONS WITH VARYING TIME SCALES

BONITA A. LAWRENCE AND RALPH W. OBERSTE-VORTH Marshall University One John Marshall Drive Huntington, West Virginia 25755, USA E-mail: [email protected], [email protected] We consider families of dynamic equations zA = f ( t , z) with the initial condition z(t0) = 20 over different time scales, treating time scales as a parameter of such a family. Our goal is to understand the behavior of the solutions of such the same initial value problems over different time scales as bifurcations and limits over their underlying domains, Le., the time scales. In a general setting, we show that the limit of solutions over convergent sequence of time scales converges to a solution over the limit time scale. A well known sequence of bifurcations occurs when f ( t , z ) = 42(3/4 - z) between R+ with its asymptotic continuous solutions, and R+, with its chaotic solutions.

1. Introduction

Consider the logistic differential equation 2’ = 4x

(; - .)

and the logistic difference equation

both with the initial condition x ( 0 ) = XO. For the differential equation (l),xo = 0 and xo = 3/4 are constant trajectories. In forwards time (ie., as t + +m), for xo E ( O , l ) , the solutions tend towards 3/4, for xo < 0, the solutions tend towards -00, and, for xo > 1, the solutions tend towards 3/4. Solutions of the difference equation (2) are far more complicated. Solving equation (2) is equivalent to iterating the function X ~ + I= 4

(a

~ n -).x

+ xn = 4 ~ n ( 1 -~ n ) .

453

Orbits beginning at xo for xo < 0 or xo > 1 tend towards -m. Orbits beginning at xo E [0,1] are generically chaotic with countably many periodic and preperiodic orbits, including the fixed points at xo = 0 and zo = 1. (1) and (2) can be thought of as special cases of the logistic dynamic equation

with the initial condition x ( 0 ) = XO. The time scales for the solutions of (1) and (2) are R+ = [ O , c o ) and Z+, respectively. W e propose t o use time scales t o parameterize families of equations. Our long-term goal is to try to understand how the dynamics of solutions change as the time scale varies. For example, solutions of (3) as the time scale varies from Z+ to R+. We present here two results in this direction. Theorem 5.1 examines the limit of solutions over convergent time scales. Theorem 6.1 examines the bifurcations that occur as time scales change. The italicized terms will be explained: Section 3 addresses the convergence of time scales, Section 4 addresses the limit of functions with different domains (i.e., time scales). The parameter values at which bifurcations occur are time scales. 2. Families of Dynamic Equations on Time Scales

We wish t o study equations like the following over different time scales and compare the dynamical behavior of their solutions:

xA = f ( t ,z),

z(t0) = 20.

(4)

We wish to examine the dynamics of the solutions 0

of the same equation as the time scales are perturbed and of the same equation as the time scales converge and the convergence of the solutions as the time scales converge.

The italicized terms used above require illumination. Consider sameness. When we say that we are solving the same equation (4) over different time scales, we mean that the domain of the t’s for the function f is given a priori; it is a set containing the union of all time scales under consideration, and the equation over a particular time scale uses the restriction o f f to that time scale.

454

This is completely natural in some sense. Yet, in another sense, it is quite peculiar.a Consider the equation

xA = t

+ -a(t),

x ( 0 ) = 20

over a collection of subsets of R+, including R+. What is the domain of -a? Either it is determined by the union of all time scales under consideration or it changes as the time scale changes. In the former case, -a(t)= t in the time scale R+; no matter what time scale we consider a t the moment, g refers to the forward jump operator on R+. In the latter case, -a(t)depends the time scale that we are considering at the moment--a refers to the forward jump operator on that time scale. In the latter case, the time scale becomes a variable: perhaps we should write xA = f(t,x,T). Such families surely behave similarly however. 3. Convergence of Time Scales

We are interested in the set of time scales, i e . , the set of nonempty closed subsets of R. In general, the set of nonempty closed subsets of a topological space X is denoted CL(X) and is called a hyperspace. For a topological space X , there are two well-known ways of topologizing CL(X), due to Hausdorff [4](priority actually belongs to Pompeiu [7]) and to Vietoris [8]. Hausdorff defined a metric topology on CL(X) and Vietoris defined a topology by giving a subbasis. These two topologies often agree, but not in general. In fact, if X is TI and the Vietoris topology on CL(X) is metrizable, then X must be compact and metrizable. Hence, these topologies differ on CL(R); see [5]. We will use the Hausdorff metric. Suppose that X is a metric space. Let d be a bounded metric on X . For every x E X and every S E CL(X), define d(z, S ) = inf { d(z, s) I s E S } . Next, for every r

> 0 and every S

E CL(X), define

N,d(S>={x€X Id(x,S)
N,d(S) is called a generalized open d-ball in X about S of radius r . The Hausdorff metric o n CL(X) induced by d is denoted Hd and defined by H d ( S , T ) = inf { r

> 0 I S c N,d(T)and T

C

N,d(S)}

"Perhaps this is analogous to whether one thinks of a function as being defined by sets or a n equation.

455

for all S,T E CL(X). S and T are at most Hausdorff distance 6 apart if and only if every point of S is no more than distance 6 from some point of T and vice verse. On CL(R), we will use the Hausdorff metric H derived from the usual bounded metric d on R defined by

for all z, y E R. Henceforth a limit or the convergence of time scales means with respect to the Hausdorff metric H . 4. Limits Over Time Scales

Next, we wish t o consider the convergence of functions over different domains (e.g,, time scales). In particular, we wish to generalize the idea of pointwise convergence of functions. Over a common domain, we have that a sequence, { f,}, of real-valued functions converges to f pointwise if, for every x in the domain, lim f,(z) = f(z).

,--too

In our case, with varying time scales as the domains, this does not make sense since every x may be some of these domains but not in others. However, if the sequence of time scales that are the domains of the functions in the sequence converges to a time scale, it may be reasonable to use sequences { x, } that converge to any given x in the limit time scale so that lim fn(xn) = f(x).

n+oo

We make this precise below and verify the uniqueness of such limits. Let T be a time scale and let T,, for all n E W, be time scales. Let t E T. Suppose that { t , } is a sequence of real numbers.

Definition 4.1. { tn } converges t o t through the T , ’s if and only if t, E T, for every n E N and t, converges to t in R. Of course, such sequences generally are not unique. Let f : T -+ R be a function and let f, : T, -+ R, for all n E N,be functions. The following should be thought of as pointwise convergence of functions on varying time scales.

Definition 4.2. The sequence of functions { f n } converges to f as T, converges to T if and only if for every t E T there exists a sequence { t , } converging to t through the T,’s such that { f n ( t n ) }converges to f ( t ) .

456

In fact, pointwise convergence of functions on time scales is often welldefined (always, for our purposes). Proposition 4.1. Suppose that T , c T for all n. Suppose that there exists M such that for e v e y t E T,there exists a neighborhood U ( t ) o f t , such that, for all n and all u E U ( t ) ,lf,"(u)I5 M . If { f, } converges to f as T , converges to T,then, for every t E T,for every sequence { t, } converging to t through the T, 's, { f,(t,) } converges to f ( t ) .

Proof. Let { t , } and { t k } denote sequences converging to t through the Ti',%, where the latter comes from the definition of convergent functions. For n sufficiently large, all t, and t; are in U ( t ) . Then I f n ( t n ) - fn(t',)I

I M I L - tkl + 0.

0

5. Convergence of Unique Solutions We come now to the first result, that the solutions of the same dynamic equation over convergent time scales converge pointwise to the solution of that equation over the limit time scale. The result requires uniqueness of the solutions as the following examples illustrates. Example 5.1. Suppose that x' = f (t,x ) with ~ ( 0 = ) xo has (at least) two distinct solutions y:R+ + R

and z : R + + R .

Consider the constant sequence of time scales { R+ } with alternating choices of y and z. These solutions do not converge. However, there are convergent subsequences in this case. Example 5.2. Consider the initial value problem

x' = 6, x(0) = 0, over R+. This familiar equation has lots of non-unique solutions. The constant sequence of time scales with any choices of solutions still has convergent subsequences of solutions. (The solutions are characterized by the point, if any, where the solution leaves the line x = 0; every infinite subset of the reds has a limit point, possibly infinite.)

As these examples suggest, we expect that the existence of convergent subsequences is always the case for an equation with non-unique solutions. We now examine the case of unique solutions.

457

Theorem 5.1. Suppose that, for every n E N, T, is a time scale with to = inf T,. Suppose that Tn converges to a time scale T with respect to the Hausdorff metric H . Iff is rd-continuous and xn : T, -+ R, for every n E N,and x : T -+ R are the unique solutions of

x A = f ( t , x ) , z(t0) = 20 over the respective time scales, then the sequence { x, } converges to x. Before beginning the proof of Theorem 5.1, let us recall the Induction Principle on time scales from [l].Let T be a time scale and to E T. If

{ s(t)J tE [ t o , M ) n T } is a set of statements such that

S(t0) is true; if t E [to,co)n T is right-scattered and S ( t ) is true, then S ( a ( t ) )is true; if t E [to,M ) n T is right-dense and S ( t ) is true, then there exists a neighborhood U of t such that, for all s E U n ( t ,M ) , S(s) is true; and if t E ( t o , M ) n T is left-dense and S(s) is true for all s E [to,t ) , then S ( t ) is true, then S ( t ) is true for all s E [to,m)n T .

Proof. We wish to show that xn(tn)converges to x ( t ) for every t E T and a corresponding sequence { t , } through the T,’s that converges to t. That is, we wish to show that, for every E > 0 and for every t E T, there exists N such that if n 2 N , then 1xn(tn)- x(t)l < E .

(5)

Statement (5) depends on E and t. Let us fix an arbitrary E > 0. Let S ( t ) denote statement (5) for this fixed E . We will show that S ( t ) is true for every t E T by induction. Condition (I). Suppose that t = to. By the initial condition, xn(to)= 20 = z(t0) for all n E N. Condition (II). Suppose that t is right-scattered in T. Let { t , } be a sequence that converges to t through the Tn’s. By hypothesis, x,(t,) converges to x(t). Let s = a ( t ) E T and let { s n } be a sequence that converges to s through the T,’s. Note that since t < s , t , -+ t , and sn -+ s, we have, for all sufficiently large n, 1 2

tn < -(t

+ S) < sn.

458

Define sequences { tk } and { sk } for large n by

and

s:, = a&). For sufficiently large n,

t, 5 t:, < a&)

= s,:

5 s,.

{ t k } and { sk } converge through the T,’s to t and s, respectively. Since - z,(tL) - z(s)

z,(sn) - z(s) = z,(s:,)

+ z,(s,)

- z,(sL)

+z(t)

+ zn(tL) - zn(tn)+ zn.(tn)- 4 t )

and z(s) - z ( t ) = z A ( t ) ( s - t ) ,

we have that Izn(sn) - 4 S ) l

L

IzCn(4)

- zn(tL) - zA(t)(s- t)I

+ IZn(Sn) - %(S:,)I

+ 12n(tk) - zn(tn>l+ Izn(tn) - 4t)l;

each term on the right hand side can be made arbitrarily small for sufficiently large n. Condition (III). Suppose that t is right-dense in T. Let { t , } be a sequence that converges to t through the T,’s. Since t is right dense, we may choose each t , 2 t. By hypothesis, zn(tn)+ z ( t ) . Set M = IzA(t)l+ 1 and choose 6 > 0 small enough so that sup { IzA(u)I I u E T n [t,t

+ 61 } 5 M .

Select any s E T n ( t ,t + 6). So It - SI < 6. Let { s, } be a sequence that converges to s through the T,’s. For sufficiently large n , Is, - t,l 5 6 since we can arrange that s,, t, E [t,t + 61 . For sufficiently large n , we have that Izn(s,) - (.)I

I IZn(Sn) - zCn(tn)l + Izn(tn) - z(t)l+ Idt) - 4 s ) l < M6 + lzn(tn)- Z ( t ) l + M6.

Choosing 6 sufficiently small first and then n sufficiently large, each term on the right hand side can be made arbitrarily small. Condition (IV). Suppose that t is left-dense in T. Let { t , } be a sequence that converges to t through the T,’s. By hypothesis, for every s E [ t o ,t ) n T and corresponding sequence { s, } that

459

converges to s through the T,’s., z,(s,) + z(s) for a sequence { s, } converging to s. Since f is rd-continuous, for some 6 > 0,

M = SUP { I X ~ ( U )I Iu E [t - 6, t + 61 n T } exists. For sufficiently large n, IGL(tn)

- 4t)I I Izn(tn) - z,(s,)l + IzrL(sn) - 4 s ) l + 1 4 s ) I MIS, - t,l + lzn(sn)- z(s)l + Mlt - SI.

-

4t)I

Each term on the right hand side can be made arbitrarily small by choosing s sufficiently close to t. Remark 5.1. Theorem 5.1 requires rd-continuity and unique solutions. We have no doubt that these hypotheses can be weakened. 6. Bifurcations over Time Scales

Let us return to the logistic initial value problem

xA = 42

(i

- z)

,

over the time scales Iw+ and pZ+ for 0 delta-derivative, for all t E pZ+,

z(0) = 20

<

p

I 1.

By definition of the

Hence, the solution of the problem is found by iterating the function

z(t + p ) = 4pz(t)

(:

) + z(t)

- - z(t)

(

= 4pz(t) 3p4; - 4t))

(7)

starting from z(0) = 20. When p = 1, pZ+ = Z+and (7) is the iterated form of the solution of the logistic difference equation. On the other hand, as p + 0, the solutions of ( 6 ) converge to the solution of the logistic differential equation on Iw+ by Theorem 5.1. Let us consider the solutions (7) of the logistic dynamic equation (6) over the time scales pZ. Theorem 6.1. T h e solutions of the initial value problem ( 6 ) over pZ+ are topologically conjugate to the quadratic polynomials Q,(x) = x2 c, where every p E (0,1] corresponds exactly to one value ofc E [-2,1/4).

+

460

Proof. Let

By (7), we know that the solution of (6) over pZ+ is found by iterating the function L, starting from ~ ( 0=) 2 0 : x(t + P ) = L,(x(t)). The function L, is topologically conjugate to Qc for c = -1( 1 - 9p2 ) 4

by the homeomorphism h ( z )= -4px

3p+ 1 +2 .

Note that (8) expresses c as a decreasing function of p for 0 < p /J = 1, c = -2. AS /L + 0, c + 1/4.

5 1. When 0

The interval [-2,1/4] is exactly the real part of the Mandelbrot set. That is, as we go through the time scales pZ from the logistic difference equation to the logistic differential equation, we go through all of the interesting dynamics of real quadratic polynomials and all of its bifurcations! Acknowledgments

We thank Marcus McWaters for reminding us of the theory of hyperspaces and John Hubbard for a helpful discussion nonunique solutions. Theorem 5.1 generalizes both the main result of the proceedings article [2] and the stronger result actually presented by Elizabeth Duke at the same conference in Corfu, Greece in August 2004 and at the American Mathematical Society meeting in Atlanta in January 2005. Theorem 6.1 appeared in the proceedings article [3] and was presented by Kelli Hall at the same conferences. References 1. M . Bohner and A. Peterson, Dynamic Equations o n T i m e Scales: An Introduction with Applications, Birkhauser, Boston, 2001. 2. E. Duke, K. Hall, and R. Oberste-Vorth, Changing time scales I: T h e continuous case as a limit, Proceedings of the Sixth WSEAS International Conference on Applied Mathematics (CD-ROM), N. Mastoraki, ed., 2004.

46 1

3. E. Duke, K. Hall, and R. Oberste-Vorth, Changing time scales 11: Bifurcations in second degree equations, Proceedings of the Sixth WSEAS International Conference on Applied Mathematics (CD-ROM), N. Mastoraki, ed., 2004. 4. F. Hausdorff, Grundzige der Mengenlehre, Verlag von Veit, Leipzig, 1914. 5. A. Illanes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances, Marcel Dekker, New York, 1999. 6. D. Pompeiu, Sur la continuite' des variables complexes, Ann. Fac. Sc. de Toulouse 7,Sdr 2 (1905), 265-315. 7. L. Vietoris, Bereiche zweiter Ordnung, Monatsh. Math. 32 (1922), 258-80.

462

TOTALLY DISCRETE AND EULERIAN TIME SCALES

KELLI J. HALL AND RALPH W. OBERSTE-VORTH Department of Mathematics Marshall University Huntington, West Virginia 25755, USA E-mail: halll42Qmarshall.edu, oberstevorthQmarshall.edu

We show two results concerning approximation via “totally discrete” time scales. In the first, we show that any time scale can be approximated by a totally discrete time scale. In the second, we show the bifurcations in the parameter space for a simple two parameter family of dynamic equations-one parameter is the time scale itself, derived from Euler’s method for solving differential equations. These results speak to the nature of numerical approximations.

1. Introduction

Time scales have been studied for nearly 20 years; see, for example, [l]for an introduction. The theory attempts to connect the areas of differential and difference equations-the continuous and the discrete. From a numerical point of view, many connections between the continuous and discrete are well-known and anticipated. These include the notion of approximating continuous phenomena by discrete ones such as Euler’s method for differential equations. From a time scales perspective, this master-servant view appears t o be highly unbalanced. However, we examine exactly this situation in this paper. When trying to model a problem, such as a population x at time t , we may wish to allow parameters in our equation to vary. Factors such as hibernation patterns or incubation durations may suggest time scales; however, these may not be easily measured, leading us to treat the time scale as a parameter. In fact, this seems like an excellent idea in general. When trying t o solve a first order dynamic equation with an initial condition such as

we know that determining x ( t ) depends on the time scale. There are (at

463

least!) two classes of time scales that deserve a name: those with no isolated points and those with only isolated points. Among those time scales with only isolated points, the subclass of those with constant graininess (one can think of this as stepsize) is important and well-known. To wit, a time scale T is continuous if and only if T contains no isolated point; totally discretea if and only if T contains only isolated points; and Eulerian if and only if p ( t ) = D ( t ) - t is constant on T".

+

For example, R+ and UF=o[n,n $1 are continuous time scales, iZ+ and (2"ln E Z+}are totally discrete time scales, and pZ+, for p > 0, is eulerian (as well as totally discrete). Note that solving (1) over the time scale pZ+, for p > 0, is the same as approximating a solution of the differential equation 2'

= f(t,z),

z(t0)

= zo,

numerically using Euler's method with stepsize p. 2. Approximation in the Space of Time Scales

2.1. The Hausdorfl Metric

In the theory of hyperspaces, the set of time scales may be topologized by In this setting, the metric space of the Hausdorff metric. (See [3] and [4].) time scales is denoted by CL(R). We will denote the subspaces of eulerian and totally discrete time scales by CLE(R) and CLD(R), respectively. The Hausdorff metric H on CL(R) is derived from the standard bounded metric d on R defined by

for all z,y E R. Note that if 1z - yI = 6, then d(z,y)

5 min{ 1,6}.

would prefer-perhaps from the difference equations and iteration theory point of view-the simpler descriptor discrete to totally discrete, but that would be inconsistent with the literature. For many authors, a time scale is discrete if it contains no intervals of positive measure. This seems natural from a differential equations or integration theory point of view. Hence, the q-linear grid is discrete but not totally discrete. Some authors use isolated instead of totally discrete. We prefer not to name a set after a description of its elements. Moreover, a time scale may be isolated in CL(R). We thank the referee for helpful comments in this regard.

464

For every z E

R and every B E CL(R), define d(z, S ) = inf { d(z, s) I s E S } .

For every

T

> 0 and every B E CL(R), define NT(B)={SEX I d ( z , S ) < r )

Then the Hausdorff metric H on CL(R) is defined by

H(B,T) = inf { T > 0 I B C N,(T)and T C N,(B) } for all $, T E CL(R). B and T are at most Hausdorff distance 6 apart if and only if every point of B is no more than distance b from some point of T and vice verse. Note that when B C T,

H ( B , T ) = i n f { r > O (TcN,(B) ) . 2.2. Approximation b y Totally Discrete Times Scales

We will show that the set of totally discrete time scales is dense in the space of time scales under the Hausdorff metric. Of course, this is not surprising. Theorem 2.1. CLD(X)is dense in C L ( X ) . Proof. Choose a time scale T E CL(X). We use the bounded metric d on R and the Hausdorff metric H on CL(lR). Given E > 0, we will construct a totally discrete time scale B E CLD(X) such that H(B,T)< E .

T

U

-

b

U+€

w

w

s

nE

ao

a2 a3 an Figure 1. Discretizing an interval.

First, for every interval [a,b] that is a connected component of T,choose N such that b-a -<€

n

and define ak = a0

+ -k b -na

465

for Ic = 0 , 1 , 2 , . . . , n. We include the following n Figure 1):

+ 1 points of T in 8 (see

a = uo,u1,u2,a3,. . . , a , = b.

T

a

-

b

A

A

w

-

A

s

A

w

a

Figure 2.

C

A

b C Isolated points not too close to a dense point.

Second, for every isolated point t E T, include t in 8. Now 8 c T and for every t E T, there exists s E T such that I s - t l < E . However, 8 may not be a time scale since accumulation points of sequences of isolated points have been left out; this is corrected in the next step. See Figure 2.

t

T

A

A

s

t Figure 3.

w

w

a

A

.. A

A

t+E

Discretizing at dense points not contained in an interval.

Third, at every left- or right-dense point t of T that is not contained in a real interval in T (as in the first step), do the following. Choose 6 such that 0 < 6 5 E and T n [t - 6, t 61 contains no intervals; for E sufficiently small, we can set 6 = E . Now remove all elements of T n [t - 6, t 61 from 9 and add t to 8. Let us compute H(8,T). Since 8 c T,for all x E 8, d(x,T) = 0. On the other hand, 8 was designed so that for all x E T, d(z, 8) < E . 0

+

+

3. Parameterized Families of Dynamic Equations

When constructing dynamical systems to model phenomena, it is not always possible t o know all parameters precisely. Therefore, one may compare the results of the model for certain parameter values against experimental measurements and make adjustments to the parameters in order t o attempt

466

to improve the predictive value of the model. In our situation, we treat the underlying time scales as just some parameter. It is possible that the time scale parameter may tell us something such as the duration of incubation or dormant stages of a species in a population model. 3.1. A n Example: Parameter Space f o r Quadratic Equation

chaos

-28

c

Figure 4. Parameter space for zA = z2- z

+ c with % ( t o )=

-2

2

20.

Let us examine the simplest nonlinear problem. Consider the dynamical behavior of the solutions of the following quadratic equations: xA = z2 - x

+ c,

% ( t o )= 20.

(2)

467

We will completely describe the parameter space consisting of pairs of the equation parameter, c, and the graininess, p, of the eulerian time scales pZ+. We will restrict our attention to ( c , p ) E R x (0,1]; see Figure 4. At the end of the section, we will describe the missing boundary piece; of course, p = 0 makes no sense, but the second parameter is actually the time scale, not its graininess! 3 . 2 . Review of the Dynamics of Quadratic Polynomials

+

Note that, solutions of (2) over Z+are given by iterating Q c ( x ) = x 2 c beginning at 2 0 . The dynamics of the family of quadratic polynomials is well understood (see [2] for example). In fact, every quadratic polynomial has exactly the same dynamics as Q c for some c. Let us make this more precise. Let X and Y be topological spaces. A function f : X + X is topologically conjugate to a function g : Y + Y if a homeomorphism h : X + Y exists such that hof =goh, i .e., the following diagram commutes:

x

k X

lh

hl Y+Y 9

Topological conjugacy is an equivalence relation. From a dynamical point of view, the following are consequences: x is an attracting (respectively repelling) periodic point for f if and only if h ( x ) is an attracting (respectively repelling) periodic point for g. Therefore, it is possible to understand the dynamics of a complicated function by showing that it is topologically conjugate to a better understood function. The most obvious representation of the family of quadratic polynomials on R (or C ) is as the three parameter family given by fa,b,d(x)

= ax2

+ bx + d

where a , b, and d are real (respectively complex) parameters with a # 0. If c = ad

then

fa,b,d

+ -2b - -,b24

is topologically conjugate to QCby b h ( x ) = ux -. 2

+

(3)

(4)

468

The dynamics of the functions Qc are well understood. For c > 1/4, the orbits of all initial points tend to 00. At c = 1/4, there is a saddlenode bifurcation (also called a tangent bifurcation): the fixed points turn from complex (and repelling) to real, with one attracting and one repelling fixed point for -3/4 < c < 1/4. At c = -3/4, there is a period doubling bifurcation: an attracting periodic cycle of period two develops around the attracting fixed point as it stops being attractive. There is a period doubling bifurcation from period two to period four at c = -5 14. These two bifurcations are the beginning of a sequence of period doubling bifurcations leading to chaotic dynamics. Further along there is a region of attracting cycles of period three (and many others!). By theorems of SharkovskiY in [6] and Li and Yorke in [5],period three implies chaos. At c = -2 there is a homoclinic bifurcation: for c < -2, all orbits tend to 00 except starting on an invariant Cantor set. On this Cantor set, the polynomial is topologically conjugate to the one-sided shift on two symbols. These can be thought of as one-dimensional horseshoe mappings; for c > 1/4, the polynomials on C can be thought of as one-complex-dimensional horseshoe mappings. 3.3. Solutions over pZ+

Next, let us consider solutions of (2) over pZ+ for constant 0 < p definition of the delta-derivative, for all t E pZ+,

< 1. By

Hence,

x ( t + CL) = P ( m 2 + (1 - P M t ) + PC and the solution of (2) over pZ+ is found by iterating the function

L,(x) = pz2

+ (1 - p). + pc

starting from x(0) = xo. Note that when p = 1, pZ = Z and L, = Q c . By (3) and (4), the function L, is topologically conjugate t o Q-, for 1- p2

by the homeomorphism

h(x) = px

1-P +2 .

469

3.4. Description of the Parameter Space The curves y = constant in the c-p parameter space are lines of “equidynamics.” In particular, the curves y = 1/4, y = -3/4, y = -5/4, . . . , y = -2 are bifurcation curves in the parameter space. The bifurcation curve y = 1/4 has c = 1/4, a vertical segment in the c-p parameter space. This is shown in Figure 4. Let y < 1/4. The corresponding curve has equation 4y-1+p2 4P2 Since 47 - 1 < 0, as p tends to 0, c tends to -GO. Figure 4 shows the curves for y = -3/4, y = -5/4, and y = -2. C=

3.5. What Happens A s p Tends To Q ?

Notice that as p tends to 0, pZ+ tends to R+ in CL(R+) with the Hausdorff topology. It is clear that as pZ+ tends to R+, the solutions over pZ+ tend to the solution over R+. The solution of (2) over R+, for c < 1/4, has two constant trajectories (fixed points) at 1AZJF-G 2 . Note that these are exactly the fixed points of L, also. The fixed point (1+4=)/2 is repelling. When p i s small, the fixed point (l-J1=)/2 is attracting. This agrees exactly with what happens for the differential equation. In fact, the logistic equation considered in [4] corresponds exactly up to topological conjugacy. This is not a surprise: the convergence of the eulerian solutions to the solution of the differential equation is the basis of using Euler’s method to approximate solutions! Acknowledgments

This paper is based on the talk by the first author entitled Change in Tame Scales: The Effects of Parameterization on Dynamics at the International Conference on Difference Equations, Special Functions, and Applications in Munich, Germany in July 2005. References 1. M. Bohner and A. Peterson, Dynamic Equations on T i m e Scales: A n Introduction with Applications, Birkhauser, Boston, 2001.

470 2. R. Devaney, An Introduction t o Chaotic Dynamical Systems, 2nd ed., Westview Press, Boulder, Colo., 2003. 3. A. Illanes and S. Nadler, Jr., Hyperspaces: Fundamentals and Recent Advances, Marcel Dekker, New York, 1999. 4. B. Lawrence and R. Oberste-Vorth, Solutions of dynamic equations with varying time scales. 5. T.-Y. Li and J. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985-992. 6. A. Sharkovski:, Coexistence of cycles of a continuous m a p of the line into itself, Ukrain. Math. Zh. 16 (1964), 61-71 (Russian); Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), 1263-1273 (translated by J. Tolosa).

471

A GENERALIZATION OF THE DISCRETE UC HIERARCHY AND ITS REDUCTIONS

Y . OHTA Department of Mathematics, Kobe University, Rokko, Kobe 657-8501, Japan

We discuss about a generalization of the UC hierarchy which was recently introduced by Tsuda as an extension of the K P hierarchy. By generalizing the determinant expression of universal characters with the independent variable transformation given by Tsuda, the bilinear equations for the multicomponent determinants are constructed through the Laplace expansion technique. It is also shown that the periodic reductions derive discrete integrable 2+1 dimensional equations.

1. Introduction

It is well-known that most soliton equations are classified in the so-called K P hierarchy which has been studied for many years from various standing points of view. For example, the classification based on the symmetry of affine Lie algebra is given in Ref. 1. Recently the requirement to extend the K P hierarchy has arisen in the scenes of both mathematical theories and physical applications, and various methods of extension have been proposed. Amongst them, Tsuda2 introduced the UC hierarchy as an extension of the K P hierarchy by using the idea of the universal character. The UC hierarchy has unfavorable features, that is, the equations in the UC hierarchy are given in the form of infinite order differential equations for infinite number of independent variables, and consequently they are coupled systems of infinite number of equations. However in Refs. 3 and 4, Tsuda and Masuda showed that some of the q-PainlevQequations are satisfied by the q-discrete version of universal character with some restrictions, and in these cases, they proved that the equations are of finite order and finite coupled form. This means that in some reduced cases of the UC hierarchy, the equations can be written in smart forms. In this paper, we will reformulate Tsuda’s q-discrete UC hierarchy3 in the language of direct method5 of soliton theory, and propose its general-

472

ization for nonautonomous (i.e. random or non equi-interval) discrete and multicomponent case. In our general framework, the hierarchy of equations is given by finite order infinite coupled systems. By applying periodic reductions to this hierarchy, we can construct a lot of new systems of 2+1 dimensional integrable equations in the form of finite order finite coupled equations. After all the formulation turns out to be so simple that it is more transparent to start with the general setting of determinant expression of T function for the generalized discrete UC hierarchy rather than starting from the q-discrete UC hierarchy and generalizing it. Examples of 2+ 1 dimensional integrable systems derived from both the nonautonomous discrete UC hierarchy and the generalized one are explicitly given in the form of Lotka-Volterra type coupled equations. 2. Generalized Discrete UC Hierarchy

As is well known, the r function for continuous/discrete K P hierarchy is given in the form of Wronski/Casorati determinant,

where 'pij) 's are functions of continuous independent variables X I , x 2 ,2 3 , . . . and discrete ones kl ,k z , k 3 , . * * ,

'py =

&Xl,

X2,X3,.

. . ;kl ,k z , k 3 , . . . ) ,

which satisfy the following dispersion relations:

and

Here al(k1) is the difference interval for kl. Since al depends on k l , the above r function gives the solution for the nonautonomous discrete K P hierarchy6 including the q-KP hierarchy.

473

Now we introduce the r function of generalized discrete UC hierarchy. Starting from the r function for K P hierarchy,

we generalize it by applying the row-respective discrete independent variable transformation in the following form:

\

,

where sil's are integer parameters and p?)'s satisfy the same dispersion relations with the case of K P hierarchy. It is clear that rKPis a special case of r g u c by taking sil = 0 and ni = n for all i and 1. Since the dependence of matrix elements on k1, k2 , . . . is modified for each row in different ways by this generalization, the dependence of r on these discrete variables is broken and normally one can not expect any discrete equations satisfied by the r function in (1).However as we will see below, it is possible t o construct nontrivial bilinear equations and their solutions r by taking some specific choices of the parameters in T. In the case of nonautonomous discrete UC hierarchy, the r function is defined by

where the elements of determinant are separated into two components, upper and lower, and p?)'s satisfy the same dispersion relations with the K P case. It is easy t o see that the above r,"" is a special case of rEzEz,... ,nN in (1)just by taking sil = 0 for upper component, sil = 1for lower component and ni = 0 for upper component, ni = n for lower component. We should comment that the q-discrete universal character in Ref. 3 can be derived from r,"" as a special case, in the same manner with the derivation of the q-KP hierarchy from the nonautonomous discrete K P hierarchy.6 Now it is clear that there are finite order differential/difference equations satisfied by r,"". For simplicity, let us change the notation as 2 1 -+ 2, 2 2 -+ 9 , 2 3 + t , ki -+ k, k2 -+ 1, k3 + m, ai(ki) -+ a k l az(k2) -+ bi, a 3 ( k s ) -+ cm and r,"" -+ 7 ., Then the r function of nonautonomous

474

discrete UC hierarchy is given by

and the discrete linear dispersion relations,

Some of the typical bilinear equations satisfied by this r function are given as

This hierarchy of bilinear equations includes the K P hierarchy as part of it. All of the above bilinear equations are easily proved by using the Laplace expansion technique for both continuous and discrete ~ a s e s . ~Here - ~ it should be noticed that these bilinear equations are still infinite coupled systems except for the subsystems which already appeared in the K P hierarchy.

475

3. 2+1 Dimensional Integrable Systems

In order to derive finite coupled systems from the systems of infinite coupled equations constructed in the previous section, we apply periodic reductions for some components. Then 2+1 dimensional integrable systems can be written in closed forms. Let us consider a periodic reduction for the lower component of the 7 function of discrete UC hierarchy given in (2). For the reduction of period p , we impose the condition,

Pi('+PI

e qp),

i E (lower component) ,

where e means equal upto gauge (typically upto multiplication of a constant). Then T~ in (2) satisfies

For simplicity, we consider the case of constant gauge, i.e.,

where G is a constant. In the case of period 2 reduction, p = 2, by rewriting as ro = f and 71 = G1I2g, we obtain for example the bilinear equations,

+ (a:- - il)

g(lc - l,z - l)g(k,Z) = 0 ,

which form a finite order finite coupled 2+1 dimensional integrable system. Through the dependent variable transformation,

the above system of bilinear equations is transformed into the nonlinear

476

form, (u(k

il) ($ i )

+ l,z - 1) - U ( k , q z = (ak:l - -

(v(k + 1 , l - 1) - v(k,l))z =

-

-

(ev(k,l-1)-a(k,l)

(- L) ($ $) 1

-

ak+l

-

-

(,a(k+l,l-l)-v(k+l,l)

-

l)

' -

(eU(k+l,'-l)-u(k+l,l)

bl

(eu(k,l-1)-W(ktl)

9

*

This is an example of the 2+1 dimensional soliton equations in the nonautonomous discrete UC hierarchy. The solution is given by the T function in (2). We note that if a k , bl are independent of k, I respectively, v = u and u depends on T = 2k I only, then the above system reduces to

+

which is equivalent with the Loth-Volterra equation, d zFT

= FT(FT+1 - FT-1)

under the variable transformations,

Next let us consider a simple example of the equations in the generalization of nonautonomous discrete UC hierarchy. By choosing s i l = 0 for all I and n i = 0 in the upper component, and s i l = 1, s i ~= 2 and ni = n in the lower component, and using the same change of notation with the case of discrete UC, we have the following r function:

where 'p?)'s satisfy

477

Then this r function satisfies, for example, the bilinear equation,

which can be proved by using the Laplace expansion technique.?-’ As an example let us apply the period 3 reduction to the lower component, i.e. by imposing the condition, ‘pi (’+3)

Vaj’

,

i E (lower component),

we get the periodicity for r,

By the dependent variable transformation,

we obtain the 2+1 dimensional nonlinear integrable system,

This is an example of the equations derived from the generalized nonautonomous discrete UC hierarchy by applying a periodic reduction, which does not appear in the original UC hierarchy. Suppose u,v,w, a k and bl are

478

independent of k and I, then the above system reduces t o the Loth-Volterra equation,

4. Single Bilinear Equation of Discrete UC Hierarchy

So far each of the bilinear equations of q-discrete UC hierarchy relates more than one universal characters, that is, they are intrinsically coupled equation^.^ ,4 Thus a natural question arises: Are there bilinear equations satisfied by one universal character only? Let us consider the r function of nonautonomous version of discrete UC hierarchy,

r(k,Z,rn,n)=

det IliJIN

‘Pp(k,2, m, n) ....................... (j+k+l+m+n)

(‘Pi

(h2,m,n))

,

where (p6j)’ssatisfy

‘Py(k,Z,m,n)- p?’(k - l,Z,m,n) - ‘Py)(k,z,m,n) - ‘Pi( j ) (k,Z- l , m , n ) bl

ak

-

‘Pp(k,Z,m,n)- ‘ P p ( k , z , m - 1,n) Cm

p p ( k , l , r n , n )-‘Pp(k,z,m,n- 1) (j+l) (k,1, m, n) * = ‘Pi dn

Then the discrete UC hierarchy tells that the lowest order bilinear equation satisfied by the above r function is

- bl)(Cm - dn)T(k - 1 , l - l , m , n ) T ( k , I , m- l , n - 1) (ak - c,)(bl - dn)T(k - 1,z,m - l,n)r(k,2- 1,m,n - 1)

(CA~

-

+ (a~-dn)(b~-cm)r(k-l,Z,m,n-l)~(k,l-l,m-l,n)=O. This gives a single equation for the universal character. The above bilinear equation appears in the nonautonomous discrete K P hierarchy also, however the above r function does not satisfy any of the lower order bilinear equations in discrete K P hierarchy. It looks that the bilinear equation is 4 dimensional but actually it describes the 3 dimensional evolution since k + Z + m + n is invariant under the propagation in the 4 dimensional space.

479

5. Concluding Remarks The q-discrete UC hierarchy which originally appeared in the study of qPainlev6 equations, is generalized into the nonautonomous discrete case and the multicomponent version. By applying periodic reductions, various 2+1 dimensional nonlinear integrable systems can be constructed, and some examples are explicitly given in a similar form to the Loth-Volterra equation. This reminds us that some of the (discrete) Painlev6 equations and the so-called Noumi-Yamada systems can be written simply in symmetric form of (discrete) Lotka-Volterra type equations.l0l1l The direct derivation of these ordinary differential or difference equations from the generalized UC hierarchy has not yet been fully clarified. The generalized nonautonomous discrete UC hierarchy contains quite rich classes of equations as its reductions. Their solutions describe solitons with a lot of internal freedom which came from the periodic reductions for some components. It may be interesting to investigate the classification of equations including the case of similarity reductions and to study the concrete behavior of the interactions of solitons with the internal freedom. Details will be reported elsewhere.

Acknowledgments The author would like to express his sincere gratitude to Dr. Tsuda for valuable discussions, crucial comments and telling the essential informations from which the idea of this work is originated. He is also grateful to Dr. Masuda for kind comments.

References M. Jimbo and T. Miwa, Publ. RIMS, Kyoto Univ. 19,943 (1983). T. Tsuda, Comm. Math. Phys. 248,501 (2004). T. Tsuda, Comm. Math. Phys. 260,59 (2005). T. Tsuda and T. Masuda, Comm. Math. Phys. to appear. R. Hirota, The Direct Method in Soliton Theory (Cambridge Univ. Pr., Cambridge, 2004). 6. R. Willox, T. Tokihiro and J. Satsuma, J. Math. Phys. 38,6455 (1997). 7. J. Satsuma, J. Phys. SOC.J p n . 46,359 (1979). 8. N. C. Freeman and J. J. C. Nimmo, Phys. Lett. 95A, 1 (1983). 9. Y. Ohta, R. Hirota, S. Tsujimoto and T. Imai, J. Phys. SOC.J p n . 62, 1872 (1993). 10. M. Noumi and Y. Yamada, Comm. Math. Phys. 199,281 (1998); Funkcial. Ekvac. 41,483 (1998). 11. K. Kajiwara, M. Noumi and Y. Yamada, Lett. Math. Phys. 62,259 (2002).

1. 2. 3. 4. 5.

480

THE DICHOTOMY CHARACTER OF

XN+1 =

+ +

PNXN ^ / N X N - ~ AN B N X N

WITH PERIOD-TWO COEFFICIENTS

C.H. GIBBONS,

s. KALABUSIC,

AND

c. B. OVERDEEP

Mathematical Sciences Department Salve Regina University Newport, RI 02841 Department of Mathematics University of Sarajevo Sarajevo, Bosnia and Herzegovina Department of Mathematics Saint Martin's University Lacey, WA 98503

We extend the known results of the nonautonomous difference equation in the title to the situation where (i) the parameters P,, and ^I, are period-two sequences of nonnegative real numbers with y-, not identically zero; (ii) the parameters A,, and Bn are period-two sequences of positive real numbers; and (iii) the initial 10 E (0,m). conditions 2-1 and 20 are such that Z - ~ , Z O E [ O , o o ) and 2-1

+

Key words and phrases. boundedness, difference equation, periodic coefficients, periodtwo solutions 2000 Subject Classification: 39A10, 3 9 A l l

1. Introduction and Preliminaries

Our aim in this paper is to continue with the investigation of the trichotomy character and the asymptotic stability of solutions of the nonautonomous difference equation G+1

=

Pn%

+ 'YnZn-1 ,

An + B n x n

12=0,1,...

48 1

where the parameters P,, y,, A,, and B, are period-two sequences of real numbers with Pn,yn nonnegative but y, not identically zero and with A,, B, positive; and the initial conditions, 2-1 and X O , nonnegative and such that 5-1 xo > 0. This paper is intended to complement the work in [GO] and cover many of the remaining regions in the parameter space not covered in [GO]. This paper, together with [GO], extends the behavior of solutions of the autonomous case.

+

The autonomous version of Eq.(l) given by

is investigated in [KLP] when the parameters p, y, A, and B are positive and the initial conditions 2 - 1 and xo are positive. A change of variables reduces the number of parameters to two, thereby allowing one to work with the simpler equation

Note that zero is an equilibrium of Eq.(3), and when q exists a unique positive equilibrium solution,

+ r > 1 , there also

3 = q + r - 1,

of Eq. (3). We summarize, in the form of Theorems A, B, and C, some pertinent known results from [KLP] on the zero and positive equilibria of Eq.(3). Theorem A

(a) Assume q+r

5 1.

T h e n the zero equilibrium of Eq.(3) is globally asymptotically stable. (b) Assume q+r

> 1.

T h e n the zero equilibrium of Eq.(3) is unstable. More precisely, it is a saddle point when 1-ql+q.

482

+

Theorem B Suppose that r=i and let 2 - 1 xo > 0. Then the positive equilibrium y = q of Eq.(3) is globally asymptotically stable. Theorem C

(a) Assume T=l+q.

Then every solution of Eq. (3) converges t o a period-two solution. (b) Assume 1< r

< 1 + q.

T h e n every positive solution of Eq. (3) converges t o the positive equilibrium of Eq. (3). (c) A s s u m e T>1+Q.

T h e n Eq. (3) has unbounded solutions. By the change of variables

where Ao,A1,Bo, B1

> 0 , we can reduce Eq.(1) to

where the initial conditions

x-1

+ X O > 0 and

For Theorems A, B, and C, the q, r- parameter space is partitioned into four regions by three lines (see Figure 1 at the end of the paper). With the nonautonomous Eq.(4), we consider two parameters spaces, one for 91,ro and the other for 40, r1. Each parameter space is partioned into four regions by three lines (see Figures 2 and 3 at the end of this paper).

483

In [GO], the behavior of solutions of Eq.(4) is studied for those cases in which T O , Q ~and ~ 1 , q ocome from corresponding regions (i.e., ( r 0 , q l ) E Ri,( T I , qo) E 725,i = j ) . For example, TO

> q1 + 1

and

> qo + 1.

TI

In this paper, we consider those cases when from corresponding regions. For example, TO

> q1 + 1

and

TI

T o , q1

= q1

and

r1, qo

do not come

+ 1.

Equation (4) can be decoupled into a difference equation representing the odd terms {zZn+1} and the even terms ( ~ 2 of ~ )the solution. Let a1 a2 a3

a4

= = = =

TO

a5

4041 + T O - qoro -q&71 - 40Tl - qoro

a6

qo'Oror1

a8

a7

= = = =

q1

+1

-4041 - T1 - qo

+1

TOT1

-40

Then, for the odd terms of the solution, we have the equation

For Eq. (5), the equilibria are the nonnegative solutions of (q1+

+

+ qoro - qo + r0rl - r0 - r1 + + 4241 + qorl + qoro - q o T o r i ) y = 0.

1 - rO)Y3 (-2qOq1

+ (-40

(6)

The possible equilibrium values for Eq.(6) are

9 = 0, Y = 40, and y = 4041 -

(To

q1

- l ) ( q- 1)

+ 1 - To

,

when

q1

+ 1 # r0.

Similarly, for the even terms of the solution of Eq.(4), we have the equation

where

484

For Eq. (7), the equilibria are the nonnegative solutions of

The possible equilibrium values for Eq.(8) are

z = 0,

and

The following identities, which we state without proof, will be of importance.

We may consider Eq.(4) as a system of difference equations as follows:

485

Then

where

+ 4041 + 7-11 / ((40 + 1). + TOU + 1 ) 2

486

Remark 1.1. We are interested in those conditions on the parameter pairs 41, TO and qo, T I of the form

Ri u R[i for i # j , where (Ql,TO) E

Ri,

(Q0,Tl)E

R[i,

i.e., the conditions where 41,TO comes from a region in the plane that does not correspond t o the region qo, T I comes from (see Figures 2 and 3). Note that we are able t o cut in half the number of cases we actually study, because the equations for the odd and even subsequences of Eq.(l), Eqs. (5) and (7) are exactly of the same form except that the indices of all their parameters differ by 1. 2. Character of Solutions

We are interested in the following six cases not covered in [GO]: Casel. q 0 + ~ ~ 5 1 a n d l - q 1 < r ~ < q ~ + l . Case 2. qo T I 5 1 and TO = q1 1. Case 3. qo + T I 5 1 and TO > q1 1. Case 4. 1 - qo < T I < QO 1 and TO = q1 1. Case 5. 1 - qo < T I < qo 1 and r0 > q1 1. Case 6. T I = QO 1 and TO > q1 1.

+

+

+ +

+ +

+

+ +

(Note that Remark 1 in Section 1 allows us to cut in half the actual number of remaining cases that are not covered in [GO].) Below we prove results for these cases or subcases of these cases. While it appears from computer observation that Cases 2-6 all possess unbounded solutions, computer simulations further suggest that the rate of divergence for Cases 2 and 4 is much slower than that for Cases 3, 5, and 6. It remains an open problem to define in a rigorous way the two different rates of divergence. 2.1. When go Case 1 )

+TI

51

and 1 - q1

< TQ 5 1

(a subcase of

We show that when qo + r l 5 1 and 1-ql < r0 5 1,every solution of Eq.(4) is bounded. We then show that when we additionally have the condition that qoq1 > (1 - r0)(1 - T I ) , there exists a positive period-two solution which is locally asymptotically stable.

487

Lemma 2.1. Let be a solution of Eq.(4) with z-1,zo E [O,m) 20 E (0, m), and suppose that and 2 - 1

+

r1

5

1

and

TO

5 1.

Then {xn}r=-l is bounded.

Proof. Let { X ~ } F = -be ~ a solution of Eq.(4) with x-1,xo E [O,m) and 2-1 20 E (0, m). We separately consider the even and odd subsequences of {~Cn)E-l. For { Z ~ ~ }we F= have ~ ,the following two cases. Case 1. xo 5 41. If xo 5 41, then from identity (12) and our hypothesis that TI 5 1, we have xzn 5 q1 for all n 2 0. Case 2. xo > q1. If 20 > q1, then from identities (11) and (12), either {x2n}F=o is a decreasing sequence or there exists N 2 1 such that xzn 5 q1 for all n 2 N . In either case, { ~ 2 ~ }is~bounded = ~ from above. Similarly, it follows from identities (9) and (10) and our hypothesis that TO 5 1 that {x~~+l}F=~ is bounded from above. Therefore, is bounded. 0

+

Theorem 2.1. Suppose that

4041

> (1 - ro)(l - r1).

Then there exists a prime period-two solution of Eq. (4),

_ _ _ _

Y , 2 , Y , 2 , . ..

where

g = 4041 - (To 41

- 1)(r1 - 1)

+ 1- T o

and

Z=

4041 - ( T o - l ) ( T l - 1) 7 40 1 - r1

+

which is locally asymptotically stable.

Proof. See Appendix A, in which we use Mathematica to help in the computation of the characteristic equation associated with the system of difference equations represented as the map T in (13). The characteristic equation is of the form x2 - P A - q = o

488

and we show in Appendix A that

Ipl

< 1 - q < 2.

Hence (y, 5 ) is locally asymptotically stable (see Theorem l.l.l(c) in [KL]). 0

Extensive computer observations indicate that the odd terms of the solution are bounded when T O > 1. Once the bounded nature of the odd terms of the solution can be established (see Open Problems), then the condition TO 5 1 may be removed from the theorem.

5 1 and ro = 91 and 4 Taken Together)

2 .2 . When r1

+ 1 (a Subcase of

Cases 2

We show that when r1 5 1and TO = q1+ 1, there exist unbounded solutions.

Theorem 2.2. Suppose that TI

5 1, ro

+ 1,

= q1

and

T h e n there exist unbounded solutions of Eq.

qoq1

> 0.

(4).

Proof. Suppose that TI

51,

TO

= q1

+ 1,

and

qoq1

>0

in Eq.(4), and let { x ~ } , " , - ~be a solution with initial conditions

>qo

x-1

O
and

From Eq.(12), we see that ~2~

< q1

for

n

2 0.

It then follows from Eq.(4) that q052n

+ TOX2n-1

>

q052n

+ r052n-1

-

QO52n

52n+1 =

1+q1

1+q1 - q052n 1 41

+- 1 + q 1

TOX2n-I

+ + 52n-I

> 52n-1.

489

For the sake of contradiction, we assume that lim xzn+1 = Lo < 00.

n+m

Then, from Eq.(4) and our hypothesis that qo

>- ‘lq0 > o

> 0, we have

n 2 0;

for

1+L,

and, thus, from Eq.(4),

This contradicts our assumption that Lo < 00. Therefore, lim

n+cc

2.3. When r1

5 90 + 1

and

X ~ ~ += I00.

rg

> q1 + 1

0

(Cases 3, 5, and 6

Taken Together)

We show that when solutions.

r1

5

qo

> q1 + 1, there

+ 1 and rg

exist unbounded

Theorem 2. 3. Suppose that r1

51

and

ro

> q1 + 1.

Then there exast unbounded solutions of Eq.(4). Proof. Suppose that r1

5 1 and

ro

> q1 + 1

in Eq.(4), and let { x ~ } ; = - ~be a solution with initial conditions 2-1

> QO

and

0 < xo 5

41.

490

From Eq.(12) and the hypothesis that x2n

5 q1

5 1, we have

TI

n20

for

(with equality for all n if q1 = 0). From Eq.(lO), the hypothesis that > q1 1 2 1, and induction, we have

TO

+

> 40

~ 2 n + l

for

n 2 -1.

(

- Tg

Hence, To

Qn+1

- Qo

=

L

9

3%-1

1

+x2n - qo)

TO(22n-1

1+ q1

(where the inequality is an equality if qo = q1 = 0). It follows that, since TO > q1 1, we have

+

lim x2,+1 n-+m

= m.

0

Theorem 2.4. Suppose that

Then there exast unbounded solutions of Eq.

(4).

Proof. Suppose that 1

in Eq.(4), and let

< TI 5 1+ QO

{xn}F=-l

and

> q1 + 1,

be a solution with initial conditions

From Eq.(12) and the hypothesis that x2n

From Eq.(4), it follows that

TO

> q1

TI

for

> 1, we have n 2 0.

49 1

Next, we show that 22 E (q1,ro-1). Note that by hypothesis, ql+l-r0 22

- (To - 1) = Q l Z l

< 0:

+ T l Z O - - 1) + + 1- TOZl + z1 (To

1+x1 41x1 T l Z O - To -

If21

- ~l(4l+1-ro)+Tlzo-To+1

1 +x1

To-y-ql %I1 + 1- + - 1)1+ - 1) + - 1)(q - 1) = 0.

k0-1

< -

TI-1

To)

(To

Tl(T0

- 1)

21

1)(T1

47-0

+

1

(To

21

By induction, it follows that lim

n+o3

= 00

x2,+1

and

22,

E (41, TO - 1 ) .

3. Open Problems

Several open problems remain. Open Problem 3.1. The change of variables employed requires that the sequences A, and B, be positive. Investigate the character of solutions of Eq.(l) when A, and B, are non-negative period-two sequences such that the denominator is never zero. Open Problem 3.2. Rigorously define the two rates of divergence: the “fast” divergence which has been observed when r1 > qo 1 and the “slow” divergence when TI = 40 1 for Eq.(4).

+

+

Open Problem 3.3. Prove that Eq.(4) possesses unbounded solutions when 1 < TO < 41

+1

and

TI

= 40

+ 1.

Open Problem 3.4. Prove that the odd terms of the solutions of Eq.(4) are bounded when 40

+ T1 L 1,

492

r = l

Figure 1. The q , r - parameter space for Theorems A , B, and C.

Figure 2.

The q1,n-parameter space for Eq.(4)

493

40

+

T1

<1 40

Figure 3.

R1

=

R2

=

723

=

724

=

72;

=

+

7-1

=1

40

The qo,rl-parameter space for Eq.(4).

R', = 72;

=

Rk

=

Acknowledgement The authors are deeply indebted to the referee for significantly improving the clarity and quality of this paper by providing extensive recommendations.

494

References

w,

GKL. C. H. Gibbons, M. R. S. KulenoviC, and G. Ladas, On the Recursive Math. Sci. Res. Hot-Line 4(2)(2000), 1-11 . Sequence yn+l = GO. C. H. Gibbons and C. B. Overdeep, On x,+1 = 0n.nAn+Bnxn +ynzn - 1 with Periodtwo Coefficients, to appear. KL. M. R. S. KulenoviC and G. Ladas, Dynamics of Second Order Rational Dijference Equations, Open Problems and Conjectures, Chapman&Hall/CRC Press, 2001. KLP. M. R. S. KulenoviC, G. Ladas, and N. R. Prokup, On the Recursive SeISz, quence %+1 = crzn+Ozn-l , J. Differ. Equation Appl. 6(5)(2000), 563-576.

Appendix A. Proof of Theorem 1 Using Mathematica The purpose of this section is to show the algebraic proof of local asymptotic stability. Muthematicu was used extensively to assist in the factoring of several algebraic expressions. Results from Muthemuticu were copied and pasted into this paper: expressions with q0, q l , r0, r l were copied from Mathemutica output; expressions involving qo, 41,ro, TI were done by hand. Evaluating Eqs. (14)-(17) at u and in Muthematicu yielded:

r 0 ( 1 + qo r l ) af au qo + qoqi + ro rori -

--

-

_ as -

av

+

+ rO(-1+

+qlrl)

-

(1 q l - rO)(qO(l

/

+ rO(-1 + 2rl))) [q02q1(1+ 41) - rO(-l+ r l ) ( q l + r l - r 0 r l ) + qO(q12 + r l - r 0 r l + q l ( 1 + ro + r l - 2rOrl))] -(-1+

7-1)

rl)(l - rl

The characteristic equation was found using Mathemutica:

(A.4)

495

+ [-qO(1 + 41) - a(-1 - q0)rO + (-1 - q0)qOqlrO + (-1 (-1 ql)(-1 + ~ 1 )qOq1((1+ ~ ql)rl 6rOrl

A2

-

-

- qO)r02

+

-

+

-2qOrOrl- 2qlrOrl+ qOqlrOrl 4r02rl +2qOrO2rl + 4rOrl2 2qlrOrl’ - 3r02r12)X / [q02ql(l+ 41) - rO(-1+ rl)(ql rl - r0rl) qO(q12 + rl - r0rl + ql(1 + ro rl - 2rOrl))l + [(rOrl + q0rOrl qlr0rl qOqlr0rl - r02rl - qOrO2rl - r0rl2 - qlrOrl2 rO2r121 / [qOZql(l+ q1) - rO(-1 + rl)(ql rl - r0rl) + qO(q12 + rl - r0rl ql(1 + 7-0 rl - 2rOrl))l

+

+

+

+

+

+

+

+ +

+

If we write the general characteristic equation

x2 - p x - q = o , we see that

+

41) - 2(-1- q0)rO (-1 - q0)qOqlrO (-1 - qO)r02 (-1 - ql)(-1 + ~ 1 -)qOql((1 ~ + q1)rl - 6rOrl - 2qOrOrl - 2qlrOrl-t qOqlrOrl + 4r02rl 2qOrO2rl 4rOrl2 + 2qlrOrl2 - 3r02r12) [q02ql(l+ 41) - rO(-l+ rl)(ql + rl - r0rl) + qO(q12 + rl - r0rl ql(1 ro + rl - 2rOrl))l

+

/

+

+

p = -[-qO(1+

+

+

+

and q=

/

+

+

+

-[(?-Or1 q0rOrl qlr0rl qOqlrOr1 - r02rl -qOrOZrl - r0rl2 - qlrOrl2 + rO2r121 [q02q1(1 41) - rO(-1+ rl)(ql + rl - r0rl) + qO(q12 rl - r0rl ql(1 ro + rl - 2rOrl))l

+ +

+

+

Observe that p and q share a common denominator. For the case we are considering, the denominator is positive:

denom = ro(1- rl)(ql

+TI

- rorl)

+ qo[r1 - TOT1 + 41(1+ To + 7-1

-

2ror1

+ 41 + qo + qoq1)]

496

The numerator will determine the sign of p and q. We will show that p is positive and thus we do not need to consider the absolute value of p when testing p < 1 - q. Using Mathematica’s Simplify command on the numerator of p gives us:

+ q0(ql2rl + ql(1 + + r l - r0rl) - (-1 + rO)(l - ro + 2rOrl)) -(-1+ rl)(ql(l - r l + 2 ~ 0 ~ 1 (-1 ) + ~ 0 ) ( 1- T O - r l + 3 ~ 0 ~ 1 ) )

qO2qlr0

TO

This can be rewritten as Pnum

= qo[q1(1 + T O

+ r1 - TOT1 + q l r l + qoro) + (1 - ro)(l - 7-0 + 2ror1)l

+ (1 - r1)[q1(1 - TI + 2rori) + (1 - ro)(1

-To

-

+ srori)]

Thus, p is positive for the case under consideration. To show the first part of linearized stability, we want 0 < 1 - q - p . Using Mathematica’s Simplify command on the expression 1 - q - p and ignoring the common denominator as it is positive leads to

+

(1 q l - r O ) ( l + qo - r1)(-1+ q0ql

+ ro + r l - 7-07-1)

The condition 4041 > (1 - r o ) ( l -

Thus, p

< 1 - 4.

TI)

is equivalent to

qOql+ rO

+ r l - rOr1- 1 > O

0

To complete the proof, we must show that 1 - q < 2 or equivalently, 0 < 1 q. Using Mathematica’s Simpli f y[1+ 41 and ignoring the common denominator gives us

+

ql

+ qOql + q12 + q0ql2 + q l r 0 + r l + q l r l - 27-07-1 - 3 q l r O r l + r02rl =ql

+ qOql + q12 + q0ql2 + qlrO + q l r l - 3 q l r O r l + r l ( r 0 - 1)2

+ q0 + q l + qOql + T O + r l - 3rOrl) + rl(r0 - 1)2 Note that 1+ r0 + r l - 3rOrl > 0 because r0 < 1,rl < 1. = ql(1

Thus, 1 - Q

< 2.

0

497

INTEGRAL COMPARISON THEOREMS FOR SECOND ORDER LINEAR DYNAMIC EQUATIONS

LYNN ERBE, ALLAN PETERSON, AND PAVEL REHAK Department of Mathematics University of Nebraska- Lincoln Lincoln, N E 68588-0130 U S A E-mail: lerbeOmath.unl.edu, [email protected]. edu Mathematical Institute Academy of Sciences of the Czech Republic Brno, Czech Republic E-mail: [email protected] Some integral comparison theorems for scalar second order linear differential equations are extended to dynamic equations on time scales. Several examples and applications to difference equations are given.

Key words and phrases. comparison theorems, linear oscillation, Hille-Wintner, time scale 2000 Subject Classification: 39A10

1. Preliminary Results Consider the pair of linear equations &[Z] := ( T l ( t ) Z A ) A

+ Q l ( t ) Z U = 0,

(1)

We will be interested in obtaining comparison theorems of integral type for (1)and (2). However, for the sake of simplicity, we assume T I T Z and therefore restrict our attention t o the pair of equations

498

since the comparison of r1 and 7-2 in the HilleWintner theorems below will follow from the Sturm type result. For details concerning calculus on time scales and other pertinent definitions, we refer to the books [l],[2], and [9]. We start with several auxiliary lemmas which are crucial in the proof of the main results. The following two statements are usually referred to as the Riccati technique. Denote

Lemma 1.1. ([lo],[4]) The equation

~ ~ ~ := ([ r (zt ) z] A ) A + q(t)z' = o

(5)

is nonoscillatory if and only if there is a function z satisfying the Riccati dynamic inequality z A ( t ) q ( t ) S [ z ] ( t )5 0 with r ( t ) p ( t ) z ( t ) > 0 for large t .

+

+

+

If the assumptions are strengthened, then nonoscillation of (5) can be expressed in terms of the solvability of a Riccati integral inequality involving improper integrals. Lemma 1.2. ([ll])Let

Sam l/r(s) As = 00

with r ( t ) > 0. Assume that

q(s)As 2 0

(and $ 0 for large t). Then (5) is nonoscillatory if and only if there is a positive function z satisfying the Riccati integral inequality

z(t) 2

4"

4 s ) As +

lm S[zl(s) As

for large t . Remark 1.1. We note that under some additional conditions, nonoscillation of (5) implies the existence of a solution z of the Riccati equation zA(t)+q(t)+S[z](t) = 0 with limt-tm z ( t ) = 0, which consequently satisfies the integral equation z ( t ) = q ( s ) As S[z](s)As. One such sufficient condition is contained in the following lemma, whose proof is similar to the proof of [Theorem 2.1,[5]]:

stm

+ stw

In the statement of the result, we will need to consider sequences of points in the set of right-scattered points, 'f?, defined by

'f? := {t E T : p ( t ) > 0).

499

Lemma 1.3. Suppose that Lx = 0 is nonoscillatory on [a,m)a, Sam &At = 00, r ( t ) > 0 , q ( s ) A s 2 0 (and $ 0 for large t). As-

stm

%<

sume there exists a sequence { t k } C ?? such that t k + 00 with M for some M > 0. Then L x = 0 has a solution x satisfying x ( t ) x A ( t )> 0 for large t and

The following lemma will be useful in the case when verges.

Sam l/r(s) A s con-

Lemma 1.4. Assume h E C,l,(T) with h ( t ) # 0 . Then x = hu transforms = 0 into the equation LRQ[u]:= ( R ( t ) u A ) A Q(t)u" = the equation LTq[x:] 0 with R = rhh" and Q = h'[(rhA)A+qh"]. This transformation preserves oscillatory properties.

+

Proof. To show that h"L,,[z] = LRQ[u],when x = hu, it suffices to show that

s,"

h"(s)L,q[X](s) As =

s,"

LRQ[u](S) As.

(6)

Using integration by parts of the left-hand side of (6) yields

which is the same as the expression obtained when integrating by parts the right-hand side of (6). Since h # 0, LTq[x] = 0 is oscillatory if and only if LRQ[U] = 0 is oscillatory. 0 2. Main Results Because of Lemma 1.2 we can now easily prove the following important comparison theorem.

Theorem 2.1. Assume r ( t ) > 0 with

Jam1/r(s) As

= 00. Further assume

500

Loo

(with q1($ As f 0 ) f o r large t . Then if Ll[x]= 0 is nonoscillatory, it follows that L2[x]= 0 is nonoscillatory.

Proof. From Lemma 1.2, there exists a T E [a,~O)Tand a positive function z satisfying the Riccati inequality roo

roo

on [T,W)T. Hence,

Then by Lemma 1.2 we have that the equation L ~ [ x=] 0 is nonoscillatory on [a,W ) T . 0

In the next result, we are able to conclude that the corresponding solution of the Riccati equation for Lz[x] = 0 has limit 0 at infinity, if an additional assumption is made on some sequence of points in ?. Since the proof is similar t o the proof of Theorem 1.1 of [5] , we shall omit it.

Theorem 2.2. Assume r ( t ) > 0 with

soml/r(s)As =

00,

and assume that

Assume in addition that there exists a sequence { t k } C ? such that with 5 M f o r some M > 0 , and suppose that L l x = 0 is nonoscillatory o n [a,m ) ~T.h e n L2x = 0 is nonoscillatory o n [a,o o ) ~ and has a solution y satisfying y ( t ) y A ( t )> 0 f o r large t with

tk

+ 00

3

Under some additional further restrictions, we are able to obtain the following extension of the more general version of the Hille-Wintner Theorem.

Theorem 2.3. Assume that r ( t ) > 0 and

Sam l/r(s) As = 00.

Let

50 1

and let x denote the characteristic function of there exists rn > 0 such that W ( t )I dt)x(t),

$1.

Further suppose that

t E T,

(8)

and (10) for large t . Then i f L1[x] = 0 is nonoscillatory, it follows that L ~ [ x=]0 is nonoscillatory. Proof. Using Lemma 1.2, there exists a T E [a,M)T and a positive function z satisfying (7) on [T,oo)~. Define

v ( t ) :=

Lm

q2(s)As

Lm + 1"

+

S[z](s)As.

(11)

Note that (using (lo)), for t E [ T , ~ o ) T , Iv(t)l

I

II"

qz(s)AsI

LW

I

41(s)As +

S[4(S)AS

LW

SC.I(s)As

= z(t).

We would like t o prove that

SkI ( t ) 2 S[vI ( t ),

(12)

for large t. This inequality is clearly true for any point which is right-dense or where v(t) 2 0. Hence it remains to show that (12) is true for all large right-scattered points where z ( t ) > -v(t) > 0. Note that this may happen only for t E 9. First we show that

+ lu(t)v(t>> 0

(13)

for large t E T. Clearly we may restrict our consideration only t o t E 91 (even to the right-scattered t's at which v(t) < 0 which form a subset of $ 1 ; we denote f?u := { t E $1 : v(t) < 0)). Inequality (13) is then equivalent t o r ( t ) / p ( t ) > -v(t), which however holds for large t because of ( 8 ) , since v(t) + 0 as t + 00. For t E k ' , note that (12) is equivalent to

d t ) ( z ( t )+ v(t>>2 - l u ( t M t ) z ( t ) .

(14)

502

+

Since z ( t ) v ( t ) > 0, (14) is equivalent to

If there is no sequence {t,} of points in T, such that limn+oo t, = 00 and

4 t n ) = -1, liminf ,--too

z(tn)

then, since limt+oo v ( t ) = 0, it would follow that

v(t) = 0. t--too, t € f 1 + v ( t ) / z ( t ) lim

Since for t E

9, we have

it would follow that (15) and hence (12) holds for all sufficiently large t. It remains to prove that (16) does not hold for any sequence in 'If, tending to 00. We do this by contradiction, so assume there is such a sequence { t n } with

we get a contradiction to (17). Therefore it follows that (12) holds and this implies that vA(t)

5 -q2(t)

- S[ v](t).

Consequently, it follows that La [x]= 0 is nonoscillatory.

0

Remark 2.1. Conditions (9) and (10) can be replaced by one condition: There exists E > 0 such that

503

for large t (where X ( t ) denotes the characteristic function of

$1).

Next we consider the complementary case, namely that

L"

-As T;s)

< 00, r ( t ) > 0,

holds.

Theorem 2.4. Assume r ( t ) > 0 and

lmql(s)

Jam

l/r(s) As

< co,and that

(J" L A T ) ~ A S > ~ * I J ~(1" (S) LAr)'As>O u(s)

u(s)

r(T)

Jt"

As $ 0 for large t). T h e n if L l [ x ]= 0 is (with 4 1 (s) (J" nonoscillatory, it follows that L ~ [ x=] 0 is nonoscillatory. Proof. First note that by Lemma 1.4, the transformation x = hu with h(t) = Jt" l / r ( s ) As transforms Li[x]= 0 into

(R(t)uA)A + Qi(t)u" = 0,

i = 1,2,

(18)

where

and

Since

we get Jam l/R(s) As = 00, and so we may apply Theorem 2.1 to equations (18). The statement now follows from the fact that oscillatory properties of transformed equations are preserved. 0 Similarly, applying Lemma 1.4 and Theorem 2.3 we obtain the following. Theorem 2.5. Assume that r ( t ) > 0 and

{

qr := t E T : p ( t ) > 0 and

Jam

qz(s)

l/r(s) As

(1" U(8)

T(7)

< 00. Let

Ar)'As


504

and let xT denote the characteristic function of there exists m > 0 such that

'kT. Further

suppose that

and

for large t . Then if L I [XI = 0 is nonoscillatoy, it follows that nonoscillatory.

L2

[XI

= 0 is

q"0

:=

Next we give an application of the Hille-Wintner theorem. Example 2.1. (Hille-Nehari type criteria) Let { 1,q, q2 . . . }, q > 1. Then the q-difference equation xAA +p(t)xU= 0

T

=

(19)

is nonoscillatory if rm

1

for large t , and is oscillatory if

for large t , and some y. To verify this we use the fact that the Euler-Cauchy type q-difference equation

is oscillatory if and only if y > l/(& + 1)2, see [3]. Since the Euler-Cauchy type equation (22) is nonoscillatory for y = then, using (20), we have 9+1)

505

for large t. It follows from Theorem 2.1 that (19) is nonoscillatory. Conversely, if (19) is nonoscillatory and (21) holds for some y > then it would follow from Theorem 2.1 that (22) is also nonoscillatory, which is a contradiction. Hence, (19) is oscillatory if (21) holds. Note that l/(&+l)2 is different from 1/4 which is the well-known critical constant from the continuous and the discrete cases. Also recall that in lo it was shown (even for the half-linear case) that 1/4 is the critical constant in Hille-Nehari type criteria (i.e., involv-

[&ill2,

ing the expression (J: l / r ( s ) As) J,"p(s) As) for the dynamic equation ( r ( t ) y A ) A+p(t)y' = o for all time scales satisfying

If r ( t ) = 1, then this condition reduces to p ( t ) / t -+ 0 as t -+ 0, and in the case when T = q"0 this condition is not satisfied since p ( t ) / t = q - 1. This justifies somehow the presence of the additional condition (23), since from the above we know that the critical constant 1/4 is not universal for all time scales. Another application concerns the oscillatory behavior of an equation which is compared with that of a conditionally oscillatory equation whose oscillation constant is known. We recall that the equation

r ( t ) > 0 , is conditionally oscillatory if there exists a constant 0 < XO < co such that (24) is oscillatory for X > XO and nonoscillatory for X < XO. The value XO is called the oscillation constant of (24). Since this constant depends on the coefficients of the equation, we often speak about the oscillation constant of the function q with respect to r. If the equation is oscillatory (resp. nonoscillatory) for every X > 0, then this equation is said to be strongly oscillatory (resp. strongly nonoscillatory). Using Theorem 2.1 it is not difficult to see that the following statement holds. Theorem 2.6. Let q1 and q 2 be nonnegative functions. Further assume l/r(s) As = 00. Denote by XI (0 < XI < co) the oscillation constant of q1 with respect to r . If A and B are defined by

saw

A = lim t+w SUP

(lm As/ lm As) q2(s)

41 (s)

506

and

then the equation L ~ [ x=]0 will be oscillatory if B if A < X i . Since A = B = 1 if

and nonoscillatory

~ ( t3 )q 2 ( t ) , both bounds are evidently sharp. If

41(t) =

JF

> XI

1/r(t)

(s,“ l l r ( s ) As) ( s y 1/r(s)As) ’

(1:

-1

then ql(s) As = l/r(s) As) and the above theorem reduces t o the Hille-Nehari type criteria provided the oscillation constant is known. In a similar manner we may apply Theorem 2.4 for the case when Jta &As < 00. We leave the details to the interested reader.

References 1. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001. 2. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Editors, Birkhkuser, Boston, 2003. 3. M. Bohner, M. Unal, Kneser’s theorem in q-calculus, J . Phys. A : Math. Gen. 38 (2005), 6729-6739. 4. L.Erbe, Oscillation criteria for second order linear equations on a time scale, Canad. Appl. Math. Quart., 9 (ZOOl),1-31. 5. L. Erbe and A. Peterson, Comparison Theorems of Hille-Wintner Type for Dynamic Equations on Time Scales, Proc. Amer. Math. SOC.,to appear. 6. L. Erbe, A. Peterson, and P. RehBk, Comparison Theorems for Linear Dynamic Equations on Time Scales, J. Math. Anal. Appl., 275 (2002),418-438. 7. Stefan Hilger, Analysis on measure chains - A unified approach to continuous and discrete calculus, Results Math., 18 (1990),18-56. 8. Einar Hille, Nonoscillation theorems, Trans. Amer. Math. SOC.,64 (1948), 234-252. 9. B. Kaymakqalan, V. Lakshmikantham and S. Sivasundaram, Dynamical Systems on Measure Chains, Kluwer Academic Publishers, Boston, 1996. 10. P. RehBk, Half-linear dynamic equations on time scales: IVP and oscillatory properties, J. Nonlinear h n c t . Anal. Appl., 7, (2002), 361-404. 11. P. RehBk, Function sequence technique for half-linear dynamic equations on time scales, Panamer. Math. J., to appear. 12. Choy-Tak Taam, Nonoscillatory differential equations, Duke Math. J., 19 (1952),493-497. 13. A. Wintner, On the comparison theorem of Kneser-Hille, Math. Scand. 5 (1957),255-260.

507

NONOSCILLATIONS IN ODD ORDER DIFFERENCE SYSTEMS OF MIXED TYPE

SANDRA PINELAS* Departamento de Matem6tica Universidade dos ACores R. da M i e de Deus Ponta Delgada Portugal

The aim of this note is to discuss the existence of nonoscillations for the difference system of mixed type

e

nz).(

=

Cpiz.(

m

- i)

+CQ

i=l

~ .(Z

+j ) ,

n

= 0,i , 2 , ...,

j=1

in terms of some matrix measures envolving the matrices P; and Q j (i = 1, ...,e , j = 1, ..., m ) .Those measures are formulated in basis upon arbitrary logarithmic norms of matrices.

1. Introduction The main goal of 1 is work is t o study the existence of nonoscillations in the regard of the difference system

e m AX(TI) = C P i z ( n - i) + C Q j z ( n j ) ,

+

i=l

where 2 ( n )E Rd, Ax ( n )= z ( n C, m E

W, and for i

n = 0,1,2, ...,

(1)

j=1

+ 1)-z (n)is the usual difference operator,

= 1,...,C and j = 1,..., m the

Pi and the

Qj

are given

d x d real matrices. By a solution of this equation we mean any sequence z ( n ), of points in Rd, with n = 4, ..., 0,1, ..., which satisfy (1). * Suported by DRCT-Direceso Regional da C i h c i a e Tecnologia (Azores, Portugal).

508

This system is introduced in

'.

In this paper the authors show that

the existence of oscillatory or non oscillatory solutions of the system (1) determines an identical behavior to the differential system with piecewise constant arguments

2 ( t )=

e

m

i=l

j=1

C Pix ([t- 4)+ C Q j x ([t+ j]),

where for t E IR, z ( t ) E IRd and by [.] is meant the greatest integer function (see also

).

For given initial values z-e,

...,20,...,zm-l, denoting by I the d

x d

identity matrix, the existence and uniqueness of solution of system (l),is guaranteed by the conditions

{ (see

and

det ( I - Q I )# 0 , if m = 1, det Qm # 0, if m 2 2,

).

There are in the literature different ways of classifying a solution of a difference system like (1) as oscillatory, in basis upon the following terminology.

A sequence y ( n )is said that satisfies frequently o r persistently a given condition, (C) , whenever for every l E

N there exists a n > l such that y (n)

verifies ( C ). On the contrary, if there exists a l E

N such that y ( n ) verifies

> l , y ( n )is said to satisfy (C) eventually or ultimately. Therefore we will say that a sequence y (n)= [y1 ( n ), ...,yd (.)IT, is said

(C) .for every n

oscillatory componentwise if each sequence Yk (n)( k = 1,...,d ) is frequently nonnegative and frequently nonpositive. If for some k E { 1,..., d } the real sequence xk ( n )is either eventually postive or eventually negative, y ( n )is said a nonoscillatory (componentwise) sequence.

509

A sequence y (n)= [yl (n),...,Y d (n)lT, is said weakly oscillatory if there exists a k E (1, ...,d } such that the real sequence yk (n)is frequently nonnegative and frequently nonpositive. If for each k E {1,...,d } the real sequence yk ( n )is either eventually postive or eventually negative, y ( n ) is said a weakly nonoscillato y sequence. Another definition of oscillatory sequence regards a given family K of convex cones K verifying v . y

c Rd such that for each K

> 0 for every y

E

IC there exists a vector

E Rd

E K , in manner that U K E K K = E d . Then a

sequence y ( n )is said K-oscillatory if for every K E K, y (n)4 K frequently.

If there exists a K E Ksuch that y ( n ) E K eventually, then y ( n ) is said

K -nonoscillatory. All these ways of defining an oscillatory or nonoscillatory sequence coincide in the real case that is, for d = 1. Whenever all solutions of (1) or are oscillatory componentwise (respectively weakly oscillatory or K-oscillatory) we will say that (1)is a system os-

cillatory componentwise (respectively weakly oscillatory or K -oscillatory). Otherwise it will be said a system nonoscillatory componentwise (respectively weakly nonoscillatory or IC-nonoscillatory) . However, with respect t o (1) is possible t o prove that all these three concepts are equivalent between them and also equivalent t o say that the characteristic equation

P

(A - 1)I -

9

i= 1

has no positive real roots (see

AjQj

A-iPi -

and

j=1

).

510

Therefore (1) is nonoscillatory if and only if, with

M (A) =

e

m

i=l

j=1

C APiPi + C AjQj,

one has det ((A - 1)I - M (A)) = 0, for some real A

(3)

> 0.

We recall that for a given induced norm,

11. 1 1 , in Md (R) , we can associate

t o any matrix C E Md (R) , the derivative

usually called the logarithmic norm of C. Any logarithmic norm p has the following elementary properties:

(a1

(C E Md (R)) , (i) Ref7( C ) c [-P (-C> ,c1 (ii) CL(C1) - p(-Cz) 6 P(C1 + CZ)6 P(C1) + CL(C2) (C1,CZE Md (a)), (iii) p(rC) = yp(C), for every y 2 0 (C E Md (R)) , where by o(C) we mean the spectral set of C and Rea(C) = {Rez : z E a ( C ) }(see

and

4).

Moreover, in the lemma below we express

two relevant relations between the determinant of a matrix C and the value P (C).

Lemma 1.1. For any logarithmic n o r m , p, and any C E Md (R):

(iv)p (4) 5 0 + det (C) 2 0; (w) if d is odd, p ( C ) L 0 + det (C) 5 0. Proof. (iv) Let u ( C ) = { z ~ , z g,...,a}. By (i), if - p ( - C )

2 0 then

Rezk 2 0, for every k = 1,...,d. Since det C = z1zz ...z d , then det C 2 0. ( v ) By (iv), we have that 0

5 det (-C)

d is odd we obtain then det (C) 5 0.

= (-l)ddet (C) . Therefore, if 0

511

For a given finite sequence of real matrices, C1, ...,C,, and with respect to a logarithmic norm, p, we define the following matrix measures wich will have some relevance in the sequel:

These measures give rise to the matrix measures Q and ,B already introduced in

as follows:

Whenever refered, the values a (-Ck), b (-ck), cx (-ck) and implicitly assume the values above for the sequence -C1,

P (-ck)are

..., -C,.

Notice that by property (ii) above, the measures a and b are related with the corresponding logarithmic norm p in the following way: k

v

i=l

i=k

for every k = 1, ..., v.

With respect t o the measures

(Y

and ,B the following lemma holds (see

9. Lemma 1.2. Let C1,...,C,, be a finite sequence of d x d real matrices. (a) If y1 2 ... 2 y, 2 0 is a nonincreasing finite sequence of nonnegative real numbers then

(b) If 0 i y1 5 ... 5 yv is a nondecreasing finite sequence of nonnegative

512

real numbers then

In view of the examples which will be given in the sections below we recall the following well known logarithmic norms of a matrix C = [ C j k ] E Md

(R):

2. Nonoscillatory odd order systems

In this section we will assume that the order, d , of the system (1) is an odd integer. Under this assumption, logarithmic norms can play some role, through the Lemma 1.1,on the existence of nonoscillatory solutions of (1). In fact, considering for X

> 0, the matrix K (A) = (A - 1)I - M ( A ) ,

notice that, if for some pair of positive numbers, XI,

then by the Lemma 1.1it is det ( K (XI))

X2,

one has

5 0 and det ( K (Xz)) 2 0. There-

fore, by continuity, there will exist a positive

A0

such that det CK (A,)) = 0

and consequently (1) is nonoscillatory. In order to conclude that (1) is a nonoscillatory system, this argument will be used in the sequel.

Theorem 2.1. Let i o E { 1,.., !} and j o E { 1,..., m} be such that b(Pio)# 0 , b(-Qjo)

# 0 and b(Pi) = b ( - Q j )

= 0 for every i

> i o and

j

> j o . The

513

system (1) is nonoscillatory if b(Pio)< 0 and either jo

> 1 and

b(-Qjo)

< 0,

(8)

< -1.

(9)

or j o = 1 and b(-Q1)

Proof. Taking into account that p (*I) = f l , by properties (ii) and (iii) of the logarithmic norms, one has for every real A,

and

By the inequalities ( 6 ) and ( 5 ) , we have for 0 < X

<1

In the same way, from (11) we obtain for X 2 1 p ( K (A))

5 X - 1+

e

j0

i= 1

j=1

C A - i ~ (-Pi) + C XjP

(-Qj)

.

(13)

Notice that if io = C then

On the other hand, if io

< l , we have P (Pi) = 0 for every i > io,

taking into account that P (Pi)= b (Pi)- b (Pi+l)and b (Pi)= 0 for every

i > io. Moreover, in such case,

514

By the same arguments, we conclude also that

p (-Qj)

>jo,

= 0 , for every j

and that, by (8) and (9), either a) j o

> 1 and p ( - Q j , )

<0

= b(-Qj,)

or b) j o = 1 and

@(-&I)

< -1.

= b(-Q1)

Having p(Pio)< 0, the right-hand member of (12) is such that

1- X

io

m

i=l

j=1

+ cX-ip(P,) +x X j a ( Q j ) +

and so for some real

A1

> 0 one has p ( - K

as X

-03,

+ O+,

< 0, for every 0 < X 5 X I .

(A))

If a) holds, the right-hand member of (13) is such that

+

Under b) this expression reduces t o X (1 8 , which as above goes to

-03,

as X

conclude the existence of a real X2

+oo. i

A2

(-91))- 1+'&

Apia (-Pi),

Therefore in both situations we

> 0 such that

p ( K ( X ) ) < 0 for every

X2.

Hence by (7) the system (1) is nonoscillatory.

0

We observe that in basis upon the inequalities (lo), ( 5 ) and (6) we have also for X

> 1, p ( - K (A))

5 1- x +

e

C i= 1

m

+ C X ~ (P~

(Pi)

Moreover from ( l l ) , (5) and (6) one has analogously for 0 < X

i=l

.

j )

j=1

j=1

< 1,

515

Therefore similarly to the Theorem 2.1, by use of the same arguments, is possible to state the following theorem. Theorem2.2. L e t i o E {l,..,l} a n d j o E {l,...,m} besuch thatb(-Pi,)

0, b ( Q j o )

# 0 and b(-Pi)

= b ( Q j ) = 0 for every i

b(-Pio) < 0 and

> io and j > j o . I f

< 0,

b(Qjo)

#

(14)

then (1) is nonoscillatory. The examples below illustrate these results. Example 2.1. Let the system ( l ) , with d = 3, l = m = 2,

PI=

[

0 1 0 -4-21 -1 2 -3

-2

, Pz=

[

-5 1 0 1 -1 1 1 -2 0 -2

and

4 1-3

&I=

[ii i ] ,

Q2=

[,'

2 -10

21.

With respect to the logarithmic norm P I , we have

So in the Theorem 2.1, one has io = j o = 1. The remaining conditions are also satisfied and the corresponding system (1) is then nonoscillatory. Example 2.2. Let the system (l),with d = m = 3 and l = 2, in manner

that p1=

[ul ;

;]

6 -33 5 2-3 ; ] , P ~ = [;I;

516

and Q3=

[

-4 0 0 0 -3 1 1 . -1 0 -4

With respect to the logarithmic norm p-, we have

So, in the Theorem 2.2 one has io

=

1 and j o

=

3. As a consequence the

corresponding system (1) is nonoscillatory.

As is described in the following corollaries, the Theorems 2.1 and 2.2 can be simplified in certain circunstances, by using only the logaritmic norm p.

Corollary 2.1. Let vo E { 1,.., l } and

TO

E { 1,...,m} be such that p (P v o )#

0 , p(-QTo) # 0 and p(Pi) = p ( - Q j ) = 0 f o r every i

system (1) is nonoscillatory if p(Pvo)< 0 and either

or T~

and

=

1 and p(-Q1)

< -1.

> vo and j > TO. The

517

This means that the values

io 2

VO

and j o 2

TO.

i0,jo defined

in the Theorem 2.1 are such that

On the other hand, as

e

c

b(Pio)5

1-1(Pk)= P(Pi0) 5 0 ,

k=io

one has necessarily b(Pio)< 0. Analogously, since m

b(-Qjo)

5

CP

= A-Qj0),

(-Qk)

k=jo

if j ,

> 1, one has p ( - Q j o ) < 0 and if j o = 1 then b ( - Q 1 ) 5 p(-Q1) < -1.

Thus all the assumptions of the Theorem 2.1 are fulfilled and so the system (1) is nonoscillatory.

Corollary 2.2. Let vo E (1, ..,e } and

# 0, and j > TO. If

p(-Pv0)

P(QTO)

0

TO

# 0 and p ( - P i ) =

E (1, ..., m }

be such that

p ( Q j ) = 0 f o r every i

> v0

then (1) is nonoscillatory.

Remark 2.1. We notice that the conditions of these corollaries are not fulfilled by the Examples 2.1 and 2.2. The following example illustrates the later corollary.

Example 2.3. Let the system (1) with d = 3, C,m E N,such that -5

0

With respect t o the logarithmic norm 1-11, we have

1

518

and by the Corollary 2.2 the corresponding system (1) is nonoscillatory. The results above can obviously be applied t o the scalar case of ( l ) ,

where the pi and the

qj

e

m

i= 1

j=1

are real numbers, for i = 1, ..., m, and j = 1, ..., l ,

with both pe and qm different of zero. Noticing that for every logarithmic norm, p, and any real number, c, one has p ( c ) = c , the corollaries before, as well as the Theorems 2.1 and

2.2, give, respectively, for (17) the following corollaries. Corollary 2.3. I f pe

< 0 and either m

> 1 and

qm

> 0,

or m = l andql >1,

then the system (17) is nonoscillatory. Corollary 2.4. If pe

> 0 and

qm

< 0,

then the system (1 7) is nonoscillatory. Example 2.4. By the Corollary 2.3 the equation

A X ,= is nonoscillatory.

- 2,-2

+ 22,+1

519

Example 2.5. By t h e Corollary 2.4 the equation Axn

= x,-1

+

~ n - 2- X,+I

- xn+2

is obviously nonoscillatory.

References 1. I. Gyori, G. Ladas and L. Pakula, Conditions for oscillations of difference equations with piecewise constant arguments, S I A M J. Math. Anal. 22 (1991) 769-773. 2. I. Gyori and G. Ladas, ”Oscillation Theory of Delay Differential Equations,” Oxford Univ. Press, 1991. 3. W. A. Coppel, “Stability and Asymptotic Behavior of Differential Equations” , Heath, Boston, 1965. 4. C. A. Desoer and M. Vidyasagar, “Feedback Systems: Input-Output Properties”, Ac. Press, 1975. 5. J. Kirchner and U. Stroinsky, Explicit oscillation criteria for systems of neutral equations with distributed delay, Differential Equations and Dynam. Systems 3 (1995) 101-120. 6. J. M. Ferreira and S. Pinelas, Oscillatory mixed difference systems, Applicable Analysis (submitted).

520

SUPERINTEGRABILITY AND QUASI-EXACT SOLVABILITY. THE ANISOTROPIC OSCILLATOR

G. S. POGOSYAN CUCEI, Universidad de Guadalajara, Jalisco,

Departamento de Matematicas,

Mexico In this note we show that two-dimensional superintegrable model - nonisotropic oscillator with the ratio of frequency 2 : 1 generate both exactly and quasi-exactly solvable problems in one-dimensional quantum mechanics via separation of variables in Cartesian and parabolic coordinates. The work done in collaboration with E.Kalnins and W.Miller. Jr.

1. Introduction Almost twenty years ago a new class of spectral problems was discovered in quantum mechanics. The main peculiarity of such systems is the fact that only finite part of the eigenvalues and eigenfunction can be found algebraically. These phenomena called quasi-exact solvability (this term first time introduced by Turbiner and Ushveridze in [l])and have been intensively discussed in literature in the late 1980’s. The crucial example that stimulated these investigation of quasi-exactly solvable systems is the Hamiltonian

K=

1 d2 --C-+ 2 ax;

V(Xl,XZ,

..., X N )

i=l

with the anharmonic potential = -w 2 z6

+

+

2pWzz4 ( 2 p -~~ 1 x 2 , (1.2) 2 where w, p, and E are the constants. As it was notice by many authors [2, 31, the solutions of this systems for special values of constant E = w ( 2 N 2 f 1 / 2 ) , ( N = 0 , 1 , 2...) split into polynomial and nonpolynomial sectors (each of these sectors are non complete). The fact of quasi-exact solvabilily was explain in [4] in term of algebra d ( 2 , R). It means following: the one-dimensional Hamiltonian (1.1) after suitable changes of variable v ( X )

+

521 z = <(z) and ',gauge transformation" H = e-"(")Xe"(") can be written in

the form

C

H =

+

CabJaJb

a,b=O,f

C

CaJa

(1.3)

a=O,f

where the first-order differential operators { J*, Jo} span to the commutation relations for algebra s l ( 2 , R ) [4, 51. The main idea of this note is to show that quasi-exact solvability of the one-dimensional system (1.2) is directly related with multiseparability of two-dimensional superintegrable system, namely - anisotropic oscillator. Let us remind that the N-dimensional nonrelativistic quantum system described by the Hamiltonian (1.1) is called superintegrable (or maximally integrable) if there exist 2 N - 1 [6, 71 linearly independent and global integrals of motions &, commuting with the Hamiltonian (1.1)

[Lk, ?']!I

= 0,

k = 1,2, ...2 N - 1,

(1.4)

but not necessarily with each other. 2. Anisotropic oscillator

The Schrodinger equation XQ = EQ with the potential of anisotropic oscillator 1 V(Z, y) = -w2(422 y2). (2.5) 2 admit separation of variables in two systems: C a r t e s i a n a n d parabolic coordinates. Cartesian wave function have following form

+

where nl, 722 = 0,1,2, ..., the energy spectrum given by

E=w and degree of degeneracy for fixed n is odd. In parabolic coordinates

,

n=2n1+n2=0,l72

,... (2.7)

(s+ l ) ,for even n and ( y), for

522

the Schrodinger equation is

The substitution Q ( [ , q ) = X ( J ) Y ( v )separates equation (2.9) into two ordinary differential equations: d2 X

-+ ( 2 E p - ,266)

x = -AX,

(2.10)

-+ (2E172 - ,2176) Y = +AY

(2.11)

dt2

d2Y

dV2

where X - is parabolic separation constant. It is easy to see, that equations (2.10) and (2.11) are transformed into one another by change 5 t)iv. They can be written in unique form

[

d2

+ (w2p6 - 2 E p 7

--

dP2

1

Z ( p ;E , A) = AZ(p; E , A),

(2.12)

where at p E (--00, -00) we have the equation for the function X(<)= Z ( p ) , and at p E [O,i-00) - for Y ( q ) = Z ( i p ) . Therefore the ”physical” regions in the complex p domain is just two lines I m p = 0 R e p = 0, I m p > 0. Consider equation (2.12). To solve it we make the substitution

Z ( p ; E , A ) = exp

( - 3 4 )

W(p;E,A),

(2.13)

and obtain the differential equation d 2-W dP2

dW 2wp3dP

+ [A + (2E - 3w)p2] W = 0.

(2.14)

The solution of the last equation split into two classes (2.15) s=o 00

W ( - ) ( pE; , A) = p

C azs+l(E,A) P”,

(2.16)

S=O

where the first is even, and the second is odd under the inversion p -p. i The substitution of expansions (2.15) and (2.16) in equation (2.12) leads to the following three-terms recurrence relations for the coefficients az8 and

523

a29+1 A(+)

(s

+ l)(s + 1/2)a2s+2 + 4 a 2 . 5 1 + -[E - w(2s - 1/2)]azs_2 = 0, 2

(2.17)

A(-) 4a2s+1 1 (2.18) -[E - w(2s 1/2)]a2,-1 = 0, 2 with the intitial conditions a--2 = a-1 = 0. To understand the behavior of series (2.15) and (2.16) we use the following method [8]. Consider first the recurrence relation (2.17). For large (S

l)(S

-t3/2)a2s+3

+

+

S

a2s+2 azS --+--a2s For s-l

<< 1

--

a2+-2

a2s+2

A(+)

a2s

4

m - 2

C1 co + + ...)

a28

a2s

a2s-2

c1 + 0 Js-i = co + &

(2.19)

(2.20)

&

and also - - c o + A

w s = 0.

(i).

(2.21)

After substituting eqs. (2.20) and (2.21) in (2.19) we have (2.22) and therefore the function W ( + ) ( p )as a solution of equation (2.14) have two asymptotic regimes

Now it is easy to see that for p2 > 0 [the case of equation (2.10)]

and for p2 < 0 [the equation (2.11)]

(2.25)

524

Thus, the function W ( + ) ( p )cannot converge simultaneously at p t f c m and p + icm and therefore the series (2.15) should be truncated. More detailed mathematical analysis of the three-term recurrence relations of type (2.18) and (2.19) as been done in [9]. The analysis of the second recurrence relation (2.18) leads to the analogous result. The condition for series (2.15) and (2.16) to be truncated results in the energy spectrum (2.7). The three-term recurrence relations (2.17) and (2.18) represent the system of N 1 homogeneous algebraic equations for coefficients ~2~ and

+

n = 2N

n=2N+1.

The requirement of the existence of a non-trivial solution leads to a vanishing of the determinants

DjV+’(A)=

=0

(2.28)

DL)(A) =

=0

(2.29)

and

+

Equations (2.28) and (2.29) determine all ( N 1) eigenvalues of parabolic A!,+,, where the index k are separation constant A(+) z Ak2N and A(-) 2q and 2q + 1 correspondingly, and q = 0,1,2, .., N .

525

In the result the physical admissible solutions of eq. (2.14) are

Mlc,2&(p)

= w(+)(p; E , A) =

c N

a2sp,

n = 2N

(2.30)

s=o

N

Mlci&yl(p) W ( - ) ( pE; , A) = p

C

~ 2 ~ p2’, + 1

n = 2 N + 1. (2.31)

s=o

and the corresponding solutions of eq. (2.12) are

z,”x4 = exp (-iP4)

Mki&(p),

W

z,”&yl(P> = exp (-4P4)

Mlc;;y,(P)

n = 2N, n = 2N

+ 1.

(2.32) (2.33)

Note, that the even values of quantum number n corresponds to the even under the transformation p -+ - p solutions of eq. (2.12), but for odd values of n - odd ones. Now it is possible to construct the complete solutions of our problem. For p = 5 the functions (2.32) describe the solutions of eq. (2.10) and for p = iq the solutions of eq. (2.11). In relation of formulas (2.32) and (2.33) the parabolic basis split into two bases

IE@ ( I ,77)

=

c?g)2,”;

( I )2,”; (4,

(2.34)

n n=2N, D = N + l = - + l 2 @ );

( I ,77)

=C Q

z,”;yl( I )z;;:l(i77),

(2.35)

n+l n=2N+1, D = N + l = 2 . Here D - are the number of state at the fixed n, and the normalization constants C$ determined by the conditions Po0

Po0

3. Conclusion We have shown that the Schrodinger equation for the potential of anisotropic oscillator can be investigated in two different ways, namely in Cartesian and parabolic coordinates. In the first one the separation of variables lead to two independent exactly-solvable equations with the complete wavefunction in form (2.6). On the second way the separation procedure

526

reduce to one-parametric (parabolic separation constant) Schrodinger type differential equation (2.12), living in the complex plane. The requirement of convergence for solutions of eq. (2.12) at the singular points p = f m and p = im leads to only polynomial solutions (2.32) with restriction for energy spectrum E in form (2.7) and at the fixed n give the spectrum of separation constant as root of nth-degree algebraic equation. At the other hand side, the substitution of the formula E = En = w ( n 3/2) in (2.12) bring us to equation

+

(3.37)

+

which for real p and /3 = 0, n = 2 N and n = 2 N 1 completely coincides with the one-dimensional spectral problem (1.2). Now it is easy to understand the phenomena of quasi-exactly solvable systems. This problem are much more easily understood in term of multidimensional Schrodinger operators and multiseparability where one family of separable solutions is exactly solvable and another is not, rather than as separate one-dimensional Schrodinger eigenvalue problems. References 1. A.V.Turbiner and A.G.Ushveridze. Preprint ITEP-169 Moscow (1986). 2. V.Singh, S.N.Biswas and K.Dutta. Phys. Rev. D18 (1978)1901-1908. 3. A.G.Ushveridze, Quasi-Exactly solvable models in quantum mechanics Institute of Physics, (Bristol 1993). 4. A..Turbiner. Comm. Math. Phys. 118 (1988) 467-474. 5. M.A.Shifman. Int. J.Mod. Phys. A126 (1989) 2897-2952. 6. S.R.Wojciechowski. Phys. Lett. A 95 (1983) 279. 7. N.W.Evans. Phys.Rev. A 41 (1990) 5666. 8. C.A.Coulson and P.D.Robertson. Proc. Phys. soc. London 71 (1958) 815827. 9. W.Miller Jr., E.G.Kalnins and G.S.Pogosyan. J. Math. Phys. 47, (2006) 033502-1 - 033502-30; ArXiv: math-ph/0412035.

527

ON THE ERGODIC AND SPECIAL PROPERTIES OF GENERALIZED BOOLE TRANSFORMATIONS

ANATOLIY K. PRYKARPATSKY*AND JACEK FELDMAN The AGH University of Science and Technology Department of Applied Mathematics Krakow 30059 Poland

The invariant ergodic measures for generalized Boole type transformatins are studied making use of the invariant quasi-measure approach, based on some special solutions to the Frobenius-Perron operator.

Key words and phrases. generalized Boole transformations, ergodic dynamical systems, invariant quasi-measures, Frobenius-Perron operator

2000 Subject Classification: Primary 34A30, 34B05 Secondary 34B15

1. Introduction We will consider the following generalized Boole transformation

c N

R3y

+ cp(y) := ay + a - pj E R, j=1 Y - bj -

-

where a and bj E R, j = 1 , N , are some real and a,pj E R+, j = 1,N, are positive parameters. It generalizes that classical Boole transformation [l]R 3 y -+ cp(y) := y - l / y E R, which appeared to be ergodic [2] with respect to the invariant standard infinite Lebesgue measure on R. In the case a = 1, a = 0, the similar ergodicity result was proved in [3, 4, 51 making use of the specially despised inner function notion. The related spectral properties were in part studied in [5]. In spite of these results the case a # 1 still persists to be challenging as the only relating result [6] concerns the following special case of (1): R 3 y -+cp(y) := ay - p/y E R for 0 < a < 1 and arbitrary ,8 E R+. The corresponding invariant measure *the first author was supported in part by a local agh grant

528

appeared to be finite absolutely continuous with respect to the Lebesgue measure on R and equal to dp(x) := d m d x 7r[x2(1--)+/3]'

where x E R. The ergodicity for the invariant measure (2) now can be easily proved. Remind here, concerning a general nonsingular mapping cp : R + R, the problem of constructing the measure preserving ergodic measures is analized [6, 71 by means of studying the spectral properties of the adjoint F'robenius-Perron operator T v p : Lz(R; R) + L2(R; R), where, by definition,

1 9 1 ,

:= y E R. Then if T v p = p, p E for any p E L2(R;R+) and J;'(y) L2(R; R+), then the expression dp(z) := p(x)dz, x E R, will be invariant, in general infinite, measure with respect to the mapping cp : R + R. Another way to finding a general algorithm for finding such an invariant measure was devised in [8,9],making use of the generating measure function method. Below we will study some another special cases of the generalized Boole transformation (l), for witch we deliver the corresponding invariant measures and prove the related ergodicity and spectral properties.

2. Invariant measures and ergodic transformations

We will start with analyzing the following Boole type surjective transformation

R3y

+ cp(y) := cuy + a - -E R Y-b

(4)

for any a , b E R and 2/3 := y2 E R+. The transformation (4) a t a = 1/2 and b = 2a E R will be measure preserving with respect to a measure like (2). Namely, the following lemma holds. Lemma 2.1. The Boole type mapping spect t o the measure

(4)

is measure preserving with re-

529

(5) where x E R and y2 := 2p E R+.

Proof. A proof follows easily from the fact that the function 7 p ( x ) := ~ [ (-x2a)2 y2]

+

(6)

satisfies for all x E R\{2a} the determining condition (3):

where, by definition, p ( y * ( x ) ) := x for any x E R. The relationship (7) is, evidently, equivalent to the next invariance condition

c

dP(Y*(X)) = dl.l(z) := I.L(dz)

f

for any infinitesimal subset dx

c R.D

The question about the ergodicity of the mapping (4) is solved here easily by the following theorem.

Theorem 2.1. The measure (6) is ergodic with respect to the transformation (4) at 0: = 1/2 and b = 2a E R as such one is equivalent to the canonical ergodic mapping R/nZ 3 s :-+ $(s) := 2s mod ( T Z ) E R / r Z with respect to the standard Lebesgue measure on R / T Z . Proof. P u t by definition R / T Z 3 s :-+ [ ( s ) = y E R, where [ ( s ) := yctgs

Then the transformation (4)at yields under the mapping (9)

Q

+ 2a,

(9)

= 1 / 2 , b = 2a E R and y2 := 2p E R+

y(cos2s - sin2s) + 2a - -tgs Y = + 2a 2 2sinscoss cos2s = y+ 2a = ytg2s + 2a := C(2s) szn2s

p(y) = p([(s)) = -ctgs 7

2

for any s E R/TZ. The result (10) means that the transformation (4)is conjugated [7, 51 with the transformation

R/xZ 3 s :-+ $ ( s ) = 2s

mod ( T Z) E R / r Z ,

(11)

530

that is the following diagram is commutative:

It is easy now t o check that the measure (5) under the conjugation (12) transforms into the standard normalized Lebesgue measure on R/ITZ:

-

1 sin2s. sin-2s d s = .rr-'ds, cos2s sin2s

IT

+

where s E R/ITZ.The infinitesimal measure r-lds on R/nZ 3 s as well as the infinitesimal measure (5) on R are normalized, being thus probabilistic. Now it is enough to make use of the fact that the measure .rr-lds on R/ITZ3 s on the interval [0,IT] N R/ITZis ergodic [6, 7 1 . ~ 0 3. Ergodic measures: an innert function approach

Assume that there exists an holomorphic in parameter w E C+ function pw E H2(@+;@), satisfying the following identity

T,Pw

= &(w)

(14)

valid for any w E C+ for some induced transformation @+ 3 w + @ ( w ) E @+. If we now take w := w E @+ being a fixed point of the mapping $ : @+ + @+, then an easy conclusion from 14 gives rise to the condition T,p, = pa, meaning that the expression

d / ~ ( z:= ) Imp,(z)dz,

(15)

is a searched invariant measure for the transformation 'p : R -+ R. There exists still no general rule of constructing such functions pw E H2(@+;@) analytic in w E C+ and related induced mappings @ : C+ -+ C+. Nevertheless, for solving this problem one can adapt some natural motivations related with the exact functional form of the determining F'robenius-Perron operator T, : &(R; R) + L2(R;R). To explain this, let us consider the following Boole type transformation:

531

Iw 3 p ( y )

:= a y + a

- -E R , Y-b

where a , b ER and /3 E R+. It is easy to observe that the Frobenius-Perron operator action on any pw E Hz(@+;C) can be represented as follows:

for all w E C+\{z}, z E one can take

Iw, and some parameter Ic

PW(Y+)(W - Y+) = Ic = Pw(Y-)@

E R.

As a result of (18)

Y-),

(19)

-

for all x E R and w E C+. Since the root functions y+and y- : R satisfy, by definition, the same equation

4Y*(%))

= 2,

-+ R

(20)

for all x E R, the following identity for all w E @+ easily infers from (20) owing to the general form of (16):

Whence, taking into account the expression (17), one gets that

532

for all x E R and w E @+.Therefore, the induced mapping 9 : @+ + @+ is exactly the transformation cp : @++ @+,extended naturally from the axis R on the complex plane @+. Let now LZI E @+is a fixed point of the induced mapping cp : @+ -+ @+, that is cp(0) = 0 E @+. Then from (23) one finds that T v p a = pa, or the corresponding invariant quasi-measure on R has the form kdx d p ( x ) := Imw-x for all x E R and some suitable parameter k E @. As ImpG E L,(R;R+) at any D E @+\R and some k E C, then the invariant quasi-measure (24) transforms into really an invariant measure. The obtained result can be formulated as the following theorem. Theorem 3.1. The quasi-measure (24) is invariant with respect to the transformation (16) for any a E R+\{l}; for a = 1 at the condition a # 0 , Imk # 0 , it is reduced upon the set R/7rZ, being equivalent to the standard Gauss measure.

Proof. Really, the searched infinitesimal quasi-measure d p ( x ) exist if there exist at least one fixed point of the equation cp(w) = w for w E @+.If a # 1, this equation is equivalent to that ( a - 1 ) d - w [ ( a - 1 ) b - a] - (ab f ,B) = 0,

(25)

always possessing a solution ij E @+, for which cp(0) = 0. At a = 1 the unique solution 0 = (ab+,B)/a E R exists only at a # 0, Imk # 0 , at which the quasi-measure (24) becomes degenerate, reducing to the standard Gauss measure [7, 31 on R/TZ.D 0 The Theorem 3.1 states only that the quasi-measure (24) is invariant with respect to the transformation (16), thereby the ergodicity if any still needs to be proved separately, on which we plan to stop in detail in another place. Below we will proceed to study the general case of the transformation (l),searching for a suitable invariant quasi-measure becoming a measure at some 0 E @+/R, k E @.

533

4. Invariant measures: the general case Consider the following equation

(26)

cp(Y) = 2,

where x,y E IR and the mapping cp : C+ + C+ is given by expression (1) at some still fixed integer N E Z+\{l}. The equation (26) one can rewrite down as

+

for all x,y E Iw and some functions yj : IR + IR, j = 1,N 1. The relationship (27) is naturally extended on the complex plane C+ as

n.

N+ 1 Q:

N

(w - Yj) = [cp(w) -

4

j=1

n(.,

- bj)

(28)

j=1

for any w E C+. Consider now the relationship (14), doing similarly are in Section 3:

where we have put, as before,

PW(Yj)(W - Yj) = k, for all j = 1,N evidently, that

+ 1, w

E C+, and some parameters

PW(Y) =

k w-y

(30)

k E C.This means

534

for any y E R and w E C+. Having substituted now the expression (28) into (29), one gets easily that

for all x E IR and any w E C+. Thus, the invariant quasi-measure for the discrete dynamical system (1) will be given by the same expression (24), where ij E C+ is some fixed point of the mapping cp : @+ 4 C+. This means that N

a~ + a

-

P. = C& w - bj

ij,

(33)

j=1

or, equivalently,

n(ij N

aij

- bj)

j=1

+a

n(ij N

N

j=1

n N

- bj) -

Pj j=1

n(ij N

(ij - b k )

k#j

=0

- b j ) , (34)

j=1

for some ij E C+. Assume now that a # 1; then it is easy to find that the algebraic equation (34) possesses exactly N 1 E iZ+ roots, which can be used for constructing the invariant quasi-measure (24). The case a = 1 gives rise to the condition

+

N

N

N

j=1

j=1

k#j

which always possesses roots if N 2 2 and a E R is arbitrary. Thereby, one can formulate the next characteristics theorem for the case N 2 2. Theorem 4.1. The expression (24) at some k E C determines, in general, the infinitesimal invariant quasi-measurefor the generalized Boole transformation (1) at all N 2 2 with arbitrary parameters a, b j E R and a , P j E R+, j = 1,N+1.

It is an important now task to separate from the obtained set of invariant quasi-measures (24) those, which are positive defined and ergodic with respect t o the transformation (1) at N 2 2. The positivity condition simply means that the determining equation (34) must possess at least one

535

pair of complex conjugated roots with a nontrivial imaginary part. Having analyzed, concerning this criteria, roots of equation (34) one can find that the following statement analogous to that proved in [6] holds. Theorem 4.2. The generalized Boole transformation (1) for any N 2 1 is necessarily ergodic with respect to the measure (24) at some ij E C+\R and k E C iff cx = 1 and a = 0. If cx = 1 and a # 0 , the transformation (1) is not ergodic being totally dissipative, that is the wandering set V(cp):= uW(cp) = R, where W(cp) c R are such subsets that all sets cp-”(W), n E Z,are disjoint. Proof. (sketch). It is easy to see that for N 2 2 at cx = 1 and a = 0 the determining algebraic equation (34) always possesses exactly N 1 E Z+real roots Uj E R, j = 1,N - 1. Therefore, the invariant quasimeasure expression (24) is degenerate for L Z ~€ R, j = 1,N - 1, giving rise to the conclusion that the corresponding invariant measure dp(z) = dz, z E R, is the standard Lebesgue measure on R. Its ergodicity with respect that transformation (1) then follows from the fact that the corresponding dissipative set D(cp) = 0 and the unique invariant set subalgebra I(cp) = 0

R.D

The statements similar to the above can be formulated for the mostly generalized Boole type transformation

R3y

4

p ( y ) := cxy+a+

s,

d’(S) E R, S-Y

where a E R, cx E R+ and a measure u on R has the compact support suppu c R, being such that the following natural conditions [3]

hold. Concerning the extension of the transformation (36) on the upper part C+ of the complex plane C in such a way that Imcp(w) 2 0 for all w E C+, the following representation

holds [ll,31, where a measure d a on R is closely related with the measure du.

536 The general properties of the mapping (38) were in part studied in [6] in the framework of the theory of inner functions. T h e corresponding to (36) invariant measures and their ergodic properties can be also treated effectively making use of the analytical and spectral properties of the associated Frobenius-Perron transfer operator (3). These aspects of the problem being as important as interesting are planned t o be studied elsewhere.

Acknowledgements T h e authors are cordially appreciated t o professors D. Blackmore (NJ,USA), F. Pszytycki (Warsow) and T. Downarowicz (Wroclaw) for valuable discussions of the ergodic measure properties related with generalized B o d e transformations.

References 1. Boole G. O n the comparison of transcendents with certain applications to the theory of definite inegrals. Philos Transaction, Royal SOC.London, 1957,

v.147, p.745-803 2. Adler R. and Weiss B. The ergodic, infinite measure preserving transformation of Bool. Israel Journal of Math., 1973, 16, p.263-278 3. Aaronson J. Ergodic theory for inner functions of the upper half plane. Ann. Inst. H. Poincare, 1978, BXIV, p.233-253. 4. Aaronson J. A remark o n this existence of inner functions. Journ. LMS, 1981, 23, p.469-474 5. Aaronson J. The eigenvalues of nonsingular transformations. Israel Journal of Math., 1983, 45, p.297-312 6. Aaronson J. An introduction to infinite ergodic theory. AMS, v.50, 1997 7. Katok A. and Hasselblatt Introduction to the Modern Theory of dynamical systems. Cmbridge Univ. Press, 1999 8. Prykarpatsky A.K. On invariant measure structure of a class of ergodic discrete dynamical systems. Journal of Nonlinear Oscillations, 2000, v.3, N1, p.78-83 9. Prykarpatsky A.K. and Brzychczy S. O n invariant measure structure of a class of ergodic discrete dynamical systems. Proceedings of the International Conference SCAN 2000/Interval2000, September 19-22, Karlsruhe, Germany 10. Pollycott M. and Yari M. Dynamical systems and ergodic theory. London Math. Society, Cambridge University Press, Student Texts, v.40, 1998 11. Ablowitz M.J. and Fokas A.S. Complex variables: introduction and applications. Cambridge University Press, 1997

537

FOCK REPRESENTATIONS FOR A QUADRATIC COMMUTATION RELATION

c. CORREIA R A M O S ~N., MART INS^ AND J. SOUSA R A M O S ~ Department of Mathematics, University of Evora, Rua Rom6o Ramalho, 59, 7000-671 Evora, Portugal E-mail: ccrOuevom.pt Department of Mathematics, Instituto Superior Te'cnaco, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: [email protected] E-mail: [email protected] In this paper we study representations associated with the positive solution of a certain difference equation, with initial condition 0. These are representations that arise as Fock representations associated to a quadratic commutation relation. We define a space of parameters for these Fock representations and we determine the regions in these parameter space where the representations are bounded.

2000 MCS Primary 37315; Secondary 37305,46LO5,81R15 Keywords: Quadratic maps, Hilbert space, irreducible representation, Fock representation, commutation relation 1. Introduction

In this paper we characterize the Fock representations of the *-algebra generated by the operators X ,X* satisfying the algebraic relation

xx*= a(X*X)2+ px*x+ y l

(1)

where I is the identity operator and a , p, y are real numbers. To give a representation for the relation (1) we mean precisely to give an operator X on an Hilbert space that satisfies that relation. The study of representations is done up to unitary equivalence. We say that two operators X,Y are unitary equivalent if there is a unitary operator U with X = UYU*.As it is shown in Ostrovskyi-Samoilenko2,the unitary classes of representations associated to the relation (1) are closely related with the dynamical system

538

+ +

(& ,F ) , where F ( z ) = a x 2 ,Bz y. This kind of questions can be found, for example, in Ostrovskyi-Samoilenko1j2and Popovych-Maistrenko3. The Fock representation is a representation in Z2(N) and is directly associated with the positive solution of the difference equation xrs+l = F(z,), with initial condition xo = 0. When the solution is periodic of period n we can interpret the Fock representation as a finite dimensional representation in en, see the Theorem below. Any aperiodic or eventually periodic solution gives always an infinite dimensional representation. If the solution is bounded we obtain a bounded representation. Theorem 1.1. (Ostrovskyi-Samoilenko) Let F be a continuous map in R. 1 ) For each positive non-periodic orbit ( z k ) & N with zo = 0, zh+1 = F ( Z k ) (Fock orbit) there is an infinite dimensional irreducible representation in Z2(N) given by Xek = G e k - 1 ,

X e o = 0.

2 ) For each positive periodic orbit (xo,x1,...,x,-I)

F

(xk)

with, 20 = 0 , x k + 1 = there is a finite dimensional irreducible representation given by Xek = A

e k - 1 ,

Xeo = 0

This is a complete list of inequivalent irreducible Fock representations f o r the given relation. In this paper we will define a space of parameters for the Fock representations of relation (1) and will determine the regions in these parameter space where the representations are bounded. 2. The quadratic family

Let F be a quadratic map and c its critical point of F. Let A = {L,C,R } be the alphabet set. For each real number z there is an unique infinite sequence in A called the itinerary of z and is denoted by it(x) = ( a d ( x ) ,a d ( F ( x ) ) , a d ( F 2 ( x )...), ) , where a d ( x ) , the address of x, is defined by ad(%)= L if x < c, ad($) = C if x = c, and ad(x) = R if x > c. The itinerary of F ( c ) is called the kneading sequence of F and is denoted by KF. This sequence characterizes the dynamical behavior of F , see Sharkovsky et a1.5. The orbit of 0 under the map F is denoted by orbF(0) and is called the Fock orbit of F whenever every point in the orbit is positive.

539

Consider the general family of quadratic maps

F(2)= E z 2 + P z + y parametrized by the real numbers E , B , r . Consider the linear map h ( z ) = k z , with k > 0, and the correspondent conjugated quadratic map given by -

a hoFoh-l(z) = -z2+pz+kT. k

If we have a Fock orbit for F with itinerary i t ~ ( 0then ) we have a Fock orbit for h o F o h-' with the same itinerary, i.e., i t ~ ( 0= ) itho~oh-i(0). We can choose k satisfying ky = 1, i.e., k = l/r. Thus it is sufficient to analyze the family fa,p (z) = a x 2

+ pz + 1, with a,p E I&

regarding the structure of the orbit of the point 0. We note that the above conjugation must be done by a linear map h with h'(z) > 0 in order to preserve the positivity and the existence of the Fock orbit. Let B be the set of points (a,p) in R2 such that the orbit orbf,,,(O) is positive and bounded. If we determine the structure of B,the description of the bounded Fock orbits for the general quadratic can be obtained as follows. Let F ( z ) = ax2 Pz y be a general quadratic map. Let B, be the set of points (a,p)in R2 such that o r b ~ ( 0 is ) positive and bounded. By the above argument we have (a,,@E B, if and only if ( a y , p ) E B. Moreover i t ~ ( 0=) itho~oh-~(0).Thus if we set

+ +

.r, : R2 + R2,

T,(Z,

Y) = (Z/Y,Y)

we have 23, = .,(a). Note that we do not allow y = 0 otherwise we would have F ( 0 ) = 0 obtaining only the trivial Fock representation.

Remark 2.1. We observe that we can look the problem of describing the Fock orbits of the quadratic fa,p(z)= az2 pz 1 from another perspective. Let fa,p (2) = a x 2 Bz 1 with ( a ,p) E R2 and ga(z) = 4bz(l - z) with b E R We can transform by conjugation, with an affine map h ( z ) = A z B , the quadratic map fa,p(z) into the quadratic map ga(z). Thus we can convert the problem of describing the behavior of the orbit of the point 0 under fa,p, depending on two real parameters a , p, into the problem of describing the behavior of the orbit of a real point z o under ga, depending on a real parameter b. The condition on the positivity of orbfp,a(0) must be replaced by an equivalent condition on orb,, (zo).

+ +

+

+ +

540

3. Types of Fock representations

Let orb,,p(O) := orbfa,,(0) be the Fock orbit associated to fa,p with parameters a,p. We denote by X,,p the Fock representation associated to the orbit orb,,p(O). The type of Fock representation (finite/infinite or bounded/unbounded) depends on the periodicity/aperiodicity, boundeness/unboundeness of the Fock orbit. Nevertheless, the behavior of the orbits is more diverse and this diversity is reflected on the representation. We will introduce some usual notions of operator theory in order to characterize the Fock representations. Consider a bounded operator T in Z2(N). Let a(T)be the spectrum of T , i.e., a(T)= {A : T - X non-invertible}, ae(T)be the set of eigenvalues of T , i.e., ae(T)= {A : exists v E Z2(N) with Tv = XU} and a,(T)= a(T)\ue(T) be the set of points in the spectrum of T which are not eigenvalues. Note that the closure of oe(T)is equal to o(T).Let p ( T ) be the spectral radius of T , i.e., p ( T ) = sup{ 1x1 : X E a(T)}.Consider the norm defined by

llTll := p(T*T). The w-limit set of a point x is the set of limit points in the orbit of 2 under fa,p, see Sharkovsky et d 5 ,and it is denoted by w,,p(x) := wfa,,(x). We have always a,(X&X,,p) = orba,p(0), see Ostrovskyi-Samoilenko2, and IlX,,pll = suporb,,p(O). We have the following types of points and correspondent orbits, with respect to the dynamics of f,,p: i) Periodic points, ii) Pre-images of periodic points, iii) Asymptotically periodic points, iv) Points correspondent to aperiodic orbits with closure equal to a countable set, v) Points correspondent to aperiodic orbits with closure equal to a Cantor set, vi) Points correspondent to aperiodic orbits with closure equal to an interval. The w-limit set in the first three cases corresponds to a periodic orbit. The case iv) gives a w-limit set equal to an infinite countable set the case v) gives a w-limit set equal to a Cantor set and the case vi) gives a w-limit set equal to an interval. Although the w-limit set does not distinguish the first three cases we must distinguish them since the properties of the associated Fock representations are different. For example the Fock representation is finite dimensional if and only if the Fock orbit is periodic, which are not

541

the cases ii) and iii). Let us give a detailed description of these cases. Let (a,/?)E B be fixed: in the case i) we have orb,,p(O) = w,,p(O), which are finite sets, and o , ( X : , ~ X , , ~ ) = 0 since every point in orb,,p(O) is an eigenvalue for Xi,pX,,p. As we saw previously, a finite dimensional Fock representation arises in this case. The dimension of the representation is equal to the period of the orbit. We have also IlX,,pll = max(orb,,p(O)). In the case ii) we have wa,p(O) c mb,,p(O), which are both finite sets. We have also a , ( X * X ) = 0 and IIXa,pII = max(orb,,p(O)). In the case iii) let 8 be the periodic stable orbit for fa,p. We have that orba,p(0) is an infinite set and a c ( X * X )= w,,p(O) = 8 are both finite sets. Note that the points in 8 are not eigenvalues of Xi,,X,,p. In the case iu) we have that a , ( X * X ) = w,,p(O) are both infinite countable sets. Finally, in the cases u ) and ui) we have that ~ ( x ~ , ~ X ,is, pa )Cantor set or an interval and ‘ T , ( X : , ~ X , , ~= ) w,,p(O) is necessarily an infinite set.

Figure 1. (a) Curves of periodic points up to period 4 (b) Curves of pre-images of periodic points (c) The two previous figures together.

The parameters that give representations of types i) and ii) can be deterThe k-dimensional mined numerically as follows. Let q k ( a , / ? ) := f&(O). Fock representations arise from the solutions of the equation q k ( a ,p) = 0 which is a polynomial equation on a,@of degree 2“l - 1. The solutions are, naturally, curves in the plane ( a ,p), see Figure 1. In the same way the parameters for which 0 is a preimage of a periodic point, and which give origin to representations of type ii), are solutions of polynomial equations of the type q k l ( a , / ? ) - q k z ( a , / ? ) = 0 with kl > k-2 > 1 natural numbers.

542

4. Characterization of the set 23

In this section we give a characterization of the region 23 of R2 which corresponds to the pairs (a, p) for which there are bounded Fock representations for relation (1). First we give a simple necessary condition on the existence of a Fock representation. Proposition 4.1. If X,,p is a Fock representation, for (a,P) E R2 t h e n a + p + 1 20.

Proof First note that f&(O) mb,,p(O) is not a Fock orbit.

= a:

+ /3 + 1.

Thus if f&(O)

< 0 then

4

i

C

-2

-4

-4

0

-2

Figure 2.

4

2

Set

B.

Theorem 4.1. (1) For every (Y > 0 and 9 , 2 0 there is a Fock representation. It is Bounded i f and only if

a: 5 (1 - ,8)2/4 and p < 1. (2) Let a: only if

< 0 and p < 0 . There

i s a bounded

a + p f 1 2 0.

Fock representation i f and

543

(3) Let a > 0, p < 0. I f a > (1 - p)2/4 or Q > -p, p < -1 then there is a n unbounded Fock representation. Thus in those cases (a,P) 4 B . I f a 2 p2/4 and a I -p then (a,p) E 8. (4) Let a < 0 and 2 0. I f Q + p 1 < 0 there is no Fock representation. I f a p 1 2 0 and

+

+ +

then (a,p) E B. Proof (1) Let a > 0 and /3 2 0. There is always a Fock orbit as f&(O) > 0 for every k E N. The critical point of f,,p, c = - p / 2 a is negative and the fixed points of f a , p are given by 1-/? f J(l - p ) 2 - 4a 2a We can see that the fixed points are both positive or both negative. The orbit orb,,p(O) is unbounded if and only if A* are complex or negative. The fixed points A h are complex numbers if and only if Q > (1 - p)2/4. Thus we must assume, in order to mb,,p(O) be bounded that a 5 (1- p)2/4. On the other hand A* < Oif and onlyif a 5 (1-,B)2/4and 1-p < 0 e ,L? > 1. (2) As we saw in Proposition 4.1, a necessary condition to exist a Fock representation is that a ,8 1 2 0. We will show that for a < 0 and ,B < 0 is also sufficient. If a p 1 = 0 the Fock orbit is of period 2 and thus the Fock representation is bounded. Consider the case f&(O) = a p 1 > 0. As a < 0 and /3 < 0 the critical point c = - P / 2 a is negative. Let z be the positive point satisfying f,,p(z) = 0. Note that z > 1 = f a , p ( 0 ) and the positive fixed point A- is in [O,z]. The fixed point A- is stable as

A* =

+ + + +

f ’ ( A - ) = 1 - p - J(1-

+ +

p)2

- 4a + p = 1- J(1-

p)2 - 4a

< 1.

The restriction of fa,p to [O,z] is monotone and consequently f & ( O ) + Abeing the set orb,,p (0) bounded. (3) Let a > 0 and p < 0. As observed previously if a > (1 -p)2/4 then the fixed points of fa,p are complex numbers. As a > 0 then f t , p ( 0 ) 4 00 and mb,,p(O) is unbounded. If a 5 (1 - /?)2/4 the fixed points are real and in the region considered are positive. If f,,p(O) > A+ then the orbit is unbounded. Note that f,,p(O) = 1 and the condition f,,p(O) > A+ H A+ < 1 is equivalent to a > -p when ,6 < -1. If p > -1 and a > -p then 1 < A- < A+ and f t , a ( 0 ) + A-. In this case orb,,p(O) is bounded.

544

Now assume a 5 -p (fa,p(0)5 A+), in this case the Fock representation if exists is bounded. On the other hand fa,p(c) = 1- P2/4a. The interval [fa,&), f2,&>1 is invariant under fa,a and fa,a(O)= 1 E [fa,&>, f2,p(c)j. Thus if f c r , ~ (>_c )0 a 2 p2/4 then mb,,~(O) is bounded. (4) Let 0: < 0 and P 2 0. As observed previously if f&(O) < 0 -e= a ,f? 4-1 < 0 there is no Fock representation. Consider a -tP -i- 1 2 0. As a > 0 and p < 0 the critical point c is positive. If f 2 (c) 2 0 then necessarily orba,p(0) is bounded as the interval [f2,B(c),fcr,p(c)]is invariant under fa,b and = [f2,p(c)7 fcY9p(c)1. As f 2 , f l ( C )= m?.3-g+p+a+1 16a we have proven the claim.

++

-+

-4

-3

Figure 3.

-2

-1

(a) Set LZ with a

> 0 and @ < 0. (b) Set B with a < 0 and @ > 0.

5 . Final remarks

The regions of the parameters associated to bounded representations which we described in our paper are mainly associated to orbits which are asymptotic to periodic orbits. In future work we pretend to describe in more detail the regions presented in Figure 4 and Figure 5 where occur the finite dimensional Fock representations. In these regions a small change in the parameters can lead to a substantial change in the type of Fock representation. We will also study this unstability phenomena. Note that the unstability on that region depends on how the variation of the parameters

545

a), b) and c ) Details of the set L3 with a

> 0, @ _< 0.

Figure 5. a), b) and c ) Details of the set B with o

< 0, @ 2 0.

Figure 4 .

is done. In fact there are curves of stability in that region, for example a curve given by the solutions of the equation q k ( a , P ) = 0, see Figure 1. There is also interest in studying sub-families of the family fa,p associated to other curves in the parameter space. Examples are isentropic curves (equal topological entropy) and curves parametrized by b arising from the conjugation with g b , see remark 2.1. These curves will intersect the periodic points curves ( q k ( a , P ) = 0) with a particular order.

Acknowledgments First author acknowledge CIMA-UE for financial support. Others authors partially supported by FCT/POCTI/FEDER. We would like to thank to the referee for their suggestions.

546

References 1. Ostrovskyi, V.; Samoilenko, Yu. Structure theorems for a pair of unbounded selfadjoint operators satisfying a quadratic relation. Advances in Soviet Mathematics, 9, (1992). 2. Ostrovskyi, V.; Samoilenko, Yu. Introduction to the theory of representations of finitely presented *-algebras. I. Representations by bounded operators. Reviews in Mathematics and Mathematical Physics, 11,pt.1. Harwood Academic Publishers, Amsterdam, (1999). 3. Popovych, Stanislav V.; Maistrenko, Tatyana Yu. C*-algebras associated with quadratic dynamical system. Symmetry in nonlinear mathematical physics, Part 1, 2 (Kyiv, 1999), 364-370, Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 30, Part 1, 2, Natsional. Akad. Nauk UkrGni, Inst. Mat., Kiev, (2000). 4. Sharkovsky, A. N. Coexistence of cycles of a continuous m a p of the line into itself. Translated from the Russian [Ukrain. Mat. Zh. 16 (1964), no. 1, 61-71] by J. Tolosa. Proceedings of the Conference “Thirty Years after Sharkovsky Theorem: New Perspectives” (Murcia, 1994). Internat. J. Bifur. Chaos Appl. Sci. Engrg. 5 (1995), no. 5, 1263-1273. 5. Sharkovsky, A. N., Kolyada, S. F., Sivak, A. S. and Fedorenko, V. V., Dynamics of One-Dimensional Maps, Kluwer Academic Publishers, 1997.

547

CHAOTIC DISCRETE LEARNING SYSTEMS

ESELx

M. MERCES RAMOS AND PEDRO SARREIRA of Sciences, Instituto Polite'cnico de Lisboa, Campus de Benfica do I P L , 1549-003 Lisboa, Portugal E-mail: [email protected] E-mail: pedros Q e s e k i p L p t

- Department

Better understanding of brain functions can have implications for the study of the dynamics of learning. We apply concepts and tools of nonlinear dynamics and chaos t o the modelling and study of learning. We use the sinusoidal activation function t o adjust the sinaptic weights of a neuron and we investigate the behaviour of the network when the learning rate changes. Our results show that these systems can exhibit a very rich dynamics, from regular to chaotic, and we calculate the topological entropy.

1. Introduction

Despite considerable developments in educational research, there is, as yet, no established model of learning. Therefore, there are still no clear ideas about what is good teaching. We are of the opinion that in educational research, as occurred in the past with the Physics of the XVI and XVII centuries, we need a change in approach in order to establish appropriate mathematical models to describe the phenomena involved in learning. This is the main aim of our research. What do we have to consider in order to understand learning? The brain is the seed of the cognition phenomenon involved in learning - the network of neurons in the brain provides the ability to assimilate information. On the other hand, computer development has made possible the birth of artificial intelligence and the design of neural networks is motivated by brain analogy. Recently, neurobiologists have looked to artificial neural networks as a research tool for the interpretation of neurobiological phenomena. Progress in mathematical models will make it possible to describe and understand much better the human brain functions such as learning, especially learning in the classroom. Thus, because they are inspired by the

548

anatomy and physiology of the brain, neural networks are the mathematical models for a specific brain process such as learning. The architecture of the neural network consists of several processing units (neurons), which are interconnected by channels (synapses). Information flowing through a synapse changes by way of a multiplicative factor, designated a connecting weight. Although learning is one of the brain attributes crucial to environmental adaptation, there are many notions associated with learning. For the present study we consider, like Haykin12, that “Learning is a process by which the free parameters of a neural network are adapted through a continuing process of stimulation by the environment in which the network is embedded. The type of learning is determined by the manner in which the parameter changes take place”. The previous definition implies the following sequence of events: 0 0

0

The neural network is stimulated by an environment. The neural network undergoes changes as a result of this stimulation. The neural network responds in a new way to the environment, because of the changes that have occurred in its internal structure.

Consider a pair of node signals xj and vk connected by a synaptic weight where xj is the output of neuron j (pre-synaptic activity); vk is the internal activity of neuron k; g k is the corresponding value of the actual response of this neuron (post-synaptic activity); and ‘ w k j is the connecting weight for the synapse j attached to the input neuron k at time t .

wkj,

,

Neuron

j

I

I

Neuron

k

,

Figure 1. A pair of neurons.

At time t an adjustment Awkj(t) is applied to the synaptic weight the updated value W k j ( t l),

W k j ( t ) , yielding

+

549

The shape of the adjustment a w k j defines the learning algorithm. What shape would it assume? The shape of the weight adjustment determines the activation function. There is a great variety of learning algorithms (which differ from each other simply in the way in which the adjustment a w k j to the synaptic weight is formulated) (see Haykin” and Botelho-Jamison3). In this paper, we choose the Hebbian learning algorithm and the unsupervised learning (Botelho-Jamison3 and Li et ~ 1 . ~ ) . Recently, much research has shown that the neural network and biological neuron exhibit a nonlinear and chaotic behaviour (for ex., Li et al.?, Lysetskiy-Zurada‘, Wang et al. 15). Cognitive and language development (for ex., van Geert14 and Smith13) as well as learning, in the classroom (Ramos’), seem to have a complex nature, following a nonlinear dynamic. Farhat’ considers that the basic functional unit for higher-level processing in cortical networks can be modelled by a bifurcation processing element like the parametrically driven logistic map, or any other one-dimensional driven map on the unit interval. So we must consider activation functions that can display a very rich dynamics: from regular to chaotic. This paper is organized as follows. In section 2 we describe the activation functions - in practice, a sinusoidal mapping will be introduced as a nonlinear activation function. In section 3 we analyze the dynamic of the map - the bifurcation diagram provides an insight into how slight fluctuation (in the parameter) of an input may be able to change the global dynamic of the system (Farhat’ and Lysetskiy-Zurada‘). Topological entropy is a reliable variable to measure the system’s complexity, so we compute the topological entropy associated with the behaviour of the neuron and we see the behaviour of topological entropy when the learning rate changes. In section 4, we discuss the results. 2. Mathematical model

According to the Hebbian postulate of learning (Haykin”), the synaptic adjustments are given by scaled products of the incoming and outgoing signals. So the increment must be a function of both post-synaptic (output) and pre-synaptic (input) activities

550

The simplest rule for a change in a synaptic weight, w k j ( t ) ,is expressed as a product of the input and output. We can represent

(4

A W k j ( t )= PYk ( t ) q

7

where the positive scalar p is the learning rate. We can rewrite the equation (1)

w(t

+ 1) = w ( t ) + p?j(t)z(t).

This algorithm leads to an exponential growth and so to a synaptic saturation (Botelho-Jamison3, and Li et d7). To limit the growth, Oja (1982) proposed a modification that is plausible in physiological terms

W(t

+ 1) = w(t>+ PLY(t)[4t)- Y(tMt)l.

(2)

, neuron model with only one We consider, like Li et a ~ the~ simplest weight (one input) and with constant input 1 and we choose as nonlinear activation function the sinusoidal function

= sin(w(t)), because many neural systems composed of a great number of interacting neurons exhibit self-sustained oscillatory behaviour leading to various space-time patterns (which may be regular or chaotic). Now we can write (2) as

w ( t + 1) = w ( t ) + p s i n ( w ( t ) ) [ l sin(w(t))w(t)] or W(t+l) =

fp(wt).

3. Topological entropy We can limit our study of the iterates of l-parameter family of functions fp in the interval I = [a, cg] to the convenient region R = [pl,p 2 ] of the parameter p , where q,i = 1,..4, are the critical points, see Figure 2, and p 1 = 1.5 (for example), and p 2 is such that fP2(c1) = a or fP2(cz) = c3, and in this region the map f, is like a symmetric cubic map, where c1 = -c2

551

-2

-1

L 2

-1

-2

0

1

2

Figure 2. Learning maps f p ( w )with the learning rate p = 2.8 and p = 3.2; CO,CI,CZ, c3, are the critical points.

2

1.5 I

0.5

C -0.5

-1 -1.5 I

-2

-1

0

1

Figure 3. Unimodal map for p = 2.5.

and Q = -23. Thus, we impose that the orbit of the critical points c1 and c2 remains within the interval [Q, cs], see Figure 2. For that family of the map f, in the interval I and changing p in the region R we obtain all the diversity of orbits and bifurcations that occur in the 1-parameter family with two critical points as in the bimodal (or cubic-like) map family. Nevertheless, for special subsets of R, the interval

552 w

w

1.5 1 0.5

1.5 1 0.5 0 -0.5 -1 -1.5

0 -0.5 -1

-1.5 1.75

2

2.25 2.5 2.15

3

P

1.75

2

2.25 2.5

2.15

3

P

Figure 4. Bifurcation diagrams of the learning map family f p ( w ) : left 'wo = 1 and right wo = -1.

[pl,p*]where p* is the value such that fi(c2) < 0 or fi(c1) > 0, and choosing the initial conditions properly, the map reduces to iterates of a unimodal (quadratic-like) map, see Figure 3. In Figure 4 we can observe the bifurcation diagrams for the initial conditions w o = 1 and w o = -1; the infinite diversity of periodic orbits and bifurcations that occur, is clearly visible. An exhaustive and rigorous approach becomes possible with the techniques of symbolic dynamics, as we will see below. For the study of the 1-parameter family f, of maps of the interval with two critical points, i.e. bimodal maps of the interval, see MilnorThurstonlO and Lampreia-Sousa Ftam0s5. For each value p of the parameter let f, : I + I be piecewise monotone, then the interval I is subdivided in three subintervals: L = [ Q , c ~ [ M , =]c~,cz[and R =]c2,c3], where Ci are the critical points or the extremum points, in a such a way that the restriction of f, to each interval is strictly monotone. Thus the restriction of f, to the subintervals L and R are decreasing maps and the restriction of f, to M is increasing. Each such maximal interval on which the function f, is monotone is called a lap of f,, and the number C = l (f,) of distinct laps is called the lap number of f,. On the other hand for each value p we define the orbits of the critical points c1 and c2, as:

O(Ci) = {P 3 : $1

= fj(Ci), j E

N}.

With the aim of studying the topological properties of these orbits we associate with each orbit O(ci) a sequence of symbols S = SlS2.. . Sj . . . where for each G , we have Sj = L if fi(ci) < c1, Sj = A if f$(ci) = c1,

553

Sj = M if c1 < fi(ci) < c2, Sj = B if fi(c;) = c2 and Sj = R if > c2. If we denote by nLR the number of the symbols L and R in a finite subsequence of S we can define the LR-parity of this subsequence according to whether n L R is even or odd. In what follows we define an order relation in C5 = {L, A, M , B, R}" that depends on the LR-parity (see Milnor-ThurstonlO, Lampreia-Sousa Ramos '). The sequences of symbols corresponding to periodic orbits of the critical point A and B is P = APlP2 ... PP-lA ... and Q = BQlQ2 ...Qq-l B .... We denote by S(') = SlS2.. . Sk-1C the periodic block in S, with C = A or B. The realizable itineraries of the critical point c1 and c2 for the maps previously defined are called kneading sequences (Milnor-Thurston") . In our study we use an order relation defined in C = { L , A ,M , B, R}" that depends on LR-parity. Thus, for two of such sequences, P and Q in C, let i be such that Pi # Qi and Pj = Q j for j < i. If the LR-parity of the block PI ...Pi-1 = Q1...Qi-1 is even we say that P < Q if Pi < Qi in the order L < A < M < B < R. If the LR-parity of the same block is odd, we say that P < Q if Pi < Qi in the order R < B < M < A < L. If no such index i exists, then P = Q. When 0 (ci) is a tperiodic orbit we obtain a sequence of symbols that can be described as a block of length k, S ( k ) = Sl...Sk-lCi. We restrict our study to the case where the two critical points are periodic (eventually periodic, respectively), 0 (c1) is pperiodic and 0 (c2) is q-periodic (fE(c1) = c2 or fi(c2) = c1, respectively). Note that 0 (c1) is realizable if the block P = PI ...Pp-lA is maximal, that is, ai(P) < P, where 1 < i 5 p , where a(Pi ...Pp-lA) = Pi+l...Pp-lAP1 ... is the usual shift operator. On the other hand, O ( c 2 ) is realizable if the block Q = Ql ...&,-I B is minimal, that is, d ( Q ) > Q, where 1 < j 5 q. Finally, note that the pair of sequences that are realizable satisfies the following conditions a i ( P ) > Q, 1 < i 5 p and &(Q) < P, 1 < j 5 q. The set of such a pair of sequences is denoted by C ( A , B ) .A kneading data ( P ( P ) , Q'Q)) is a pair of sequences such that P(p)= P1...Pp-lA, Q(,) = Q1...Qq-l B for some pair (P,Q) E C ( A , B ) ,a bistable sequence PI...Pp-lBQ1 ...Qq-l A, an eventually periodic sequence PI...Pp-1BQ 1.. .Qq- 1B or Q 1...Qq- 1API...Pp-1A. Now we shall consider the topological entropy. This numerical invariant measures the complexity of mappings. A possible definition of chaos in the context of one-dimensional dynamical systems state that a dynamical system is called chaotic if its topological entropy is positive. Thus, the topological entropy can be computed to express whether a map has chaotic behaviour. For the class of systems studied in this paper the topological

fi(c;)

554 h

0.8

0.6

0.4

0.2

I 1

..” +.....-‘ .....

I

j ,

,

1.15

2

<

,

2.25

,

I

2.5

2.75

,

,

k

3

Figure 5. Variation of the topological entropy hto,(,u) for f, E [1.5,3.171.

entropy off,, denoted by htop(f,),can be computed by the growth rate of The topological lap number, s(f,). Let .& denote the number of laps of entropy of f,, denoted by htop(f,), is then

fi.

htop(f,) = log s(fP) = lim l/kloglk k+w

(see Misiurewicz-Szlenkll and Milnor-Thurston1’). Alternatively, the topological entropy of the bimodal map can be calculated by the spectral radium of the transition matrix M(f,) (see Lampreia-Sousa Ram0s4>5and h o s et d9).For each value of parameter p a kneading data is determined. This kneading data determines a Markov partition of the interval to which is associated a transition matrix M(f,). Then

ht,

(f,)

= log L a x ( M ( f P ) )

where X,,(M(f,)) is the maximum of the eigenvalues of the transition matrix M (f,) . We now turn to the behaviour of the topological entropy of the chaotic system with the parameter p, i.e., the dependence of h t o p ( f p )on p E R. For the values of the parameters p E [1.5,p*], with p* E 2.6, the dynamics is described by a unimodal map, then htop(f,) E [O,log2] and for p E [p*,3.17] the dynamics is described by a bimodal map, then htop(f,) E [log2,log3]. In both these cases the topological entropy is an increasing function of the parameter p, see Figure 5. This figure represents the variation of the topological entropy with p, in the region R of the parameter space. For these values of the parameter, the topological entropy hiopis monotone increasing. In conclusion, we emphasize that in the interval p E [2.15...,3.171, we have chaotic behaviour because the topological entropy is positive.

555

4. Final considerations In this study we show how the learning rate p , in the case of the Hebbian unsupervised learning and with sinusoidal activation function, produces successive changes in the behaviour of the system. After successive duplication of the period, the system begins in a chaotic way (see Figure 4) and complex behaviour becomes apparent. In the interval where the topological entropy is 0, the system displays a regular behaviour. We can also see that topological entropy is a monotonous increasing function with the learning rate parameter p (see Figure 5). It is further evident that as the parameter p increases, the behaviour of the system, as we would expect, becomes more complex and that for certain values of parameter p the dynamics goes t o the periodic point. So the entropy, as a measure of complexity, naturally increases as the parameter p increases. The study of topological classification for bimodal maps leads to the introduction of two topological invariants. One of them is the growth number s(f,) = e h ( p ) and the other number is associated with the relative position of the maximum and the minimum of the map. Our future study deals with the properties of these topological invariants related to variation of the parameter p and another parameter Y ,that we must t o introduce. Topological entropy in itself is no longer sufficient to classify the maps introduced. We need to consider a new topological invariant in order t o distinguish maps with the same entropy, see Ramos et d 9 .

Acknowledgements Both authors thank IPL and FCT (program POCTI - Portugal) for having in part supported this work. They would like to thank J. Sousa Ramos for his help in the calculation of topological entropy and for many stimulating discussions on the subject of this paper.

References 1. M. Courbage, V. B. Kazantsev, V. I. Nekorkin, and M. Senneret, Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons, Chaos, 14 (2004) 1148-1156. 2. N. H . Farhat, Biomorphic Dynamical Networks for Cognition and Control, Journal of Intelligent and Robotic Systms, 21 (1998) 167-177. 3. F. Botelho and J. E. Jamison, A learning rule with generalized Hebbian synapses, J. Math. Anal. Appl. 273 (2002) 529-547.

556 4. J. P. Lampreia and J. Sousa Ramos, Computing the Topological Entropy of Bimodal Maps. Proceedings of ECIT 87, World Scientific Publishing, Singapore, 1989. 5. J. P. Lampreia and J. Sousa Ramos; Symbolic dynamics of bimodal maps, Portugal. Math. 54 (1997), no. 1, 1-18. 6. M. Lysetskiy and J. M. Zurada, Bifurcating neuron: computation and learning, in Neural Networks. 17 (2004) no. 2, 225-232. 7. C. Li, X. Liao and J. Yu, Generating chaos by Oja’s rule, Neurocomputing 55 (2003) 731-738. 8. M. Ramos, A entropia como medida da complexidade e estabilidade do conhecamento e m contextos de ensino e aprendizagem, Universidade de Lisboa, PhD these, 2001. 9. M. Mercb Ramos, C. Correia Ramos, R. Severino and J. Sousa Ramos, Topological invariants of a chaotic pendulum, Int. J. Pure Appl. Math. 10 (2004), no. 2, 209-226. 10. J. Milnor and W. Thurston; O n iterated maps of the interval. Dynamical systems (College Park, MD, 1986-87), 465-563, Lecture Notes in Math., 1342, Springer, Berlin, 1988. 11. M. Misiurewicz and Szlenk, Entropy of piecewise monotone mappings. Studia Math. 67, 45-63 (1980). 12. S. Haykin, Neural Networks: A Comprehensive Foundation, Mcmillan, New YorkJ994. 13. L. B. Smith, Stability and Variability: T h e Geometry of Children’s NovelWord Interpretations, in Chaos Theory in Psychology, Praeger, Londres, 1995. 14. P. V. van Geert, Dynamic Systems of Development, Harvester Wheatsheaf, London, 1994. 15. L. Wang, H. Zhao, G. Dong and J. Li, O n the complexity of finding emerging patterns, Theoretical Computer Science. 335 (2005) 15-27.

557

O N THE ASYMPTOTIC BEHAVIOR OF THE MOMENTS OF SOLUTIONS OF STOCHASTIC DIFFERENCE EQUATIONS *

J. APPLEBY School of Mathematical Sciences Dublin City University, Dublin 9, Ireland E-mail: [email protected]

G. BERKOLAIKO Department of Mathematics, Texas A&M University, U.S.A . E-mail: [email protected]. edu A. RODKINA Department of Math/CSci, University of the West Indies, Jamaica E-mail: alexandra.rodkina8uwimona.edu.jm

We investigate the asymptotic behavior of the a-th moment n H EX: of the solution ( X , ) of a stochastic difference equation with independent noises. Depending on (Y E (0,1] and on the ratio u H 2f(u)/g2(u)(where f is the intensity of the deterministic term and g is the intensity of the stochastic term), EX: tends to 0 or to infinity. The analysis applies t o a weak Euler-Maruyama approximation of a stochastic differential equation. To obtain our results we make use of an elementary lemma about the estimation of a positive continuous function from below by a positive continuous convex function.

*This work is supported partially by the Enterprise Ireland International Collaboration Programme (Grant number IC/2004/003).The first author is also partially supported by an Albert College Fellowship awarded by Dublin City University’s Research Advisory Panel. The first and third authors also gratefully acknowledge the support of the Boole Centre for Research in Informatics, University College Cork, where the research was partly conducted.

558

1. Introduction The problem of stability of stochastic difference equation has been investigated in a number of papers. We refer the readers to Refs. 1-6. Some results on the asymptotic behavior of the moments was obtained in Ref. 7. The stability of stochastic difference equation is also connected with the stability of numerical methods for stochastic differential equation, SDE, in particular with the stability of a weak Euler-Maruyama simulation of a SDE (e.g. Ref. 8). For a linear scalar stochastic differential equation, the meansquare asymptotic stability of the numerical methods has been studied e.g. in Refs. 9, 10. More recently, results on the exponential mean-square stability of numerical methods to general nonlinear finite-dimensional SDEs were obtained in Ref. 11. A contribution of the present paper is that its analysis covers equations which do not have leading order linear behaviour close to the equilibrium, and also studies the asymptotic behaviour of the a-th moment for a E ( 0 , l ) . We consider the equation

where h > 0 is a nonrandom parameter, the functions f and g are nonnegative, continuous and bounded, and the random variables are independent with E<, = 0 and E<E = 1 for n = 0 ,1 ,. . .. We denote by (R, F,{Fn},€~, P) a complete filtered probability space, where filtration {Fn},€~ is naturally generated: F,+1 = a{
Lemma 2.1. Let F : [0,m) -+ [0,m) be a continuous function with F ( 0 ) = 0 such that for all c > 0 there exists C = C(c) > 0 such that

inf

t>c

F(t) > C. t

(2)

Then there is a continuous convex function p : [O,w)-+ [O,m) such that 0 < p(t)5 F(t), t

> 0,

p(0) = 0.

(3)

Proof. It is simple enough to construct the function p explicitly. The condition (2) means that there exists a decreasing sequence {Cn}r==l satisfying F ( t ) > C,t for t > l / n . It is easy to see that the linear functions

559

pn defined by cpn(t)= Cnt - Cn/n satisfy cpn(t) 5 F ( t ) for all t: indeed cpn(t)< Cnt 5 F ( t ) when t 2 l / n , while cpn(t)< 0 5 F ( t ) when t < l / n . Now we can define a function cp, which satisfies the conditions (3), namely

~ ( t=)SUP pn (t). n2l

As cp is a supremum of linear functions, it is a convex function. It is positive for t > 0 because cp,(t) > 0 for all n > l / t . Since {C,}~=, is decreasing, 0 p(O) = supn>l - vn(O)= - infn>l Cn/n = 0. 3. Main result

We consider equation ( 1 ) and suppose that noise is bounded from below: there is a

> 0 such that

i$,2

n E N.

-a,

(4)

Condition (4)implies that for each y E ( 0 , l ) there is a small h such that for all h 5 h ( a , y ) 1

+ h f (Xn) + h g ( X n ) t n + , 2 1 - h - a

h=

=

h ( a ,y)

> 0,

(5)

which means in particular that the solution of ( 1 ) remains positive for hI h(a,y). Below we formulate some conditions.

For all c > 0 there is C = C ( c ) > 0 such that inf

uLc

For all c

> 0 there is C = C ( c ) > 0 such that

There is

~1 E

( 0 , l ) such that

sup UER\tOI

There is

~2

E [0,1) such that

inf uER\{OI

If I).(

> C > 0.

inf [g(u)[> C

> 0.

U L C

{-}
{ ""}

g2(.2L)

(6)

(7)

(8)

> 1 - E2.

(9)

Theorem 3.1. Assume that there is H > 0 such that E1<,I3 < H for all n E N. Let (Xn) be a solution to (1) with positive initial value XO> 0.

a) If condition (6) holds and f (u)< 0 fo r all u E R\{O} then EXn+O

as n + m .

(10)

b) If condition (6) holds and f (u)> 0 for all u E R\{O} then

EXn-+oo as

7240;).

(11)

560 C)

If (7) and (8) hold then for all a E (0, &I)and all h suficiently small EXZ+O

as n + c o .

d) If (7) and (9) hold then for all a E

(12)

( E Z , 1) and

all h suficiently small

E X z ~ c oas n - + c o .

(13)

We make some comments on this result before giving its proof. First, we notice that the result shows that the presence of noise can not only stabilise a difference equation almost surely (as shown in Ref. 12), but that moments of small order of the solution can also be stabilised. For instance, if f(z) > 0 for all z > 0, solutions of the stochastically unperturbed equation

+

zn+l = zn(l h f ( z , ) ) , n = 0,1,2,. . tend to infinity as n -+ 00; however, if (8) holds, and a > 0 and h > 0 are sufficiently small, then E[X,"]+ 0 as n --+ co. (1) may also be viewed as a weak Euler-Maruyama approximation of the stochastic differential equation

d X ( t ) = X ( t ) ( f( X ( t ) )dt

+ S ( X ( t ) )dB(t)l

(14)

where B is a standard Brownian motion (e.g. Ref. 8). We recall that the Euler-Maruyama (EM) numerical method computes approximations X n ( h ) M X ( n h ) by

Xn+l(h) = X n ( h ) - h X n ( h ) f( X n ( h ) )+ xn(h)S(Xn(h))ABn+lI (15)

+

where h > 0 is the constant step size and ABn+l = B ( ( n 1)h) - B(nh). We see that when

L+l =

+

B ( ( n 1)h) - B ( n h )

en

fi

1

(15) coincides with (l),where are standardised normal random variables. It is therefore to be hoped for h sufficiently small that solutions of (1) have similar asymptotic properties to those of (14). In fact, it is known that the condition (8), which guarantees moment asymptotic stability of (1) also guarantees almost sure asymptotic stability for (14); and condition (9), which ensures that a moment of the solution of (1) tends to infinity, also guarantees the almost sure instability of the zero solution of (14) (a statement of these asymptotic results for stochastic differential equations can be found in e.g., Ref.12). This observation and Theorem 3.1 suggest

561

that t H E[X(t)'Y]and n H E[XE] should have similar asymptotic behaviour. It also suggests that for sufficiently small a > 0 that there is a close connection between the almost sure and a-th moment asymptotic behaviour for both nonlinear stochastic difference and differential equations. We hope to explore these connections in later work. We now return to the proof of Theorem 3.1.

Proof. To prove parts a) and b) we first take the conditional expectation with respect to F, on both sides of the equation (1) to get E(&+I IF,)

+ hX,f(X,) + E(&Xng(Xn)
=

x, + hX,f(X,).

(16)

Taking the expectation on both sides of (16), we arrive at

EXn+1 = EX,

+ hE(Xnf(Xn)).

(17)

In case a), since uf(u)5 0 for all u 2 0, and condition (6) is satisfied, the function F defined by F ( u ) = - u f ( u ) , u 2 0 satisfies all the conditions of Lemma 2.1. Hence there is a continuous convex function cp, satisfying (3), such that cp(0) = 0 and 0 < cp(u) 5 - u f ( u ) for all u > 0. From (17) we obtain

EX,+1 5 EX, - hEp(X,) 5 EX, - hv(EX,).

(18)

Then for zn = EX, we have z,+1 I 2 , - hcp(z,).

(19)

Since cp(u) 2 0 for all u 2 0, inequality (19) implies that the sequence of nonnegative numbers {z,},~N decreases. Therefore limn--rmz, =: L 2 0. If L > 0, by passing to the limit in (19), we obtain the contradiction L 5 L - hcp(L) < L. Thus L = 0 and therefore case a) is proved. In case b), Lemma 2.1 holds true when we define the function F by F ( u ) = u f ( u )for u 2 0. Then there exists a continuous convex function cp, satisfying (3), such that (p(u) 5 uf(u) for all u 0. From (17) we obtain

E&+i L EX, and for z,

+ hEp(&)

2 EX,

+ b(EX,),

(20)

= EX, Z,+l

2 zn

+ hcp(z,).

(21)

Since cp(u) > 0 for all u > 0 and zo > 0 the sequence {z,},>o - is positive and increasing. If it has a finite limit L < 00, then passing to the limit in

562

+

(21), we get L 2 L hip(L) > L , which is a contradiction. Hence, we must have z, -+ 00 as n + 00, proving case b). For the proof of c) we expand (1+ u ) ~a, E ( 0 ,l),up to the fourth term and get

Therefore for u = h f ( X , )

=:

+ fig(X,)t,+l

we have

X:Ma,,+1.

We define @, = E[Ma,,+1l.Fn] - 1 and pn+1 = X,*(Ma,,+l - @ , - l ) ,so that recursive inequality (22) can be written as x:+1

5 x:

+ x:@,+ P n + l .

(23)

We note that Epn+l = 0. Expanding @, in powers of h we get

where o( 1) + 0 as h 4 0 uniformly in X,. This step is justified by virtue of the boundedness of the functions f , g and of the third moment of Itn/. Let a < ~1 and h be small enough. Then (24) together with (8) imply that @n

Ia b 2 ( X n )

2

and

+7 + o ( l ) =: -&T a-1 )

< 0.

We note that since condition (7) holds, by defining the function F by F ( u ) = ug2(u1/")for u 2 0, we can apply Lemma 2.1 and conclude that there is continuous nonnegative convex function ~ ( usuch ) that (25)

563

Taking the expectation on both sides of (23) and letting z, = EX: we get

In case d) we apply Taylor expansion of (1 third term: (1

+ +)a = 1+ au + a(a2

-

1)u2

+

+ u ) ~a ,E (0, l),up t o the

a(a 6( 1

+

+

- 21+3 <)a+3

.

Here 5 is between 0 and u. Let u, = h f ( X , ) fig(X,){,+l and h ( a , y ) be defined in (5). Therefore for h 5 h(a,y), we have 1 un 2 y > 0 and

+

Then, because y E (0, l ) , we have l+&2 y,and so by using the elementary inequality (z yI3 5 4((zI3 lyI3), for h I h ( a , y ) , we get

+

+

and

We note that the expansion of 8 , in powers of h is similar t o that (24):

Let

a!

> E Z and h be small enough.

Then (28) together with (9) imply that

564

and a-1 +f f h( 7 2 1-E2

+o(l)

)

=: &;

> 0.

where cp is the nonnegative convex function satisfying (25) for a > €2. Using the inequalities (26) and (29) we can now complete the proofs of c) and d) in the same way as we did for parts a) and b). 0

Remark 3.1. We note that similar results can be obtained when no restrictions of type (4)are imposed on the noise term, but we leave this for a later treatment. 4. Examples

In this section we present simple examples to illustrate parts a)-d) of Theorem 3.1. Define f1 by fl(u)= Cu2/(1 u2), and f2 by 2u sin 1 fz(u)= u3 sin -, 1 u E (0,1], f2(0) = 0, f2(u) = - u > 1. U 1+u2'

+

Also, define g1 by g1(u) = u

/ d m for u 2 0, and g2 by

We note that for f l , the condition (6) holds, but for f2 it does not. For both g1 and g2 the condition (7) holds. We therefore have the following case distinctions, which cover the four cases covered by Theorem 3.1. If in equation (1): i) f coincides with fl for C < 0 and g is an arbitrary bounded function, then (10) holds; ii) f coincides with f l for C > 0 and g is an arbitrary bounded function, then (11) holds; iii) f coincides with fz and g coincides with gz, then (12) holds; iv) f coincides with fi with C > (1 - ~ 2 ) / 2and g coincides with 91, then (13) holds.

Acknowledgments The authors would like to thank Dr C. Kelly for his valuable suggestions.

565

References 1. V. Kolmanovskii and L. Shaikhet, Dyn. Syst. and Appl., World Sci. Ser. Appl. Anal. 4, 397 (1995). 2. A. Rodkina, X. Mao and V. Kolmanovskii, Stochastic Anal. Appl. 18(5), 837 (2000). 3. A. Rodkina and X. Mao, J. Differ. Equations Appl. 7(4), 529 (2001). 4. A. Rodkina and H. Schurz, Advances i n Difference Equations, 3,249 (2004). 5. A. Rodkina and G. Berkolaiko, Proc. of The Int. Conf. "2004-Dynamical Systems and Applications ", 614 (2004). 6. A. Rodkina and H. Schurz, SACTA 6(1), 23 (2004). 7. H. Schurz, Discr. Cont. Dynam. Systems 7(3), 487 (2001). 8. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin (1992). 9. Y . Saito and T. Mitsui, SIAM J . Numer. Anal., 33,2254 (1996). 10. D. J. Higham, S I A M J . Numer. Anal. 38(3), 753 (2003). 11. D. J. Higham, X. Mao and A. M. Stuart, LMS J . Comput. Math. 6, 297 (2003). 12. J. A. D. Appleby, X. Mao and A. Rodkina, Discr. Cont. Dynam. Systems. 15 (3),843 (2006).

566

ORTHOGONAL POLYNOMIALS AND THE BEZOUT IDENTITY

A. RONVEAUX De'partement de Mathe'matique, Unite' d'Analyse Mathe'matique et d e Me'canique, Universite' Catholique d e Louvain, Bitiment Marc de Hemptinne, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium, E-mail: [email protected]. be A. ZARZO Instituto Carlos I de Fisica Tedrica y Computacional, Facultad de Ciencias, Universidad d e Granada, Spain and Departamento de Matemdtica Aplicada, E . T.S. Ingenieros Industriales, Universidad Polite'cnica de Madrid, Spain, E-mail: [email protected] I. AREA Departamento de Matemdtica Aplicada 11, E . T.S.E. Telecomunicacio'n, Universidade d e Vigo, 36200-Vigo, Spain, E-mail: [email protected]

E. GODOY Departamento d e Matema'tica Aplicada 11, E. T.S. Ingenieros Industriales, Universidad d e Vigo, 36200-Vigo, Spain, E-mail: [email protected]

We present a survey on the properties of the Bezout polynomials A ( z ) and B ( z ) solving the Bezout's problem A ( z ) P n ( z ) B(z)PA(z)= 1, when Pn(x)belongs to an orthogonal polynomial family. We extend results given by P. Humbert for Legendre polynomials on the several recurrences involving the four families Pn (z), PA(z), A ( x ) and B ( x ) and, from these recurrences, orthogonality of the Bezout , ( z ) )is stated. pair ( A ( z ) B

+

Key words and phrases. Orthogonal polynomials, Bezout identity 2UUU

Subject Classification: Primary 33C25 Secondary 33C20 05A10

567

1. Introduction and Motivation

Given two polynomials P ( x )and Q ( x ) ,the Bezout’s identity for them is [3]:

A ( x ) P ( x )+ B(x)Q(x) = gcd(P, Q ) , gcd(P,Q) being the greatest common divisor of P ( x ) and Q ( x ) , which is a non-vanishing constant when P ( x ) and Q(z) are relatively prime. It is well known that, for fixed (P,Q), there exists an infinite number solutions ( A ,B ) ,i.e. a infinite number of Bezout’s pairs associated with the starting pair (P,Q). However, when the degrees of A ( z ) and B ( x ) are smaller than the degrees of Q(x) and P ( x ) , respectively, then the Bezout pair is unique [13]. Of course, the solution ( A , B ) can be computed from the Euclidean division of P ( x ) by Q ( x ) (see e.g. [lo] for a computer algebra system doing this). Recently [l,21, the authors have considered this Bezout’s identity when P ( x ) := P,(x) is a monic polynomial of degree n belonging to a family of classical orthogonal polynomials and Q ( x ) := Ph(x).Interest of these settings comes e.g. from the close relation existing between the Bezout’s pair and the second kind function corresponding P,(x).In this context we should mention that, starting with

A,-z(x)P,(x)

+ B,-l(x)P?Xx)

= 1,

where P,(x)are here the Legendre polynomials, Humbert [6,7,8] studied in in order to compute the corresponding detail the Bezout pair (An-2, Legendre’s function of second kind Q,(x) and its derivative QL(x). More precisely he proved that fn-l(x) in 1 x+l Qn(x) = ,P,(x)log -- f n - l ( Z ) x-1 is linked to B,-l(x) by

and QL(x) is linked to An-2(x) from:

+

+

1 2 1 .PA(.) An-2(x) QL(x) = -PL(x) log -2 x-1 22 - 1 Here, we present a survey of the Bezout’s pair properties in the aforementioned settings giving for An-2(x) and B,-1 (x)several recurrence formulas and differential characterizations. In doing this, we first give in Section 1 the required properties of classical orthogonal polynomials. With this set

568

of properties at hand, Section 2 is dedicated to the construction of mixed recurrence formulas linking the polynomial P,(z) with B,-1 (x) and PA(z) with A,-z(z), which (as shown in Section 3) allow us to obtain recurrence formulas for each of the the Bezout’s pair elements. On the other hand, Section 4 contains some differential relations satisfied by A,-z(z) and B,-1 (z), which in fact are natural consequences of the hypergeometric differential equation satisfied by the starting classical family P,(z). Finally, in Section 5 the orthogonality of the Bezout pair is discussed and some extensions and open questions are proposed in the last Section 6 . 1. l . Basic properties of classical orthogonal polynomials

The hypergeometric differential equation =0,

&[Y(z)] := D(z)Y”(z)4-T(z)Y‘(z) A,Y(S)

(1)

has an unique monic polynomial solution P,(z) of degree exactly n if and only if

and A, # A, for m = 0 , 1 , 2 , . . . ,n-1, where ~ ( zand ) T(Z) are polynomials of degree at most two and one, respectively. The polynomial solutions P,(x)of (1) for A = A, are the classical orthogonal polynomials of Jacobi, Laguerre, and Hermite and satisfy the orthogonality relation

s,”

Pn(z>prn(z)e(z) dx = did,,,

>

where

e = o , 1 , 2 , ... ,

e(z) 2 0 ,

and 1 5 , , ~denotes the Kronecker’s delta. are monic, these families satisfy also the Assuming that {P,(Z))~~W three-term recurrence relation

zPn(.) = Pn+l(z)+P,P,(z)+Y,P,-l(z),

n 2 0,

P-l(Z)

= 0 , Po(z) = 1 ,

(3)

569

where Pn and yn (1) are [5]:

> 0 given in terms of the polynomials

a(.)

and

T(Z)

of

and T ~ ( Z = ) T ( Z ) + pa'(z). The family PA(,) also orthogonal in the same interval ( a , b ) but with the weight c(z) = a(z)e(z),satisfies the recurrence relation n zf'A(z) = =PA+l(z)

+ ,&PA(z)+ %PA-l(z),

n 2 1,

(6)

Pi(z) = 0,Pi(") = 1 , with

where the coefficients Pn and yn are given in Eqs. (4)and (5), respectively. As it is well-known [12], Pn(z)has exactly n real and different roots, which interlace with those of PA(,) and also with the zeros of P t l l ( z ) , which is the first associated monic polynomial of the orthogonal family Pn(z)of weight e(z) (classical or not classical) satisfying the t h r e e t e r m recurrence relation:

1.2. Structure relations and derivative representations for

classical families Classical orthogonal polynomials satisfy two types of derivativedifference relations.

570

1.2.1. Structure relations

1.2.2. Derivative representations

All these relations allow to prove next Theorems 2.1-2.4.

571

2. Recurrence relations between P , ( z ) and Bn-i(z),and between PA(z)and A,-z(z)

Starting here, we present the Bezout's identity in the form

An-2(x)Pn(x)

+ Bn-l(x)PA(z)

on,

1

A-1 = 0

BO=

,

(16)

where Rn is a constant chosen in the following convenient way

Then, Rn being this constant appearing in the Bezout's identity (16), the following theorems can be stated: Theorem 2.1. For n 2 2,

where bn is the coeficient in the structure relation (8). Theorem 2.2. For n 2 2,

where ,Bn is the coeficient of the recurrence relation (3) and bn Es the coefficient in the structure relation (8). Theorem 2.3. For n 2 2 ,

where bn is the coeficient in the structure relation (8). Theorem 2.4. For n 2 2 ,

Fn

(21)

where is the recurrence coeficient in Eq.(6) and bn is the coeficient in the structure relation (8). Relation (18), as a polynomial of degree 2 n - 2 is true, because it is true for 2 n - 1 points: the n zeros of Pn(x), using Eqs. (16) and (9), and also for the n - 1 zeros of Pn-1(x), using now Eqs. (16) and (8) (shifted to

572

n - 1). Relations (19), (20) and (21) are proved in the same way, using the appropriate relations (see [2]). For Legendre polynomials a ( % )= 1 - z2, ~ ( z = ) d ( z ) , these results were given by P. Humbert. Partial results for Gegenbauer, shifted Jacobi and Hermite polynomials appear also in [S, 7, 81.

3. The three term recurrence relations for the &-family and the A,-family Theorem 3.1. The family {Bn(z)}r=o satisfies the three-term recurrence

relation

Pn

and 7, being the coeficients of the recurrence relation (3).

Theorem 3.2. The family {A,(z)}F=o satisfies the three-term recurrence

relation n+l zAn-l(z) = -An(z) n+2 with the initial conditions

+ Fn+lAn-1(z) + Tn+lAn-,(z)

A-l(z) = 0 being

Fn

and

7,

( n 2 1) (24)

and A0 = 2 ,

(25)

the coeficients of the recurrence relation (6).

The trick for the proofs (see [2] for details) is t o transfer the recurrence relation (3) for P,(z) (respectively PA(.)) to a recurrence relation for Bn(z) (respectively A,(x)), multiplying Eq.(18) (resp. (20)) by z and expanding in order t o eliminate P,(z) (resp. PA(z))using (19) (resp. (21)). Polynomial P,(z) being monic, the constant 52, (introduced in (16)(17)) is therefore linked to the normalization of A n ( z ) or B,(z) and it appears in each relation (18)-(21). Fixing 52, = 1, as done by Humbert, defines without ambiguity the normalization of A , ( z ) and B n ( z ) . But introducing a constant which depends on n allows t o characterize the possible common roots of P,(z) and PA(z) when 52, = 0, which is only possible outside the orthogonality parameter’s domain.

573

Another possibility is to impose, as sometimes done for classical Jacobi polynomials = l),the value of A,(x) or B,(x) at some point a. In this spirit, the family B,(x)has the advantage of being easily computed ) = 0), knowing the value of P,(a) from from Eq.(18) a t a zero of ~ ( x (.(a) the Rodrigues formula for instance. Our choice (also considered in [l,21) has been to normalize the A , ( x ) (1) . family using the fact (see next section) that An--2(x) = x [P;_,(x)] Fixing therefore x = 1 and A0 = 2, the value of R, given in (17) appears. Notice that the factor b,/?-‘ in R, is equal to one in the Laguerre and Hermite cases.

(Pp”)(l)

4. Differential relations between A,(z) and B , ( z )

Multiplying by ~ ( xthe ) derivative of the Bezout’s identity (16)

and taking into account that, from Eq. (1) one has c(x)P:(x) -?-(x)P;(x) - AnPn(x),we get:

M(z)P,(z)

=

+ M(x)P;(x)= 0

and therefore, up t o a constant N : -

M ( z ) = -NP;(x),

M ( z ) = NP,(x),

with

M ( z ) := a(z)AL-,(x) - AnB,-1(~) = - N P ; ( x )

-

,

+

M ( z ) := ~ ( z ) A , - z ( x ) (T(z)BA(z) - T(z)B,-~(z) = NP,(x).

Differentiating again allows a full separation of the variables giving linear ordinary differential equations for each A n ( s )and B,(x) as follows:

+

~ ( x ) A g - z ( ~ (o’(z) ) - ~ ( x ) ) A L - z ( x-I-)AnAn-z(z) = -2NP:(x) ,

which is, up t o a constant, the equation satisfied by [P;-l(x)](l), giving the simplest choice A,-Z(Z) = [PA-.l(x)](l), and

574

The right hand sides are easily eliminated giving two fourth order linear differential equations for An-2(z) and B,-1 (x):

L [ A , - z ( z ) ] = 0 , G[&-i(x>I = 0 . 5. About the orthogonality of families A , ( z ) and Bn(s) Humbert already noticed and proved for Legendre polynomials, P,(z), that, up t o initial conditions, the three-term recurrence relation (TTRR for short) satisfied by Pn(z)and B,(z) coincides, as do so the TTRR satisfied by PA(.) and A,(x). Of course, at this period (1915-1920), before J . Favard and even precursor of Favard, the orthogonality property of polynomials satisfying TTRR was not known. However, Humbert wrote already the TTRR of P,(z) and also the TTRR of A , ( z ) and B,(z) in the Jacobi form: Z I P = JP,

Pt = (Po(z),P1(z),. . . ,P,(z)),

where J is the tridiagonal Jacobi matrix

built from the recurrence coefficients ,& and 7, of P,(z), given by Eqs.(4) and (5), respectively. The orthogonality of A, (z) being a consequence of the representation given in Section 4, let us now write the Jacobi matrix corresponding to the family B,(z) using a simple extension of a technique used in [ll]to modify upward a given Jacobi matrix. Let J(’) be the submatrix

family (associated of order two of P,(z)). corresponding to the Pi2’(x)

575

€+om the TTRR (22) of B,(z), it is clear that the corresponding Jacobi matrix JJB can be written as

where Bt = (Bo(z), B1 (z), . . . ,B,(z)) and zB = JBB. Now, A, p and u can be extracted easily from the initial conditions:

giving, from ABO

+ pB1(x) = zBO,

The important conclusion comes from the fact that p is positive and therefore the orthogonality weight of B,(z) is also positive definite. 6. Extensions: From continuous to discrete and q-discrete. Open problems

The Bezout problem treated here for the couple ( P ( z ) D , P ( z ) ) ,with D := d/dx, can be extended to other linear operator shifting down by one the degree of P ( x ) . For instance, the difference operator Af(z) := f(x 1) - f(z) (of(.) := Af(z - 1))or the q-difference operator D , f ( z ) := [f(qz) f(z)]/[(q- l)z],which give rise to two new Bezout’s identities:

+

A,-~(x)P,(z) + Bn-1(z)APn(z) =

(constant),

A,-2(2)Pn(2) + B,-1(z)DqP,(~) = \k, (constant), where now, the starting orthogonal polynomial family y(z) := P,(z) is, respectively, solution of

+

D~[Y(Z)] := g(x>AVy(z) ~ ( ~ ) A Y + ( ZAny(.) ) %[Y(Z)l

:= ‘T(4D,DI/,Y(4

+ +)D,Y(Z) + Any(.)

=0

7

=0>

with appropriate coefficients a ( z ) ,~ ( zand ) A, [l, 91. The theory is already developed in [l]giving representations of the A,(%) and B,(z) families in each of the A and D, situations and their own recurrence relations.

576

From the similarity properties of discrete and q-discrete families a with the continuous one, the following (expected) results are already obtained [ l ]in each extensions:

An-2(x) = [ V P , ( Z ) ] ( (discrete ~) case), A,-2 (x)= [D1/qPn(x)](l)(q-discrete case) . But relations equivalent to (18)-(21) are not yet written explicitly. They could be obtained from the A and D,-structure relations and from the discrete (q-discrete) difference relations [9] like in Section 1.2. 6.1. Extensions: Enlarging the familg P n ( x ) . Open

problems From the Favard theorem, the families A,-z(x) and B , - l ( z ) satisfying a three term recurrence relation are also orthogonal. Their orthogonality are linked to the orthogonality of the (classical) P,(z) family (of weight ~(x)) and the PA(z) family with orthogonality weight a ( z ) e ( z ) . For instance, the families Pi."'(x) and P?)(x)with P,(x)classical are and [ c ~ ( z ) ] (and ~~(z) still orthogonal with weight, respectively, [a(x>]."e(z) possibly common roots) could be investigated from the Bezout's identity

+

A , - T - l ( ~ ) P i k ) ( ~Bn--k-1(2)P?)(~) ) =

C2ik'r) .

(26)

Nothing is done yet in this direction, except the basic case r - k = 1 , treated here. Nevertheless fourth order linear differential equations for An-T-l (x)and Bn-k-l (x)can again be obtained like in Section 4,starting with the derivative of the Bezout's identity (26), multiplying by an appropriate power of a(.). The hypergeometric character of Eq. ( 1 ) allows to write

[a(x)]jP;+l)(x)= "j'(x)P,(x)

+ -w N

(x)P;(x),

which gives a starting relation as in Section 4:

M(j)(x)P,(x) + z(j)(x)P;(z)= 0 . It is interesting to notice that searching differential equations for A(%) and B ( x ) in the more general Bezout problem

A(z)P,(x) + B ( x ) P m ( x )=

(constant),

(27)

[l],q-discrete case was treated, but a misprint appears in (47) where the sign just before P,(z) must be minus, and also in the three foregoing equations.

577

where now P,(z) and p m ( z )are two different classical families (not linked by the derivative as above), the differential relations between A ( z ) and B ( z ) are no more linear. This comes from the fact that the derivative of (27) multiplied by u ( z ) F ( z )generates a relation like

+

0 = Mi(z)P,(z)+ Ni(z)PL(z) XTi(z)P,(z)

+ T i ( x ) F ; ( z ).

(28)

The polynomials A ( z ) and B ( z ) still appear linearly in the coefficients of (28), but elimination of the p,(z) and F;(z) families after two more derivations generates two other similar relations as (28) which creates mixed products in A ( z ) B ( z )killing the linearity. However, if we choose B,(z) = P,-l(z)in (27), the Bezout problem

can be solved immediately from a known relation linking (see e.g. [4,p.861) and t o be compared with (1) and Pn-2(z), (1) monic P,(z), P,-l(z),Pn-l(z) Eq. (16). Indeed, as already mentioned in the Introduction, it is well known that the polynomial (1) (z) satisfies the same three-term recurrence relation as P,(z) (see Eqs. (3) and (7)). If we consider

W , = Pn-l(zPn-l(z) (1) - P?Y2(z)Pn(z)( n 2 11,

Po(1) (z)Po(z)= 1,trivial manipulation of both recurrences for and P,(z) gives W, = ~,-1W,-1, and iterating this recurrence: Pn-l(z) (1)

SO that W1 =

Now from (29) and

we obtain A * ( z ) = -P?jz(z) and B * ( z ) = Pi91(z).Clearly, in case of monic Hermite polynomials P,(z) = H,(z), with H ; ( z ) = nH,-l(z), the Bezout's problems (16) and (29) are the same (0, = -nW, in this case) giving rise to the well known relation H,-,(z) (1) = [HA-l(z)](l),where [HA-, (z)](') is the monic associated of HApl(z) of degree n - 2 .

578

Acknowledgments

The work of I. Area, E. Godoy and A. Zarzo was partially supported by Ministerio de Ciencia y Tecnologia of Spain under grant BFM2002-04315C02-01, cofinanced by the European Community fund FEDER. A. Ronveaux was partly financed by the Commissariat Gknkral aux Relations Internationales de la Communaut6 franqaise de Belgique (CGRI). A. Zarzo acknowledges financial support from Ministerio de Ciencia y Tecnologia of Spain under grant MTM2005-08648-C02-01 and also from Junta de Andalucia (FQM-0207), and from UPM (AY05/11263) and A. Ronveaux thanks the Departamento de Matemdtica Aplicada I1 of Universidade de Vigo and Departamento de Matemdtica Aplicada, E.T.S. Ingenieros Industriales of Universidad Politkcnica de Madrid for kind invitations and financial supports. References 1. I. Area, E. Godoy, A. Ronveaux, and A. Zarzo. Extensions of some results of P. Humbert on Bezout's identity for classical orthogonal polynomials. J. Comput. Appl. Math. 196 (ZOOS), 212-228. 2. I. Area, E. Godoy, A. Ronveaux, and A. Zarzo. Bezout's identity, classical

orthogonal polynomials and second kind functions. Preprint 2005. 3. E. BBzout. The'orie ge'ne'rale des e'quations alge'briques. 1779. 4. T. S. Chihara. A n Introduction to Orthogonal Polynomials. (Gordon and Breach, New York, 1978). 5. E. Godoy, A. Ronveaux, A. Zarzo and I. Area. Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case. J. Comput. Appl. Math. 84 (1997), 257-275. 6. P. Humbert. Expression de la fonction de Legendre de seconde espke. C. R. 165 (1917), 759-761. 7. P. Humbert. Sur deux polynbmes associCs aux polynbmes de Legendre. C. R. 167 (1918), 18-20. 8. P. Humbert. Sur deux polyn6mes associBs aux polynbmes de Legendre. Bulletin de la Socie'te' Mathe'matique d e fiance 47 (1918) 120-151. 9. A,. Nikiforov and V. Ouvarov. Ele'ments d e la the'orie des fonctions spe'ciales. (Editions Mir, Moscow, 1976). version 2.1.5, Bordeaux, 2004, 10. PARI/GP, http://pari .math.u-bordeaux.fr/. 11. A. Ronveaux and W. Van Assche. Upward extension of the Jacobi matrix for orthogonal polynomials. J. Approximation Theory 86(3) (1996), 335-357 . 12. G. Szego. Orthogonal Polynomials. 4th ed. (American Mathematical Society, Providence, 1975). 13. J. Tannery. LeCons d'algdbre et d'analyse des e'lkves des classes de mathe'matiques spe'ciales. T. I . (Gauthier-Villars, Paris, 1906).

579

INFORMATION ENTROPY OF GEGENBAUER POLYNOMIALS

JULIO I. DE VICENTE Departamento de Matema'ticas Uniuersidad Carlos 111 d e Madrid Auda. de la Uniuersidad 30, E-28911 Leganks, Madrid, Spain E-mail: [email protected]. es SILVIA GANDY Fakultat fur Mathematik Technische Uniuersitat Munchen Boltzmannstr. 3, D-85747Garching b. Munchen E-mail: [email protected] JORGE SANCHEZ-RUIZ Departamento de Matema'ticas Uniuersidad Carlos 111 de Madrid Auda. de la Uniuersidad 30, E-28911 Leganb, Madrid, Spain and Instituto Carlos I de Fisica Tedrica y Cornputacional Uniuersidad de Granada, E-18071 Granada, Spain E-mail: [email protected] We present a novel analytical approach to the evaluation of the information entropy of Gegenbauer polynomials of parameter A. This method allows us to express the entropy in terms of finite sums in the case X E N. Furthermore, we obtain closed formulas when X = 1 , 2 , 3 .

1. Introduction According to Shannon's information theory', the only rigorous measure of the uncertainty or lack of information associated to a continuous probability distribution X with density function p(z) is the entropy

H(X)= -

J

p ( z )logp(z)dz .

580

In particular, when p(x) is the single-particle probability density for position of a quantum system, H ( X ) is the only rigorous measure of the uncertainty in the localization of the particle in position space. The momentum entropy H ( P ) can be defined likewise from the single-particle density of momentum y(p). The sharp i n e q ~ a l i t y ~ > ~

H(X)

+ H ( P ) 2 log(7reh)

(2)

provides a quantitative formulation of the position-momentum uncertainty principle which is stronger than the standard Heisenberg inequality. For many important quantum systems, such as D-dimensional harmonic oscillator and hydrogen atom, the calculation of position and momentum information entropies involves the evaluation of integrals of the form

where {p,(x)} denotes a polynomial sequence (degp,(x) = n) orthogonal on [ q b ] C R with respect to the weight function ~ ( z ) During . the last decade there has been an intense activity in the study of these integrals, motivated not only by their relevance to quantum physics but also by their close relationship to other interesting mathematical objects, such as the LP-norms or the logarithmic potentials of the polynomials p,(x). A survey on the state-of-the-art in this field up to 2001 can be found in Ref. 4. The calculation of the entropic integrals E(p,) is generally a very difficult task, and in most cases only asymptotic results are known4. In the Jacobi case, an efficient algorithm for the numerical evaluation of E(p,) has been developed very recently5. Closed analytical formulas for E(p,) are only known for a few particular cases of the Gegenbauer or ultraspherical polynomials CiA’.We recall that when X > -1/2 these polynomials are orthogonal on [-1,1] with respect to w ( x ) = (1 - x 2 ) X - - 1 / 2 ,

The information entropies of the Gegenbauer polynomials, on which we focus in the present contribution, are thus given by

When X is a non-negative integer or half-integer number, these integrals are especially relevant since they appear in the calculation of information en-

581

tropies in both position and momentum spaces for any quantum-mechanical system with a central potential in D-dimensional ( D 2 2) pace^?^-'. 2. Previous Results

As mentioned before, closed analytical formulas for E(p,) are only known for certain Gegenbauer polynomials. The simplest particular cases of these polynomials are the Chebyshev polynomials of the first and second kind,

For both of these families, information entropies can be computed in closed analytical form, the results being6i7

0

E(U,) = l! 2

(-

1 n+l

ifn=O -

(7)

1).

The following general expression for the entropy of the orthonormal Gegenbauer polynomials of integer parameter (1 E N) was obtained in Ref. 9:

EL')

where s,~ and r,1 are known constants, and the auxiliary polynomials P and H are defined from the family {P,}, generated by the recurrence relation ~ k + l ( z= )

( 2 ~ - 2 k - 3 ) z ~ ~ ( z ) - ( n + k + l ) ( n + 2 1 - k - l ) ( l - z ~ ) ~ k - l ( z ) (10)

from the initial values P-l(z) = 0 and PO(.) = 1, through the formulas

n

21-2

-t j ) ,

(11)

H ( z ) = c(-1)sP,1(z)Pz1-s--3(z).

(12)

P ( z ) = PZl-2(57) = an1

.(

j=1

21-2

s=o

The explicit expression of the polynomial P was found later onlo. Regretfully, Eq. (9) is not easy to use in practice. Furthermore, it is not a completely analytical formula save for small values of 1, since the zeros [j of the polynomial P have to be determined numerically when 1 2 6.

582

However, Eq. (9) allowedg to find a closed expression of the entropy in the case 1 = 2, 2(n

+ l ) ( n + 3) log(n + 1)+ n3

-

5n2 - 29n - 27 n+2

3. Our Approach

It has been recently shown" that the entropy of both Chebyshev polynomials can be easily obtained by direct calculation of the corresponding integrals using the well-known trigonometric representations

+

sin(n l ) 8 sin8 ' with x = cos8. Motivated by this observation, here we aim at evaluating E(CLA')for general values of the parameter X using representations of the same kind for the Gegenbauer polynomials. Due to space restrictions, we present our results without proof; a more detailed exposition will be given elsewhere12.

T, (cos 8) = cos n8

,

U,(COS8) =

3.1. Trigonometric Representations f o r Gegenbauer

Polynomials The most widely known trigonometric representation of the Gegenbauer polynomials is (see e.g. Ref. 13)

m=O

where

Another representation, due to S ~ e g O ~is~ t ~ ~ ,

c p (COS 8) = where

$1

(sin o

00

~ c y ~ ~ ~ s i n ( n + 2 v + 1X) >8O, , X$N, (17)

) ~ v=o ~ - ~

583

At first sight, this representation seems to be less useful than the previous one, because it contains infinitely many terms. Moreover, it is supposed not to hold when X E N. However, it is not difficult to prove that this is not the case. Proposition 3.1. The Szego representation (17) holds true when X E I n this case, it reads

N.

The importance of this remark lies in the fact that when X E N the sum in (17) terminates after a finite number of terms. This suggests that Szego's representation may be useful to evaluate the entropy of Gegenbauer polynomials of integer parameter, so in what follows we shall assume that X E N. 3.2. Evaluation of the Entropic Integral

With the change of variable x

= cos 8,

the integral (5) takes the form

Using the representations (15) and (19) for each of the two Gegenbauer 2 polynomials in (CiA)(cos6 ) ) , after some algebraic manipulations Eq. (20) can be rewritten as

where

1 7r

I ~ := A

cos(2m~)log ( c i ~ ) ( c oel> s 'de .

(22)

+

Now we need to evaluate the integrals IZA, with 0 I m 5 n A. This can be done by means of complex integration techniques, using the Szego representation (19) for the Gegenbauer polynomial inside the logarithm. The result is summarized in the following theorem:

Theorem 3.1. For X E N,

IA? = 27rlog

( n + X - l)! (A - I)! n!

(23)

584

and, when m 2 1,

The fact that the Szego representation (19) has a finite number of terms plays an essential role in the proof of Theorem 3.1. Although we are mainly interested in evaluating the integrals when X E N, it is worth pointing out that these integrals can be calculated in a similar way for all possible values of A, using the standard trigonometric representation (15) instead of the Szego representation for the Gegenbauer polynomial inside the logarithm. This generalization is contained in the next theorem.

ICL

Theorem 3.2. For X

E

R,X > --+,

and, when m 2 1,

In order to carry out the sums in (21), the next step is to obtain closed formulas for the integrals IC)n with m 2 1; fortunately, we only need them for m 5 n A. Despite its greater generality, Theorem 3.2 turns out to be less useful than Theorem 3.1, because Eq. (24) expresses the integrals in terms of the logarithm of a polynomial of 2X terms, while in Eq. (26) they are given in terms of the logarithm of a polynomial of n 1terms. As we shall see, the difficulty of obtaining a closed formula for the derivatives of such functions increases with the number of terms in the polynomial. Therefore, if we want an expression of I,,(A) , for a fixed value of X and any

+

+

n E N Theorem 3.1 is more helpful, particularly for small values of A. In the case X = 1we readily notice from Eq. (24) that, if 1 5 m 5 n + l ,

585

When X 2 2, the polynomial inside the logarithm has more terms and the above trick does not work. However, we can obtain closed formulas for the derivatives in (24) by means of Fa& di Bruno's formula for the derivatives of the composition of functions, which states that (see e.g. Ref. 15)

where the inner summation is extended over all partitions satisfying

Due to the last condition, this formula is not easy to apply. However, using (A) when X = 2 and X = 3, it we have been able to find closed formulas for Im,n

1221~= m [ 3 -

(-TI n+3 n+l

n+3 n+l

+7T-

Sm,n+Z,

1I m

I n + 2 , (30)

( n+ l ) ( n + 5) + iJ3(n + l ) ( n + 5) 12"ln= m [5-2%( ( n+ 1)(n+ 2) ( n+ 4)(n + 5) 6m,n+3r 1 I m I n + 3 , ( n + l ) ( n + 2) -7T

while for X

2 4 the integral can be expressed as a multiple sum,

3.3. Closed Form Expressions f o r the Entropy

Equations (23), (27), (30) and (31) enable us to derive closed formulas for E(CAA')when X = 1,2,3. For X = 1, after substitution of the corresponding

586

values of

&A',

av,n (A) and

d2,'n

(see Eqs. (16) and (18)), Eq. (21) reduces to

which recalling Eqs. (23) and (27) immediately leads t o (8). When X Eq. (21) takes the form

n

n+ 1

+ 1)(n+ 3)IA:A

-

4

mIg,)n - ( n

= 2,

+ 1)

m=l

(34) from which Eq. (13) is recovered by using (23) and (30) together with the well-known formula for the sum of a geometric series, x(l - xc") * l-x

m=l In the case X = 3, Eq. (21) can be rewritten as 1 E(CL3))= - - [(n l ) ( n 2)(n 4)(n 128

+

-(n

+

12

+ 5)I(g

+ 2)21f23,n]

n+2 -

+

+

(35)

m(n2

+ 6n + 7 - 2m2)121,.

(36)

m=l

Substituting (23) and (31) into the above expression, we encounter again the geometric sum, as well as a sum of the form Emm2xm. Using Eq. (35) and the summation formula16 n x ( 1 + x) - xn+l [ ( n 1)2- (2n2 2n - 1)" n2x2J m2xm= I (37) (1 - x)3 m= 1 after a tedious but straightforward calculation we arrive at the following (3) closed formula for E(Cn ), which is a new result:

+

C

{

+

+

+

+

+

+

E(CL3))= -2( n l)(n 2)(n 4)(n 5) log 128 n5 - 16n4 - 269n3 - 1200n2- 2102n - 1250

+

+

+

2(n 5)2 (n+2)(n+3)

~

[

fn+:l)(n

+ 5) + id/3(n+ 1)(n+ 5) (n

+

2n2 + 13n + 14 - i(n l)(n

+ l ) ( n + 2)

.}I)-/

+ 6)

(38)

587

When X 2 4, combination of Eqs. (21) and (32) provides an expression for the entropy E(CAA’)in terms of finite sums. Unlike Eq. (9), this expression is completely analytical for all X E N, which makes it suitable for symbolic computation. For instance, a Maple implementation of the formula enabled us t o obtain the closed analytical expressions for E(CA4)) t h a t are displayed in Table 1.

Acknowledgments This research was developed during a stay of the second author (S.G.) at the Departamento de MatemAticas of the Universidad Carlos I11 de Madrid, funded through the European Union Socrates/Erasmus Programme. The work of the third author (J.S.-R.) was partially supported by the Direccibn General de Investigacibn (MEC) of Spain under grants BFM 2003-06335C03-02 and FIS2005-00973, and the Junta de Andalucia research group FQM-0207.

References 1. C.E. Shannon, Bell Syst. Tech. J. 27, 379; 623 (1948) [Reprinted in: “The Mathematical Theory of Communication”, eds. C.E. Shannon and W. Weaver (Univ. Illinois Press, Urbana, 1949)]. 2. W. Beckner, Ann. Math. 102,159 (1975). 3. I. Bialynicki-Birula and J. Mycielski, Commun. Math. Phys. 44, 129 (1975). 4. J.S. Dehesa, A. Martinez-Finkelshtein and J. SBnchez-Ruiz, J. Comput. Appl. Math. 133,23 (2001). 5. V.S. Buyarov, J.S. Dehesa, A. Martinez-Finkelshtein and J. SBnchez-Lara, SIAM J . Sci. Comput. 26,488 (2004). 6. R.J. YBiiez, W. Van Assche and J.S. Dehesa, Phys. Rev. A 50, 3065 (1994). 7. J.S. Dehesa, W. Van Assche and R.J. YBiiez, Meth. Appl. Anal. 4, 91 (1997). 8. R.J. YGez, W. Van Assche, R. GonzBlez-FBrez and J.S. Dehesa, J . Math. Phys. 40, 5675 (1999). 9. V.S. Buyarov, P. L6pez-ArtBs, A. Martinez-Finkelshtein and W. Van Assche, J. Phys. A: Math. Gen. 33,6549 (2000). 10. J. Skchez-Ruiz, J. Phys. A : Math. Gen. 36,4857 (2003). 11. J. SBnchez-Lara and J. SBnchez-Ruiz, preprint (2005). 12. J. I. de Vicente, S. Gandy and J. SBnchez-Ruiz, preprint (2005). 13. G. Szego, “Orthogonal Polynomials” (Am. Math. SOC.Colloq. Publ., vol. 23), 4th edn. (American Mathematical Society, Providence, RI, 1975). 14. G. Szego, Math. Ann. 110,501 (1934) (in German). 15. S. Roman, Amer. Math. Monthly 87,805 (1980). 16. E.R. Hansen, “A Table of Series and Products” (Prentice-Hall, Englewood Cliffs, NJ, 1975).

588

Table 1. Exact and numerical values of the entropy E(CAA’)for X = 4 and 1 5 TI 5 15. n Exact value Numerical value -7xln(2)

1

119 + -240 x

-88.862

2 75 --7rln(20) 2

3

4

17355685 6449434961 -3003 K In (84) + 1058158080 8

--1365 x ln (120) - 1396715852287A 139218750000 2

7 -

8

z

64

x ln(165) -

-135295.739

1779685691911133495 1202109806542848

-202952.031

36234350694889865223938313068785 15613637127259094259005915136

-296836.555

-

In (455) -

x In (680) -

-3303 1.075

25537984326378849719971131 28270687046875000000000

In (286) -

-9975 x In (560) -

-56525 4

-18974.368

-87616.538

-4641 x In (364) -

13

-10296.556

31458443588344487293819 60436675052957701680

11

64

-5206.005

-54866.421

-12155 x 4 _-440895 K

1200329915 9135984

-2404.173

325291539600149215255 1172732412725203616

10

15

24757176334716125 493018566815808

- 1 9 2 5 ~In (220) -

9

-338.714 -983.613

2

6

14

+ -95 x 21

64

5

12

-13.685

-19635 x In (816) -

130243656594168370141034405A 37115886521993021558784

-424587.139

589

HIGHER GENUS AFFINE LIE ALGEBRAS OF KRICHEVER - NOVIKOV TYPE

MARTIN SCHLICHENMAIER University of Luxembourg, Campus Limpertsberg, 162A, Avenue de la Faiencerie, L-1511 Luxembourg, Grand-Duchy of Luxembourg E-mail: martin.schlichenmaierOuni.1u Classical affine Lie algebras appear e.g. as symmetries of infinite dimensional integrable systems and are related to certain differential equations. They are central extensions of current algebras associated to finite-dimensional Lie algebras g. In geometric terms these current algebras might be described as Lie algebra valued meromorphic functions on the Riemann sphere with two possible poles. They carry a natural grading. In this talk the generalization to higher genus compact Riemann surfaces and more poles is reviewed. In case that the Lie algebra g is reductive (e.g. g is simple, semi-simple, abelian, ...) a complete classification of (almost-) graded central extensions is given. In particular, for g simple there exists a unique non-trivial (almost-)graded extension class. The considered algebras are related to difference equations, special functions and play a role in Conformal Field Theory.

1. Introduction Classical current algebras (also called loop algebras) and their central extensions, the affine Lie algebras, are of fundamental importance in quite a number of fields in mathematics and its applications. These algebras are examples of infinite dimensional Lie algebras which are still tractable. They constitute the subclass of Kac-Moody algebras of untwisted affine type. If one rewrites the original purely algebraic definition in geometric terms the classical current algebras correspond to Lie algebra valued meromorphic functions on the Riemann sphere (i.e. on the unique compact Riemann surface of genus zero) which are allowed to have poles only at two fixed points. If this rewriting is done, a very useful generalization (e.g. needed in string theory) is to consider the objects over a compact Riemann surface of arbitrary genus with more than two points where poles are allowed. The

590

main problem is t o introduce a replacement of the grading in the classical case, which is necessary to construct highest weight and Fock space representations. This is obtained by an almost-graded structure (see Section 3), a weaker structure but still strong enough t o do the job. Furthermore, to obtain representations of certain types one is forced to pass over t o central extensions. Such objects (vector fields, functions, etc.) and central extensions for higher genus with two possible poles were introduced by Krichever and Novikov (31 and generalized by me to the multi-point situation [ 6 ] . These objects are of importance in a global operator approach to Conformal Field Theory [lo],[11]. More generally, the current algebra resp. their central extensions, the affine algebras, correspond to symmetries of infinite dimensional systems. Their q-deformed version (i.e. the quantum affine algebras) are in close connection with difference equation and with special functions. There is a very interesting direct relation to difference equation. Krichever and Novikov constructed higher genus analogues of Baker Akhiezer difference functions which are eigen functions of suitable difference equations. Starting from these functions, representations of higher genus affine algebras associated to the Lie algebra s[(2, C)are obtained. These are related to the two-dimensional Toda lattice [4]. In this write-up of the talk I report on uniqueness and classification results for higher genus multi-point affine Lie algebras which I recently obtained. In particular, it turns out that for the current algebra associated to a finite-dimensional simple Lie algebra (e.g. for s [ ( n , C)) there exists up to equivalence and rescaling a unique non-trivial central extension which extends the almost-grading of the current algebra. The proofs can be found in [9]. There also further references, historical remarks and corresponding results for the Lie algebras of Lie algebra valued meromorphic differential operators can be found. The results depend also on a complete classification of central extensions of scalar functions, vector fields and differential operators of Krichever-Novikov type obtained in [8]. I am indebted to Paul Terwilliger who asked me to supply explicit examples of such algebras. They can be found in Section 5 and Section 6 .

2. The classical situation and some algebraic background

Let us first consider the nowadays classical affine Lie algebras. Let g be a finite-dimensional complex Lie algebra. A special example of fundamental

591

importance is given by the algebra of traceless matrices

sI(n,C ) := {A E M a t ( n ,C ) I tr(A) = 0},

(1)

with [A, B] := A B - BA the commutator as Lie product. The current algebra g (sometimes also called loop algebra) is obtained by tensoring g by the (associative and commutative) algebra C [ Z , Z - ~of] Laurent polynomials, i.e. g = g I8 C[z,z-l] with the Lie product [z 18 zn,y 18 zm] := [z, y] 8 z ~ + ~z,,y E g, n, m E

Z.

(2)

If g is a matrix algebra, then B can be considered as matrices with Laurent polynomials as entries, e.g. a typical element of a(2,C ) can be written as

b(z,z-1)

(“(zl z-1) c ( z , z-1)

--a(&

)’

z-1)

(3)

where a, b, c are polynomials in z and 2 - l . By setting deg(z8z”) := n the Lie algebra is graded (see (2)). Clearly, g is an infinite dimensional Lie algebra. These algebras are candidates for symmetry algebras of systems with infinitely many independent symmetries. Unfortunately, in the process of constructing representations of certain types (e.g. highest weight representations) one is forced to “regularize” certain natural actions. As a result one obtains only so called projective representations, which in turn define honest representations of certain central extensions of 8. What is a central extension of a Lie algebra V ? As vector space we take = V @ C. We set t := ( 0 , l ) and d := ( a ,0) and consider the following product

c

h

[d, 61 := [a,b]

+ $(a, b) t ,

[ii]t]= 0 ,

c

(4)

with $ : V x V -+ V a bilinear map. Now is a Lie algebra if and only if $ is a Lie algebra two-cocycle of V (with values in the trivial module C). The cocycle conditions are

$(a, b) = -$(h a ) ,

$ ( [ a ,bl, c) + $ ( [ h a ) + $([c, a ] ,b ) = 0,

(5) for all a , b, c E V . Two central extensions of the same algebra are called equivalent if they are the same up to some change of basis of the type d H 6 = ( a ,4(a)). In more precise terms, given two extensions defined by $1 and $2 respectively, the two extensions are called equivalent if there exists a linear form 4 : V -+ C such that $1(a, b )

4

- $2(a,

b) =

1

bl).

(6)

592

In other words, the difference is a Lie algebra cohomology coboundary.

Fact. The set of equivalence classes of central extensions is via (4) in 1 to 1 correspondence to the space of Lie algebra two-cohomology classes H2(V,C)= Z2(V,C ) / B 2(V,C)(cocycles modulo coboundaries). How do we obtain central extensions for our current algebra? Let Q be an invariant, symmetric bilinear form for g. Invariance means that a ( [ u , b ] , c )= Q ( U , [b,c])for all a,b,c E g. For a simple Lie algebra the Cartan-Killing form is up to a rescaling the only such form. In particular for sl(n,C) it is given by a ( A , B ) = tr(AB). Then a central extension = @ Ct is defined by

a

[z @ zn, y I8 z"] = [z, y ] @ zn+"

-

a ( z ,y ) . n .",S

. t.

(7)

To avoid cumbersome notation I dropped the in the notation. It is called the (classical) affine Lie algebra associated to g. By setting degt := 0 (and using n = deg(z @ z n ) ) the affine algebra is a graded algebra. If the finite dimensional Lie algebra g is simple then defined via (7) is up to equivalence of extensions and rescaling of the central element the only non-trivial central extension of 3. In this case the algebras are exactly the Kac-Moody algebras of untwisted affine type [2]. A

3. The higher genus case

Before we can extend the construction to higher genus we have to geometrize the classical situation. Recall that the associative algebra of Laurent polynomials C [ z ,2-'1 can equivalently be described as the algebra consisting of those meromorphic functions on the Riemann sphere (resp. the complex projective line P'(C)) which are holomorphic outside z = 0 and z = 00 ( z the quasi-global coordinate). The current algebra jj can be interpreted as Lie algebra of g-valued meromorphic functions on the Riemann sphere with possible poles only at z = 0 and z = 00. The Riemann sphere is the unique compact Riemann surface of genus zero. From this point of view the next step is to take X any compact Riemann surface of arbitrary genus g and an arbitrary finite set A of points where poles of the meromorphic objects will be allowed. In this way we obtain the higher genus (multi-point) current algebra as the algebra of gvalued functions on X with only possibly poles a t A. We need gradings, central extensions etc. For this goal we split A into two non-empty disjoint subsets I and 0, A = IUO. In the interpretation of string theory, I corresponds to incoming

593

free strings and 0 to outgoing free strings. Let K be the number of points in I . See Figure 1 for an example given by a Riemann surface of genus two with I = { P I ,P2} and 0 = { Q l } .

Figure 1. The higher genus, multi-point case

Let A be the associative algebra of functions meromorphic on X and holomorphic outside of A. In some earlier work [6] I introduced {An,p

I n E z,P = 1,.. . ,K}

(8)

a certain adapted basis of A. For the exact definition I refer to this publication. Here we only note that

ordp,(An,p)= n

+ 1- d,:

VPi E I.

(9)

For genus zero and I = {0}, 0 = {co} we get A,,p = 2,. Let A, := (A,,p I p = 1,.. . ,K) be the K-dimensional subspace of A. We have A = enEzAn and there exist constants L1, L2 (independent of n and m ) such that n+m+L2

A,.A,

C

@

Ah,

Vn,mEZ.

(10)

h=n+m-L1

We call the elements of A, homogeneous elements of degree n. As long as L1 and La cannot to be chosen to be 0 the algebra is not honestly graded. It is only almost-graded. In a similar way one introduces almost-gradedness for Lie algebras. This notion was introduced by Krichever and Novikov [3] (they called it quasi-graded) and they constructed such an almost-grading in the higher genus and two point case. To find an almost-grading in the multi-point case is more difficult. This weaker grading is enough to introduce and study highest weight representations. As a remark aside: with a special choice of basis one has L1 = 0 and the L2 depends in a known manner on the genus g and the number of points in I and 0 [6].

594 -

The higher genus multi-point current algebra 3 is the tensor product

g = g @ A with the Lie product

[ x @ f , Y @ ’ 9 1 =[ X , Y I @ ( f . 9 )

(11)

and almost-grading

4. Central extensions in higher genus

The next task is to construct central extensions and to study the question of uniqueness.

Proposition 4.1. ([9]) Let CY be an invariant, symmetric bilinear form of g and C a closed contour o n X not meeting A, then $a,C(X@f,Y@’9)

:=a(x,Y)q+J

(13)

is a Lie algebra two-cocycle for the current algebra 3. Hence, it defines a central extension &c. Consequently, there exist central extensions for 8. But contrary to the classical situation, even if g is a simple Lie algebra, there will not be a unique nontrivial cocycle class. If we choose essentially different contours C the $ja,cdefine essentially different central extensions &,c. But in the classical situation we were able to extend our grading of 3 to g by assigning a degree to the central element t. This will not necessarily be true for all cocycles of the form (13).

A

Definition 4.1. A 2-cocycle is called local if there exist TI and that

$(Bn,Bm)#O

==+ T 2 I n + m I T i .

T2

such (14)

Given a local cocycle $ defining a central extension, then by setting deg(t) = 0 (or any other number) the almost-grading of 3 extends to the central extension. Vice versa, if such an extension of the almost-grading exists, the defining cocycle will be local. We use H;o,(B, C) to denote the subspace of 2-cocycle classes containing a representative which is local. In general, the cocycles $a,care not local. But if we choose as integration contour a smooth contour Cs separating the points in I from the points in 0 and which is of winding number 1,

595

then it can be shown that &,cSis a local cocycle. In fact its values can be calculated as

We call any such Cs a separating cycle.

Theorem 4.1. /9] Let g be a finite-dimensional simple Lie algebra, g 18A its (higher genus) current algebra, then dimH;oc(g,C)

=

1,

=

(16)

and a basis is given by the class of (15), where Cs is a separating cycle and Q is a multiple of the Cartan-Killing form. I n particular, there exists up to equivalence and rescaling a unique almost-graded non-trivial central extension of 3. As a side-result we obtain that every local cocycle is cohomologous to a geometric cocycle of the form (15).

Remark 4.1. The cocycles coming from Fock space representations and other type of representations are local. Hence we obtain that there exists a unique equivalence class (up to rescaling of the central element) of central extensions coming from these representations. In [9] the more general situation of reductive Lie algebras is considered. Recall that a finite-dimensional Lie algebra g is reductive if and only if it is the direct sum (as Lie algebra) g = go 3 '3 81 . . . €B B M ,

go abelian ,g1,g2,. . . ,gM simple .

(17)

If the abelian summand go is missing then g is semi-simple. In the semi-simple case it is shown that every local 2-cocycle is cohomologous to a cocycle of the type (15) where Q is an arbitrary linear combination of the individual Cartan-Killing forms of the summands (trivially extended to the rest). Vice versa, such cocycles are local. In particular, we get dim H;oc(g, C)= M In the reductive case we have to add another condition. We denote by C the Lie algebra of meromorphic vector fields on X which are holomorphic outside of A . A 2-cocycle is called C-invariant if ~ ( 5 ~ ( e . f ) , Y ~ g ) + ~ ( x ~ f f , y= ~ O , ( e vf,g . g ) )€

4

V e ' e L . (18)

596

Cocycles of the form (13) are C-invariant. In [9] it is shown that every C-invariant local cocycle is cohomologous to (15). In particular dim H'&=(g, C) = M Here the index C denotes the classes containing cocycles as representatives which are C-invariant, and m = dimgo. In the semi-simple case there is no need to pose explicitly C-invariance as in this case in every local cocycle class there is a unique C-invariant representative. Again, the condition of C-invariance is automatic if the representations under consideration are representations coming from the larger algebra of Lie algebra valued differential operators.

+q .

Example 4.1. The Lie algebra of traceless matrices sl(n,C) is simple. Hence, the unique non-trivial almost-graded central extension is given (up to equivalence and rescaling) by the cocycle

Example 4.2. The Lie algebra of all matrices gI(n,C) is the direct sum gl(n,C) = 5(n,C)@sl(n,C), where 5(n,C) is the abelian summand of scalar matrices. In particular gI(n,C) is a reductive Lie algebra. Following the general results dimH&,(gI(n, C), C) = 2. A basis is given by the elements $1 (19) and $2(A 8 f l B 8 g) = tr(A) . tr(B)

(20)

5. An example: The three-point genus zero case Let us consider the Riemann sphere S2= P1(C)and the set A consisting of 3 points. Given any triple of 3 points there exists always a n analytic automorphism of S2mapping this triple to { a , -a, co}, with a # 0. In fact a = 1 would suffice. Without restriction we can take

I := { a , - a } ,

0 := {co}.

Due to the symmetry of the situation it is more convenient to take a symmetrized basis of A: A2k := (Z - u)'(z

+ a)k ,

+

A2k+l := Z(Z - u ) ~ ( z u ) ~ ,

k

E Z. (21)

It is shown in [7] that it a basis. By more or less direct calculations one can show the structure equation for the current algebra

597

Again a = 1 could be set. The reason to keep a is that it can be seen that if we vary a over the affine line we obtain for a = 0 the classical current algebra. In particular, this family gives a deformation. In [l]it was shown that this deformation is a geometrically non-trivial deformation despite the fact that for g simple, 3 is formally rigid, i.e. there does not exists a nontrivial formal deformation. This effect is peculiar to infinite dimensional Lie algebras and is discussed in detail there. For the central extension & , a , ~ of Section 4 we obtain the defining cocycles (see [l],A.13 and A.14)

with

1

n, m even,

S,,

n, m different parity,

AndAm =

-nSGn

+ a2(-n + 1)6;n+2,

n, m odd.

(24) Of course, given a simple Lie algebra g with generators and structure equations the relations can be written in these terms. For sI(2, C) with the standard generators

h :=

(k

!l)

, e

:=

(: k) ,

f

:=

(; :)

we set en := e @ A,, n E Z and in the same way f n and hn. Recall that a(%, y) = t r ( z . y). We calculate

[en,f m l

hn+m, =

[hn em] =

hn+m

+~ ~ h ~ +

2en+mr

1

[hn7 fml =

n or m even n ~ and - 2m , odd

2en+m

+ 2a2en+m-2,

-2.f n+m -2

1

fn+m - 2a2fn+m-z1

(25)

n or m even n and m odd n or m even n and m odd

(27)

For the central extension we get

[en,f m l =

n or m even hn+m - nSgn1 hn+m a2hn+m-2 - nSkn - a 2 ( n- 1 ) 6 ~ " + ~ n, and m odd

+

(28) The other commutators stay the same.

598

6. An example: The torus case

Let T = @ / Lbe a complex one-dimensional torus, i.e. a Riemann surface of genus 1. Here L denotes the lattice L = ( 1 , ~ ) with z im T > 0. The field of meromorphic functions on T is generated by the doubly-periodic Weierstrai3 p function and its derivative pr fulfilling the differential equation

(fJ

r 2

= 4(63 - el)(@- e 2 ) b - e3) = 4P3 - Q263 - Q3.

with the ei pairwise distinct and given by 1 7 7+1 = el, = e2, p ( 7 ) = e3,

~(5)

~(5)

el

+ e2 + e3 = 0.

(29)

(30)

The function 63 is an even meromorphic function with poles of order two at the points of the lattice and holomorphic elsewhere. The function pr is an odd meromorphic function with poles of order three a t the points of the lattice and holomorphic elsewhere. It has zeros of order one a t the points 1 / 2 , 7 / 2 and (1 7)/2 and all its translates under the lattice. We consider the subalgebra of functions which are holomorphic outside of 2 = 0 and Z = 1/2. As shown in [7] a basis is given by

+

Azk

=

(P- el)k ,

I

A2k+l = 5 p r .(P-

k

E

Z.

(31)

See also [5] for a similar result in the vector field algebra case. The following is shown in [l]

Y] @ An+m, [x,Y] @ An+, + 3e1[z, Y] 8 An+m-2

1x1

[x@Anlg@AmI =

+(el - e2)(2e1

+ e2)[2, y] 8 An+m-4,

n or m even,

n and m odd.

(32) If we let el and e2 (and hence also e3) go to zero we obtain the classical current algebra as degeneration. The cocycle defining the central extension is given by ([l], Thm. 4.6) 1 Y(Z An, Y @ Am) = a ( x ,Y) . % AndAm (33)

S,,

with

-nSkn,

1

%

S,,

AndAm =

01

-726;” (+(el

n, m even,

n, m diff. parity,

+ 3el(-n + - e2)(2el + ez)(-n + 2 ) 6 ~ ~ +n,~m, odd. (34)

599

References 1. Fialowski, A., and Schlichenmaier, M., Global geometric deformations of current algebras as Kricheuer-Nouikou type algebras. Comm. Math. Phys. (2005), DO1 10.1007/~00220-005-1423-5. 2. Kac, V.G., Infinite dimensional Lie algebras. 3rd ed., Cambridge University Press, 1990. 3. Krichever, I.M., and Novikov S.P., Algebras of Virasoro type, Riemann surfaces and structures of the theory of solitons, Funktional Anal. i. Prilozhen. 21 (1987), 46-63. 4. Krichever, I.M., and Novikov, S.P., Two-dimensionalized Toda lattice, commuting difference operators, and holomorphic bundles, Russ. Math. Surv. 58:3 (2003), 473-510 5 . Ruffing, A., Deck, Th., and Schlichenmaier, M., String branchings on complex tori and algebraic representations of generalized Kricheuer-Novikou algebras. Lett. Math. Phys. 26), 23-32 (1992). 6. Schlichenmaier, M., Central extensions and semi-infinite wedge representations of Kricheuer-Nouilcou algebras for more than two points, Lett. Math. Phys. 20 (1991), 33-46. 7. Schlichenmaier, M., Degenerations of generalized Kricheuer-Nouikou algebras on tori, Jour. Math. Phys. 34, 3809-3824 (1993). 8. Schlichenmaier, M., Local cocycles and central extensions for multi-point algebras of Krichever-Novikov type, J. reine angew. Math. 559(2003), 5394. 9. Schlichenmaier, M., Higher genus affine Lie algebras of Krichever-Novikov type, Moscow Math. Jour. 3 (2003) 1395-142. 10. Schlichenmaier , M., Sheinman, 0.K ., Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, Russian Math. Surv. 54(1999), 213-250. 11. Schlichenmaier, M., Sheinman, O.K., Knizhnik-Zamolodchikov equations for positive genus and Krichever-Novikov algebras, Russian Math. Suru. 59(2004), 737-770.

600

ASYMPTOTIC TRICHOTOMY OF SOLUTIONS OF A CLASS OF EVEN ORDER NONLINEAR NEUTRAL DIFFERENCE EQUATIONS WITH QUASIDIFFERENCES

E WA SCHMEIDEL Institute of Mathematics, Faculty of Electrical Engineering, Poznari University of Technology ul. Piotrowo 3a, 60-965 Poznajl, P o l a n d e-mail: [email protected]

We consider a class of higher order nonlinear neutral difference equations with quasidifferences where sequence ( p n ) is a real sequence. The classification of nonoscillatory solutions of this equation are obtained. For a class of even order nonlinear neutral difference equations, dependent on ( p n ) , the conditions under which the eventually positive solutions of the equation can be classified into three nonempty distinct categories are given. We present sufficient condition under which the equation has a solution which converges to zero as well necessary and sufficient conditions under which the equation has a solution which tends to a nonzero constant and which diverges t o infinity. Keywords Nonlinear, neutral, difference equation, quasidifferences, higher, even, odd order AMS Subject classification 39A10

1. Introduction

Consider the difference equation

+

+

where lc is a positive integer and n E N = (7 1,7 2, ...}, and 7, a are a positive integers 77 = m a x ( ~a). , Here A is the forward difference operator defined by Ax, = x,+1 - x,, (a,(4), i = 1 , 2 , 3 , ...lc - 1 are sequences of positive real numbers, and (p,) is a sequence of real numbers. Function f : N x R R. By a solution of equation ( E ) we mean a sequence (yn) which satisfies equation ( E ) for n sufficiently large. We consider only such solutions which are nontrivial for all large n. A solution of equation ( E ) is called nonoscillatory if it is eventually positive or eventually negative. Otherwise it is called oscillatory.

601

In the last few years there has been an increasing interest in the study of oscillatory and asymptotic behavior of solutions of neutral higher order difference equations (see, for example, and s-ll) and the references cited therein. The results in this paper have been motivated by the results obtained by Li and Cheng in ', and Migda, Musielak and Schmeidel in '. Background of difference equation theory can be found in and 3-5. The main purpose of this paper is to establish some sufficient conditions for the existence of solutions of equation ( E ) which can be classified into three distinct categories and each one is nonempty under relatively mild conditions. Denote quasidifferences Li(z,), where i = 0,1,2,3, ...,k of a sequence ( z n )as follow^ Lo(zn) = z, L i ( z n ) = & ) A ( L i - l ( z n ) ) ,for i = 1,2,3,...,k. Assume that u p ) = 1. Hence equation ( E ) takes the following form L k ( z n ) + f(n,Yn-n)

= 0772 E

N,

where (z,) is companion sequence of a sequence defined by

(1)

relative to ( p n ) 7

We use the following notation

c

1 n-l 1 iz-l Sn(f(.i,.>) = (1) (2)

C

'n

iz=N ' i z

i3=N

...-

1

&*-l

(3)

Zk-1

and n- 1

il-1

1

(4)

Throughout the rest of our investigations we will assume that M

A

(hence lim S, = cm), n-cc and yf(n,p)

> 0 for all y # 0 and n E N .

602

2. Classification of nonoscillatory solutions

The next Lemma gives us the classification of nonoscillatory solutions of equation (1). Lemma 2.1. Let (y,) be a positive sequence and (z,) its companion sequence such that Lk(z,) < 0 , f o r suficiently large n, where k is a positive integer. T h e n the exactly one of the following statements holds:

there exists a positive integer 1 such that Li(z,) > 0 , f o r 0 6 i < I , and Li(z,)Li+l(z,) < 0, f o r 1 6 i < k - 1,

Li(z,)Li+l(zn) < 0, for 0 < i

< k - 1,

there exists a positive integer 1 such that Li(z,) < 0 , f o r 0 6 i < 1, and Li(z,)Li+l(z,) < 0, f o r 1 6 i < k - 1, Li(z,) < 0 , f o r 0 6 i < k .

Proof. The proof of the above Theorem follows directly from Lemma 2

’.

For k an even positive integer and for k an odd positive integer it is clear that the following corollaries hold. Corollary 2.1. In Lemma 1 assume that

k is a n even positive integer .

(5)

Hence the hypothesis of this Lemma holds when the integer 1 in Class (ii) is a n even positive integer and integer 1 in Class ( i v ) is a n odd positive integer. Corollary 2.2. In Lemma I assume that k is a n odd positive integer. Hence the hypothesis of this Lemma holds with the integer 1 in Class (ii) is a n odd positive integer and integer 1 in Class ( i v ) is a n even positive integer.

603

Depending on the sequence (p,), the nonoscillatory solutions of equation (1) may belong to two, three or all the above Classes, and the following Lemma holds.

Lemma 2.2. Assume that ( 5 ) holds. Let (y,) be a positive solution of equation (1) and (2,) its companion sequence. If p, 3 0 , for suficiently large n, then (z,) belongs t o Class (2) or (ii). If -1 6 p , < 0, f o r suficiently large n, then (2,) belongs to Class (i), (ii) or (iii). If p , < -1, for suficiently large n, then (z,) belongs to Class (i) - ( u ) . Proof. The proof of the above Lemma is obvious and hence omitted.

Lemma 2.3. Assume that ( 5 ) holds, sequence (y,) of equation (1) and (2,) its companion sequence.

is a positive solution

1")If lirn p , = 00 then lim y, = 00 and lim z, = 00. n+oo n+oo n-cc 2")If 0 6 lim inf p, < lirn supp, = co then lirn y, = 00 and lim z, = 00. n+cc

n+cc

n-cc

12-05

< limsupp, < 00 then n-+m lim y, n-icc n-cc 4")If 0 6 lim p, = p < 00 then n+cc 3")If 0 6 liminf p,

= co

and n+m lim z,

= 00.

lim yn = co and lim z, = co,

n-cc

or @)

5")If lirn p, n-icc holds:

n+cc

lim y, = y* E R+ and lim z, n+cc

71-00

=p E

(-1,O)

then exactly one of the following conditions

lim yn = 00 and lim z,

n+m

@)

n-cc

= 00,

lim y, = y* E R+ and lirn z, = y * ( l + p ) , n-im lirn y, = 0 and lim z, = 0.

n+cc

nim

6")If --OO holds:

+ p),

= y*( 1

< lim pn

=p

n-im

< -1 then exactly one of the following conditions

71-00

lirn y, = 0 and lim z,

nlcc

@) (c)

=

-00,

71-00

lim yn = y* E R+ and lim z, n-w $iryn = 0 and lirn zn = 0. n-co

n-iw

= y*(l

+p),

604

7")If -co < liminfp, 6 limsupp, < 0 then (y,) 12-00 n+cc sequence.

can be a n y positive

Proof. We will prove only cases 2") and 3"). The proof of the other cases is similar and hence omitted. The proof of Case 2"). Assume that 0 6 liminfp, = p < limsupp, = 00. n+m

n-w

Suppose to the contrary that 0 6 lirn inf y, = y* < lirn sup yn. n+m

n+cc

Hence lim inf(y, ,--too

+ pnyn-7)

= y*

+ py* < co

and limsup(y, n-+m

But n-cc lim z, = n+cc lim (y, +p,y,-,) tion implies that lim yn = co.

+ pnyn-,)

= 00.

exists and n-00 lim z,

= 00.

This contradic-

n+oo

The proof of Case 3"). Assume that 0

< liminfp,

= p < limsupp, = P < 00. 12-00

71-00

Suppose to the contrary that 0 6 liminf yn = y* < limsupy, = y**. n+cc

n+cc

Hence lim inf(y, n-+m

+ p,y,-,)

= y* (1

+ p,y,-,)

= y**(l+

+ p ) = n+w lim z,

and limsup(y, n-cc

P ) = lim zn. n-00

So, y*(l + p ) = y**(l + P ) . This implies y* = y** = 0 or y* = y** = CO. If y* = 0 then lim z, = 0. From the another hand (2,) is a positive n-cc increasing sequence then lim z, > 0. We see that y* = y** = co. The n-im proof is completed. 0 The above Lemma improves and generalizes results obtained by Migda and Migda and Li and Cheng '.

605

3. Sufficient conditions

Assume that lim p , = p E ( - l , O ) ,

n+m

and (5) holds. Under the above assumptions, the Lemma 2 states there are exactly three types of eventually positive solutions of the equation (1). We say that nonoscillatory solution (y,) of equation (1) is asymptotically zero solution if lim yn = 0, asymptotically constant if there exists n-m some nonzero constant a such that lim yn = a and asymptotically S, n-m solution if there exists some nonzero constant p such that lim L!E = p. n-cc s n From lim S, = 00 we have S, solution tends to infinity. 71-00

Lemma 3.1. Assume that conditions ( 5 ) and ( 6 ) hold and (y,) is a positive solution of equation (1) such that (z,) defined by (2) is eventually negative sequence. T h e n lim yn = lim z,

n-cy)

n-00

= 0.

Proof. Assume t o the contrary that lim y, = y* > 0. Hence lim z, n-o3

n+m

=

y*(l + p ) > 0, and (2,) is an eventually positive sequence. This contradiction give us the thesis.

It is of interest to find sufficient conditions for the existence of each type of eventually nonoscillatory solutions. First, we present a sufficient condition for the existence a solution of equation (1) which tends to zero. Theorem 3.1. Assume that conditions (5) and ( 6 ) hold, and function f ( n ,z) is nondecreasing in z E (0, m). Let

then equation (1) has a n eventually positive solution (y), to zero.

which converges

Proof. From (6) and (7), there exists a positive integer N such that

606

for n 2 N . Consider the partially ordered Banach space BN of all bounded sequences (yn) defined for n 2 N , with the usual supremum norm. Let S be the subset of BN defined by S = {(y,)

E BN : 0

6 yn 6 1,n 2 N } .

We define a partial order on BN in the usual way. Then for any nonempty subset A of S both inf A and supA belongs to A. Let us define a mapping T : S + BN as follows

for n 3 N . Thus T maps S into itself. Furthermore, it is clear that T is an increasing mapping. Therefore, by means of the Knaster-Tarski fixed point theorem, there exists w* E S such ~ , n 2 N . So, that w; = ( T w * )for

If we set

then M

Yn

z=

-Pnyn-r

+ C si (f(j,~ j - c ) ). i=n

It is easy to verify that for large n sequence (yn) is a solution of equation (1). Furthermore, note that (yn) is eventually positive and converges to zero. The proof of this theorem is completed. 0 Next we present a necessary and sufficient condition for the existence of an asymptotically constant solution of equation (1). Theorem 3 . 2 . Assume that conditions (5) and ( 6 ) hold, and function f is continuous and monotonic in the second argument. A necessary and suficient condition for equation ( 1 ) to have solution (y,) which satisfies

lim yn = a # O

n+oo

607

is that

i=l

for some nonzero constant c. Proof. Necessity Let (y,) be a nonoscillatory solution of equation (1) which satisfies (8). Without loss of generality, we may assume that a > 0. We set y = (1+ p ) a , where p is defined by ( 6 ) . Then there exists a positive numbers a1 and a2 such that yn E (alla2). Sequence ( z n ) , defined by (2), satisfied condition lim z, = y # 0.

n-cc

Then there exists a positive numbers y1 and large n. Using (1) in the following form

72

such that z, E (yl,y2), for

i=n

>From the monotonicity of the function f, we get

when f is nondecreasing and

when f is nonincreasing. This implies that

where m = 1 for f nondecreasing and m = 2 for f nonincreasing. Then the necessity of this theorem holds. Suficiency >From (9), there exists a positive integer N such that

i=n

for n 2 N . Consider the Banach space BN of all bounded sequences (yn) defined for n 2 N , with the usual supremum norm. Let S be the subset of BN defined by

s={(yn)E B N

:

< Y n < 27,n b N } .

608

It is easy to see that S is a nonempty, bounded, convex and closed subset of B N . We define a mapping T : S + BN as follows 00

(TY)n = Y -pnYn-r

+CSi(f(j,

~j-m)),

i=n for n 2 N . It is easy to see that T maps S into itself. Furthermore, since f is a continuous function we get T is continuous and T ( S ) is a uniformly Cauchy. Therefore, by means of the Schauder fixed point theorem, operator T has a fixed point (9:) = ( T y * ) , in S. One can see that sequence (y:) is a solution of equation (l),for large n, such that z: = y: p , ~ : - ~ converges to y. Hence, by Lemma 3, y: converges to a= The proof of this theorem is complete. 0

&.

+

Finally, we present a necessary and sufficient condition for the existence a solution of equation (1) which diverges to positive infinity. Theorem 3.3. Assume that conditions (5) and (6) hold, and f is a monotonic function in the second argument. If

cSdlf (A 00

CSj--(T)O< 00, (10) i= 1 for some nonzero constant C , then equation (1) has solution (y,) such that Yn lim =P

n+w

Sn

# 0.

Proof. >From (6), there exists p* E (-1,O) p* < p =

such that

lim pn.

n+w

Assume that (10) holds with C > 0. Let N be large enough that

and -pn < -p*, for n 2 N . Consider the Banach space BN of all real sequences ( x n ) defined for n 2 N with the usual supremum norm. Let S be the subset of BN defined by S = { ( z n ) EBN : P ( l + ~ ) ~ x n 6 2 P ( l + p ) , n 2 N } .

It is easy to prove that S is a nonempty, bounded, convex and closed subset of B N . We define a mapping T : S + BN as follows

609

for n 2 N . From monotonicity of f, we prove that T maps S into itself. Furthermore, it is clear that T is continuous and T ( S ) is a uniformly Cauchy. Therefore, by means of the Schauder fixed point theorem, there exists a sequence (x*)such that z: = ( T x * ) ~ It .is easy to see that lim z: = p.

n-09

Let (y:)

be defined by y: = S,x:.

Because

lim Sn = co,

n+m

then sequence (y:) is divergent. This divergent sequence (y;) is a solution 0 of equation (l),for large n. The proof of this theorem is completed. Theorems 1, 2 and 3 improve and generalize Theorems 1, 2, and 3 presented by Li and Cheng in 6 . To obtain the results presented therein we should put a:) = l , i = 1 , 2 , 3 ,..., k - 1 in equation (1). References 1. R. P. Agarwal, Difference Equations and Inequalities. Theory, Methods and Applications, Marcel Dekker, New York 2000 (second edition). 2. R.P. Agarwal, S.R. Grace and D. O’Regan, Oscillation criteria of comparison type for higher order order difference equations, Nonlinear Anal. Forum, 7, 1 (2002), 73-91. 3. R.P. Agarwal and P.J.Y. Wong, Advanced Topics in Difference Equations, Kluwer, Dordrecht, 1997. 4. S.N. Elaydi, An Introduction to Difference Equations, Springer, 1996. 5. W.G. Kelley, A.C. Peterson, Difference Equations, an Introduction with Applications, Academic press, 1991. 6. W.T. Li, S.S. Cheng, Asymptotic trichotomy for positive solutions of a class of odd order nonlinear neutral difference equations, Comput. Math. Appl., 35, 8 (1998), 101-108. 7. M. Migda, J. Migda, Asymptotic properties of solutions of second order neutral difference equations, Nonlinear Anal. (to appear). 8. M. Migda, A. Musielak, E. Schmeidel, On a class of fourth order nonlinear difference equations, Advances in Difference Equations, 1 (2004), 23-36. 9. E. Schmeidel, J. Schmeidel, Asymptotic behavior of solutions of a class of fourth order nonlinear neutral difference equations with quasidifferences, (submitted). 10. B. Szmanda, Oscillatory and asymptotic behaviour of higher order difference equations, Le Matematiche, 70, 1 (1997), 171-178. 11. E. Thandapani, P. Sundaram, J.R. Graef, A. Miciano, P.W. Spikes, Classification of nonoscillatory solutions of higher neutral type difference equations, Arch. Math., 31, 4 (1995), 263-277.

610

CRITICAL GROUPS FOR ITERATED MAPS

C. CORREIA RAMOSl, N. MARTINS', R. SEVERIN03, J. SOUSA RAMOS4 The critical group G of a connected graph is an abelian group, which is closely related to the discrete laplacian. The order of G is the number of spanning trees in the graph. Using the Markov partitions associated to the itinerary of the critical points of one-dimensional maps we define and study critical groups for m-modal discrete dynamical systems.

2000 MCS Primary 37B10, 05C35; Secondary 37305, 37315 Keywords: Discrete dynamical system, discrete Laplacian, critical group, spanning trees Dedicated to the Memory of Bernd Aulbach 1. Introduction An important classification in symbolic dynamics is the classification of shifts of finite type up to flow-equivalence, see Parry-Sullivan l o , Franks and Bowen-Franks 3 , Huang and Martins et al. 7. Bowen-Franks introduced an algebraic flow-equivalence invariant for shifts of finite type based on the fact that any shift of finite type is topologically conjugate to a graph defined by a non-negative integer square matrix A , the now called Bowen-Franks group, BF(A) = Zn / ( I - A)Z", with n the dimension of A . On the other hand, in their study of algebraic invariants of unimodal and bimodal maps of the interval, Severino-Sousa Ramos and Martins et al. computed the Bowen-Franks groups for transition matrices associated with those maps. In Martins et al. it was proved that the Bowen-Franks group of a transition matrix associated to a m-modal interval map satisfies a factorization theorem related with the number m of critical points of the map. An expression for BF(A) was obtained and given by Z,,@ . . . @ Zam, for a l , ..., a , natural numbers. Here we study another group, called critical group, which is defined on a digraph, and we will see how it can give information about the dynamics of a m-modal interval map. Our goal here is to investigate the relationship between the dynamical properties of the iterated m-modal maps and the

611

algebraic structure of the associated critical groups. The critical group appears in the work of Biggs 2 , associated t o a game defined on a graph, the chip-firing game, although it appears also in another contexts, see Bacher et al. and Wagner 1 2 . The critical group is defined with the help of a Laplacian matrix defined on the graph. Given a m-modal interval map we associate to it a transition matrix and an associated Markov digraph as in Lampreia-Sousa Ramos 6 , Martins et al. ', '. Since one important feature of these graphs is that they are directed graphs (digraphs), we must define an appropriate Laplacian matrix. Using this Laplacian matrix, we define the critical group. The critical group is a finitely generated abelian group and it is from its factorization, in the same way as the Bowen-Ranks group, that we obtain the dynamical information. 2. Definitions and Notations

Let us consider a family of m-modal maps of shape ~ ( f = ) ((-l)k 0 , . . . ,m) on an interval I = [a,b ] , and the partition

u c1 u I1 u . . . u c, u I,, induced by any such map f , where Io,I1,. . . ,I , and c1, C Z , . . . , c, I = I0

:

Ic

=

(1)

are the subintervals of monotonicity and the critical points of f,respectively. Now, consider Io,I1 , . . . ,I , and C1,C2, . . . , C, as abstract symbols of an alphabet % = { I o ,c11 I1 1 c21.. ., ern,I,},

and order the symbols in U as they lie along the real line, i.e.,

I0 4 CI 4 I2 4

c 2

4

. . . 4 c,

4 I,.

From (l),we have that each point x E I has a unique address ~ ( xE) Iu defined by the condition that x E ~ ( xc) I . Let Iu" be the set of all infinite sequences SoSlS2 . . . of symbols Si E U. Thus, each x E I has a well defined itinerary under a map f given by Zf(x) = ~ ( x~)( f ( x )b)( f o 2 ( x ).). . . Consider the sign function E : U -+ {-1, 0,1} as follows:

{

&(Io)= E(I2) = . . . = &(Ieven) = 1, E(I1) = &(Is)= . . . = &(I&) = -1, . &(el) = E(C2) = . . . = &(em) =0

Next, we introduce a partial ordering in U".

Definition 2.1. For any two distinct sequences P o P ~ P .~ and . . SoSlS2... of U", we will say that POPI. . . 4 SoSl. . . if and only if there is a Ic 2 0

612

so that Pi = Si,for i < 5, and Pk 4 s k , if the product & ( P O.). .& ( & - I ) is equal to +1, or s k 4 Pk, if the product & ( P O.). &. ( & I ) is equal to -1. Thus, using this order relation in !21N, for any points x,y E I , we have

w4 4

< Y. The itineraries Kj(f) = Zj(f(cj)),with j = 1,.. . , m, are called the kneading sequences of f , and the set K ( f ) = ( K l ( f ) ,Kz(f), . . . , K m ( f ) )of ZAY)

===+ x

kneading sequences of f will be called its kneading data. Consider a map f such that its kneading sequences have the form ~ j ( f= )

( s ( ~ ) c ~ j) "=,1 , 2 , . . . , m ,

where S ( j ) are finite sequences without any of the symbols C1,. . . , C,. Denote by p j the length of S(j)Cj. As usual, we will simplify the notation and write only the periodic blocks, i.e.,

Denote by W the set of m-modal kneading data. Next, we will associate transition matrices to m-modal kneading data. Given (S(')Cl,S(2)C2,. . . , S(m)C,) E W , consider the sets of shifted sequences ~ ( j =)

{ o'((s(~)c~), ,t = 1,.. . , p j },

j = 1,.. . , m,

where o is the usual shift operator, and the union of all these p = p 1 + . p , shifted sequences

. .+

Now, let {xi }:"=, denote the set of points of the interval whose itineraries Z(xi) are exactly the sequences X i . Let p denote the permutation on {1,2,.

. . , p } such that

a < Xp(1) < X p ( 2 ) < . . . < X p ( p ) < b and let

ui :=

x p ( i ) .Finally, take the n = p - 1 subintervals

Ji := [ui,~ i + l ] .

Definition 2.2. Let K ( f ) = (S(l)C1,.. . , S(m)C,) be the kneading data of a rn-modal map f . The matrix associated with K ( f ) is the n x n matrix Af whose entries are given by

a , . .23

'-

{

1 if f ( i n t ( J i ) )2 i n t ( J j ) , 0 otherwise

613

Let X j = (V,E ) be a digraph, the Markov graph associated to the map f through the Markov partition, with V = {1,2, ...,n } the set of vertices associated to the subintervals of the partition { J l , ...,Jn}, and E = { ( z , j ): 1, for i , j = 1,...,n } the set of directed edges or transitions between subintervals. Thus, from this definition, the transition matrix A f is the adjacency matrix of the digraph X j . Let us define the diagonal matrix Gi"(X) of the in-degrees of the vertices whose entries are given by aij =

dii =

aij. j€V

The discrete Laplacian matrix of X j is defined by

Lf = G y

-Aj.

Every row sum of the transpose of the L f matrix is zero and the determinant is also zero. Then we define the reduced Laplacian L, choosing a vertex r as root and cutting out the row and the column correspondent to the root. An oriented spanning tree leaving the root can be described as a function g : V + V\{r} such that for each v E V\{r} there is a k : g k ( r )= v and for all v E V, v 4 g(v) is an edge. We want to count these spanning trees in discrete dynamical systems. The determinant of the reduced Laplacian L, gives the number N, of spanning trees rooted in T (this is the KirchhoffTutte's Theorem for digraphs), i.e.

N, = det(L,). The total number of rooted spanning trees in the digraph X f is the sum of N, over all vertices r and we can compute it by N f = det(Qj), where Qf is obtained from LF by replacing its first column with a column of 1's. We designate Q j the extended laplacian.

Example 2.1. Let f be the bimodal map with kneading data

K (f) = RBLA. The correspondent A f , LF and Qp matrices are given by

(::a),LF=

(-12 -1-1 2 -l),Qf=

(11-1 2 -1 -1).

A, = -1-1 We have

2

1-1 2

Nf= det(Qj) = 9, the number of rooted spanning trees of X f .

614

Definition 2.3. We define the critical group CG( f ) of X f by

CG(f)

:= Z n / Q f Z n .

This is a finitely generated abelian group. By the structure theorem for finitely generated abelian groups, there are nonnegative integers a l , ..., a, such that ai divides ai+l for each 1 5 iln-land

CG(f)

= Z/UlZ @ Z / a z Z €3.. . @ %/a&

(We have Z / l Z = 0 as the trivial group and %/OZ = Z).These integers are computed by reducing Qp to its Smith normal form. We say that two n x n integer matrices P and Q are equivalent, denoted by P M Q , if and only if there exist n x n integer matrices M and N with determinant f l such that M Q N = P . That is, Q can be transformed into P by applying elementary row and column operations which are invertible over the integers. Every n x n integer matrix Q is equivalent to a diagonal matrix diag(a1,az, . . . , a,) of nonnegative integers (invariant factors) such that ai divides ai+l for each 1 5 i 5 n - 1. This is the Smith normal form of Q and we denote it by SNF(Q).

Example 2.2. Let us return to the kneading data K ( f ) = RBLA. By applying the Smith normal form to Q f we obtain the matrix diag(l,3,3) and thus we have

c q f ) = z1@ 2 3 @ z3. For each natural number u,let p,(f) denote the multiplicity for which u occurs on the diagonal of S N F ( Q f ) . Thus, we have CG(f) Y C G ( f )@ Zpo(f), where CG(f) is the torsion subgroup of X f . Now, we consider the minimal number of generators of CG(f). Let s(f) denote the minimal number of generators of the torsion subgroup of X f , so C G ( f )is the direct sum of s ( f ) nontrivial finite cyclic groups. This is a numerical invariant of f, as it depends on the properties of Q f . Thus, n ( f ) = s ( f ) pl(f) pO(f). Our goal is to prove that s(f) 5 rn(f), where m ( f )is the modality of the map f.

+

+

3. Main Theorem Consider the set of all rn-modal maps f with kneading data K ( f ) . Recall that the dimension of the matrices A f is related with the length of the kneading sequences and hence it can be very large, finite or infinite.

615

Theorem 3.1. Let f be a m-modal map and Q f be its extended laplacian. The critical group o f f is given b y

CG(f)

E!

z:* 63 z,,83 z,,63 ' . . 63 Z,,(,)

@

with n ( f )= s ( f ) + p l ( f ) +p,,(f) f o r some ai > 1, i = 1,..., s ( f ) . Then s( f ) 5 m ( f ) , where m(f ) is the modality of the map f . It is obvious that this statement strongly depends on the position of the 1 entries in the transition matrix A,. For that reason, the analysis of the general situation is extremely difficult and we will be forced to seek a different point of view, where the dynamical information will be written in an adequate way to the intended computation. Sketch of proof: The technique of the proof is based on a commutative diagram derived from the study of the homological configurations of graphs associated t o m-modal maps of the interval. From the subintervals J k defined by the itineraries of the critical points, we introduce 0 - 1 chains of complexes that translate the dynamical properties into relations of homological type. The proof will follow with some not evident and long adaptations of Martins et al. 7, for the Bowen-Franks groups. Acknowledgments First author acknowledges CIMA-UE for financial support. The other authors were partially supported by FCT/POCTI/FEDER.

References 1. R. Bacher, P. de la Harpe and T. Nagnibeda, T h e lattice of integral flows and the lattice of integral coboundaries o n a finite graph, Bull. SOC.Math. de France 125 (1997), 167-198. 2. N. L. Biggs, Chip-firing and the critical group of a graph, J. Algebraic Combin. 9 (1999), 25-46. 3. R. Bowen and J. Franks, Homology f o r zero-dimensional nonwandering sets. Ann. Math. (2) 106 (1977), no. 1, 73-92. 4. J. Franks, Flow equivalence o,f subshifis o f f i n i t e type. Ergodic Theory Dynam. Systems 4 (1984), no. 1, 53-66. 5. D. Huang, Automorphisms of Bowen-Franks groups of shifts o,f finite type. Ergodic Theory Dynam. Systems 21 (2001), no. 4, 1113-1137. 6. J.P. Lampreia and J. Sousa Ramos, Symbolic dynamics of bimodal maps, Portugal. Math. 54 (1997), no. 1, 1-18. 7. N. Martins, R. Severino and J . Sousa Ramos, Bowen-Franks Groups f o r Bimodal Matrices, Journal of Difference Equations and Applications, 9 (3/4) (2003), 423-433.

616 8. N. Martins, R. Severino and J . Sousa Ramos, Bowen-Franks Groups associated with iterated maps of the interval, Grazer Math. Ber, 9 (3/4) (2004), 423-433. 9. J. Milnor and W. Thurston, O n iterated maps of the interval. Dynamical systems (College Park, MD, 1986-87), 465-563, Lecture Notes in Math., 1342, Springer, Berlin, 1988. 10. W. Parry and D. Sullivan, A topological invariant of flows on 1-dimensional spaces. Topology 14 (1975), no. 4, 297-299. 11. R. Severino and J. Sousa Ramos, Dimension Groups of Iterated Maps, Intern. Conf. Equadiff95, Lisboa, ed. L. MagalhLes, C.Rocha and L. Sanchez, World Scientific (1998), 511-517. 12. D. Wagner, The critical group of a directed g r p h , arXiv: math. C0/0010241, 25 Oct 2000.

617

IDEAL TURBULENCE AND PROBLEMS OF ITS VISUALIZATION

A. N. SHARKOVSKY Institute of Mathematics National Academy of Science of Ukraine Kiev, Ukraine E-mail: [email protected] Ideal turbulence is a mathematical phenomenon which occurs in certain infinitedimensional deterministic dynamical systems and implies that the attractor of a system lies off the phase space and among the attractor points there are fractal or even random functions. Ideal turbulence is observed in various idealized models of real distributed systems. Development of ideal turbulence is accompanied by cascade processes of birth of coherent structures of infinitely decreasing scales and this leads t o certain problems in its visualization.

1. Introduction The term turbulence in the wide sense is used when describing spatialtemporal chaos in parameters distributed systems. The notion of ideal turbulence (IT) as turbulence in ideal media (without inherent resistance) was proposed back in 1983. Initially such a phenomenon was named dry turbulence by analogy with von Neumann’s dry water [l,21. Since then this turbulence is kept in the area of our scientific interests. As a result a mathematically rigorous definition of IT has been sharpened and an approach to the study of IT has been elaborated upon. These investigations have led to the fixation of the notion ideal turbulence as a mathematical phenomenon, and now this notion is represented, in particular, in the “Encyclopedia of Nonlinear Science” [3]. The distinguishing features of turbulence are cascade processes of emergence of structures of decreasing scales and chaotic mixing. In real distributed systems, processes of reducing structures to smaller and smaller size cannot go indefinitely because of their internal resistance. In idealized systems, cascade processes may produce structures of arbitrarily small scale

618

and even lead to stochastization of the systems. This paper is a short review of our investigations represented in [l-601, it is based on current works [53]-[59]. 2. Spatial-temporal chaos in boundary value problems Investigation of the models of I T given by boundary value problems (BVP) has became possible only in the last time due t o development the theory of continuous time difference equations based on dynamical systems theory given by one- or low-dimensional maps. There exist classes of nonlinear BVPs demonstrating appearance and development of spatial-temporal chaos and reducable to continuous time difference equations. We begin with several simple and well-known examples. Let us consider the simplest BVP

wt

-

w, = 0,

z E [0,1], t E R+,

(1)

w(1,t)

f(w(O,t)),

(2)

=

where f is a C1-smooth function. On substituting the general solution of (1) w(z, t ) = u(t z), where u is an arbitrary C1-smooth function, into the boundary condition (2) we obtain the difference equation with continuous argument (CDE)

+

u(7+1) =

f(U(T)),

7-

E R+.

(3)

Any initial condition for this BVP

w(z,O) = dz), z E [0,111 with

'p

being a C1-smooth function, gives the initial condition

4.)

=47-),

7

E [O, 11,

(4)

for the CDE (3). The solution U ( T ) of the CDE (3) with the initial condition (4) can be written in the form u(T) = f " ( ' p ( { T } ) ) ,

12

5 7 < 72 + 1,

? =I 0,1,

...,

(5)

where f" is the n-th iteration of f , and hence the corresponding solution of the BVP can be written in the form

w(z, t ) = f
+ z})),

t E R+,

(6)

619

where < . > and { . } are the integral and fractional parts of a number; in particular, w ( z , n) = fn(cp(z)). Thus, the properties of solutions of both the BVP (l),(2) and the CDE (3) should be closely connected t o the properties of the difference equation with discrete time u,+1

n. E

= f(un),

z+,

(7)

and hence, to the properties of the one-dimensional dynamical system given by the map u H f(u).Every solution u,, n E Z+, of the equation (7) is determined uniquely by the value uo E R: u, = fn(u0). Every solution u ( T ) , T E R+, of the CDE (3) is determined by its values on the interval T 2 0. Thus, every solution of the CDE [0,1): u,+,(T)= f < T ’ ( c p ( { ~ } ) ) , (3) consists of a continual family of solutions of (7).

08

OL

08

OL

0

I

I

*

.

.

a6

.

.

a6

I

I

a8

I

. .. I

X

Figure 1. [12] Typical behaviour of solutions of the BVP (l), (2) if the corresponding one-dimensional map f has an attracting cycle of the period # 2i, i 2 0.

Evolution of a typical solution of a typical BVP of the form (l),(2) is shown in Fig. 1. We see there exist points at which the slope of the solution graph tends t o 03 with the growth of time and, hence, if even the limit of such (smooth) solution exists, it can not be even a continuous function. Here in the limit (t = +GO) so-called fractal functions can appear. At the same time, we can say that the behaviour of solutions of the CDE

620

and, hence, solutions of the BVP are more simple in certain sense than the behaviour of solution of (7): typical solution of the CDE or the BVP is asymptotically periodic [12, 161. To formulate this property, it makes sense to use the metric given by the Hausdorff distance between the graphs of functions or equivalent to it the metric pa defined in the next section.

Theorem A. For almost eacha function f , almost each solution of B V P is asymptotically periodic in t with respect t o the metric pa, i.e. lim pa (w(z, t ) ,w(z, t + N ) ) -+

t+m

o

for a certain integer N = ~ ( wf ), > 0.

Moreover, i f f has periodic trajectories of periods 2', i = 0 , 1 , ..., 1 < 00, and no other periodic trajectories, then each solution of B V P is asymptotically periodic in t . The reason for the first part of Theorem A is that the asymptotic behavior of solutions just mentioned is determinated by not the trajectories of points but the trajectories of neighborhoods of these points under the map f , and for almost each function f , the trajectories of neighborhoods points are asymptotically periodic [52, 601 (which is not true, in general, for the trajectory of the point itself). Of course, it is very important to know which is the period for such asymptotically periodic solution. As shown in [12], the period N of a solution w ( z , t ) is not bigger than 2 M with M being the least common multiple of the periods of domains of influence of points from the set (4 u = 4 x 7 O)(= p(z)), z E [O, 111. To investigate asymptotic behavior of solutions of a BVP, it is usually convenient to transfer to the dynamical system (DS) generated by the shifts along solutions of the BVP on the space of its initial states. The BVP ( l ) (, 2 ) induces dynamical systems of shifts

St : p(z) H f'"+"'(p({t in particular, S(p)

+ z})),

t

E

R+,

(8)

=fop.

The analogous situation may take place in the case of the wave equation and related ones. A representative example is the BVP Wtt

- w,, = 0,

5

E [O, 11,

say that some property takes place for almost each this property is a residual set in 2.

tE

(9)

2 if the set of all t having

621

with h being a C1-smooth function on the real line R. The initial conditions

specify the phase space for the DS of shifts, associated with the BVP. Substituting the general solution for the equation ( 9 )

w(2,t)= u l ( t + 2 )

+ u2(t - 2 ) ,

into the boundary conditions ( l o ) , one can obtain the representation of the shift operator St in terms of the iterations of a certain one-dimensional map. This map, labeled f : u, H u,+1, is defined implicitly by

+ %+l).

u,+1 - % = h (%

When (10) is replaced with w(0, t ) = 0, w z ( l ,t ) = h(w,(O,t ) ) ,there arises a two-dimensional map defined by un+l- u,-1 = h(u,).

As an example we consider a idealized time-delayed Chua’s circuit (see [29]-[33],[35],[37]) which is a lossless transmission line with tunnel diode, described by the BVP

ux

=

-Lit,

i, = -Cut,

0 5 2 5 1, t E R+,

~ ( 0t ), = 0, i(1,t) = G(v(1,t ) ,i(1,t)).

(11) (12)

Here w(z,t ) and i ( 2 ,t ) are the voltage and the current along the line, L and C denote the inductance and capacitance per unit length, and the function G specifies a w - i characteristic of the diode. Under the condition u(0, t ) = 0, solution (11) is of the form u ( 2 , t ) = a(t - ./u) Z(2,

d m ,

- a(t

t ) = ( l / Z ) [ a (t 2/.)

+z/u),

(13)

+ a(t + 2 / u ) ] ,

m,

where u = Z = and a is an arbitrary function. With the new variables T = ( u t / l - 1 ) / 2 , U ( T ) = a ( 2 1 ~ / u )combining , (13) and the second boundary condition (12) gives the CDE U ( T 1 ) = f(u(7)) defined implicitly. Thus, the behavior of solutions of the BVP is determined by the one-dimensional map f : z , -+ z,+l. In case of particular function G and particular parameters, conditions to the occurrence of spatial-temporal chaos can be formulated [33].

+

622

3. Ideal turbulence: Defininions

A mathematical definition of turbulence can be given for dynamical systems on spaces of smooth or piecewise smooth functions [3, 27, 41, 47, 53, 55, 57, 591. Spatial-temporal chaotization in a DS is perceived as an evolution of such functions which leads to that their behavior becomes more and more intricate and the limit states of the DS cannot be described with smooth functions (see Fig.1). The global attractor of the DS is not contained entirely in the phase space, and the DS needs to be extended on a wider functional space so that this new space contains w-limit sets of all or almost all trajectories. The spaces of fractal and random functions are particularly appealing for use as such wider space. By a fractal function is meant a function whose graph is a fractal (in the wide sense). Let { C k ( D ,E ) ,TIS t } be an infinite-dimensional dynamical system with C ' ( D , E ) being the space of Ck-functions cp : D -+ E l D c R', E C R" (k,1, m 2 l ) , T = Rf or Zf. The phase space Ck equipped a priori with the "ordinary" C'-metric is noncompact. Therefore, for some or even for almost all trajectories St[cp],cp E C k , the corresponding w-limit sets w[cp]may be found empty. To operate with the extended phase spaces, it is, of course, necessary to use metrics which should satisfy to certain physical considerations. If, for example, we investigate the behavior of the function St[cp](x)as t -+ 03 at some point x = X I , we should consider the values of this function at some &-neighborhoodof x' too, because of the possible presence of large gradients in both x and t. For a space containing fractal functions we take the space of multivalent functions : D + E l whose graphs are closed sets (such functions are called upper semicontinuous), and a metric is taken as follows

c

where dH (., .) is the Hausdorff distance between sets, v T ( x ) = < ( V E ( x ) ) , and VE(.) is the &-neighborhoodof a point. A function : D 4 E l deterministic or random, can be interpreted as a collection of all its finite-dimensional distributions. To obtain a desired metric starting from pa, the set c ( V E ( x )should ) be replaced with the distributions of E on x ( x ) . Thus we obtain the metric

623

where

and for every t E E‘, F t ( x , t ) is the averaging of the r-dimensional distributions of 1 over &-neighborhoodof a point x E D‘.

Definitions. Let CA = C A ( D ,E) be the completion of the phase space C k ( D E , ) via the metric pa with upper semicontinuous functions and C# = C # ( D ,E) be the completion of the phase space C k ( D , E )via the metric p# with random and deterministic functions. For an initial state cp : D + E , denote by W A [ c p ] the set of the limit points of the trajectory St[’p] in the space CA and by w#[cp] the set of the limit points of St[cp] in the space C#. The initial state cp is said to generate ideal turbulence (IT) if the set ~ ~ [ c contains p ] at least one “point” that is a fractal function. The initial state cp is said to generate stochastic turbulence (SIT) if the set w#[cp] contains at least one “point” that is a random function. If there is a great deal of initial states which generate IT or respectively SIT, then IT or respectively SIT is said to occur in the DS. This classification of turbulence can of course be extended. For instance, the initial state cp is said to generate weak ideal turbulence (WIT) if ‘p does not generate IT but there exists at least one function ( E ~ ~ [ c such p ] that the set {x E D : ((x) is multivalued} is infinite. If we consider a BVP it is natural to say that IT, o r WIT, or SIT arises in a BVP if such a turbulence occurs in the corresponding DS.

The simplest model f o r ideal turbulence is the discrete DS acting on the space of smooth functions cp : D + E according to the rule

s:‘PHfop,

(14)

where f : E -+ E is a smooth function and the symbol o is for the composition of functions. The trajectory through a “point” cp can be written in the form S”[cp] = f”

o cp

or

S”[cp](z)= fn(cp(x)), n E Z+

A description of long-term properties for the DS (14) is most advantageous when f is a one-dimensional map and D and E are intervals. In this case, one has the following criterion:

624

Theorem B. Let D and E be closed bounded intervals. I. I n the DS (l4), there occurs (1) weak ideal turbulence, if the map f has periodic trajectories of periods 2’, 0 I i 5 I , with some 1 > 1, and no other periodic trajectories; (2) ideal turbulence, if the map f has a periodic trajectory of period # 2i, i = 0 , 1 , . . . . II. I n the DS (14), there occurs stochastic turbulence, if the map f possesses a smooth (i.e., absolutely continuous with respect to Lebesgue measure) ergodic invariant measure. Let f in the DS (14) be the parameter-dependent map fx : u H X u ( 1 - u),u E [0,1], 0 < X 5 4. If the parameter X varies in the interval (1 A’) with A* M 3.57 being the limit value for the period-doubling bifurcation values of A, there occurs WIT and the progression of ideal turbulence follows the period-doubling pattern. Where X > A*, the map fx has a cycle with a period differing from the powers of 2 and there occurs IT in the DS. Finally, there is a set of positive Lebesgue measure A c (X*,4] such that the map fx with X E A has an ergodic smooth invariant measure and hence SIT in the DS is a non-exclusive phenomenon. In particular, when X = 4, the map u H 4u(l - u)has an invariant measure with the density p ( u ) = 1/ and the support [0,1]. Then for almost every (p, the set w#[(p] consists of a single point which is the pure random function with x-independent distribution F ( x , u ) = :J p ( u )du = (2/7r) arcsin f i and the attractor of the DS consists of just this one point. Generally, functions that make up w-limit sets may appear to be deterministic on one subset of D and random on the other. This may take place where the map f possesses several attractors and the initial values interval (p(D) “clings to” the basins at least of two attractors. It makes sense to note that the topological entropy of the DS (14) is equal to 0 in case WIT and is equal to m in case IT [58]. Figure 2 displays a typical evolution of the vector field given by twodimensional BVP [12]

+ 6,

7rdm

1 Wt

1

=w,

+ w 1y ,

2 Wt

w

1

2

2

= -w, - w y , 2

= w ly=o,

-O3

< x < +m, 0 5 y I 1, 1

w2 = f x ( w

)ly=l

with fx(u)= X ( 1 - u2)- 1 and (wl, w 2 ) ) l t = o = (2y - 1,-22). This BVP is reduced to CDE u(x l , t 1) = fx(u(x,t)), - 03 < x < +a, t 2 -1. For X = 1.75, the one-dimensional map u H fx(u)has cycles of the periods

+

+

625

t=12

t=16

t =20

Figure 2. Evolution of flow lines for the vector field (w1,w2) given by the twodimensional BVP W: = w$+w$, ?I. = - w $ - w ~ , - m < z < +co, 0 5 y 5 1; w1 = w 2 Jy=o, w2 = X ( ~ - ( W 1) 2)-lly=l; (W 1 , W2 )It=o == ( 2 ~ - 1 , - 2 ~ ) X; = 1.75 (left) and X M 1.7963 (right).

1,2,4 only; the BVP generates WIT; we can observe polynomial increasing

the number of structures (from the left). For X = 1.7963, the map fx possesses a smooth invariant measure with the support consisting of the four intervals; the BVP generates IT and, moreover, SIT; we observe exponential

626

increasing the number of coherent structures (from the right). Summarizing, we can state: if a BVP induces an infinite-dimensional DS whose dynamics is specified by some interval map, then the theory of one-dimensional maps allows one t o understand why and how turbulence occurs in the BVP and to present scenarios for self-structuring and selfstochasticity phenomena. For ideal turbulence, the main factor is the complex topological structure of both the set J(f) of points of unstable trajectories and the basins of attracting cycles o f f . The set J ( f ) is self-similar at the points of repelling cycles and a t their preimages and, if the map f has a cycle whose period is different from a power of 2 (i.e., the topological entropy ent (f)is positive), the Hausdorff dimension of J ( f ) is positive. For stochastic turbulence, the key property is the existence of a smooth ergodic invariant measure for the map f. 4. Bifurcations leading to ideal turbulence

Infinite-dimensional DSs, whose dynamics is determined by some onedimensional map, possess the full spectrum of bifurcations inherent in onedimensional DSs: period-doubling bifurcations, tangent bifurcations, bifurcations subordinated to Farey’s rule, period-adding bifurcations. The bifurcations of two last kinds hold in BVPs reducible t o CDEs with a many-valued nonlinearity and most often they do not give rise to turbulence. These bifurcations take place in BVPs simulating real distributed parameters systems, for example, the transmission line with tunnel diode (that was mentioned above). We will talk about typical bifurcations causing ideal turbulence in BVPs reducible to CDEs of the form u(t 1) = f x ( u ( t ) ) ,t E R+,with the parameter X E A where A is some closed interval. In such case the bifurcation properties are determined by the 1-D map u H fx(u).We suppose that there exists a closed invariant interval I: fx(1)C I for all X E A, and that the initial data for all solutions belong t o I .

+

It is useful t o do several remarks concerning the properties of bifurcations realizable in dimension 1 and inherentable by bifurcations in BVPs. The period-doubling bifurcation of cycle is a “soft” bifurcation and the tangent bifurcation of cycle is a “hard” one. The “soft” bifurcation occurs for a value X when the map X H Q( fx), as a map from A in 2R, is continuous (in the Hausdorff metric), where R(f) is for the set of nonwandering points of the map f . The “hard” bifurcation happens for a value X a t which the

627

map X

H

R(fx) is discontinuous and so-called R-explosion takes place.

It makes sense to consider also two types of bifurcations of cycles of intervals ‘. Bifurcations of cycles of intervals are also classified as “soft” or “hard” according to whether the map X H f i ( f x ) is continuous or discontinuous at a bifurcation value of A. Bifurcations of the first type take place when the boundary of a cycle of intervals is not an invariant set. If the boundary is invariant (in this case, the extremum point is homoclinic to the cycle belonging to the boundary), then there occurs a “hard” bifurcation and we again have the R-explosion. It is significant that the set of the parameter values to which correspond “hard” bifurcations of cycles of intervals is countable, and the set of those to which correspond “soft” bifurcations of cycles of intervals is uncountable and, moreover, has positive Lebesgue measure. NQWwe discuss two simple examples, in which turbulence appears as result of a parameter change in the first case and of an initial data change in the second case. Consider the BVP (*) [50] Wt

- w, = aw, 1 4 =

z E [O, 11,

f(w)Ix=O.

(15) (16)

The general solution of (15) is of the form w ( z , t ) = e-azu(t +x), and this BVP is reduced to the CDE u ( t + l ) = f x ( u ( t ) )where fx = Xf and X = ea. We will assume that f x ( u ) is a C3-smooth unimodal convex function that depends smoothly on the parameter X E [a,p] and has negative Schwarzian. We also assume that dfx/dX > 0 for all y and fl;(c) # 0 with c = c(X) being the extremum point of fx(u). For one-dimensional continuous maps, the ordering of the natural numbers 1 4 2 4 22 4

... + 5 . 2 4 3 . 2 4 ...4 9 4 7 4 5 4 3

(17)

determines the universal order of bifurcations of periodic trajectories. Let mx be the supremum of the periods of periodic trajectories of fx according to the linear order (17) (in particular, the equality mx = 00 bBy a cycle of intervals is meant a collection of closed intervals Js, s = 1, ..., m, with disjoint interiors, such that f x ( J s ) = J ( s + l ) m o d m and U ~ = l J scontains an everywhere dense orbit.

628

means that the map has periodic trajectories of periods 2i and has no trajectory of period # 2i for any i 2 0). For any m, ma 4 m 4 mp, denote by X(m) the infimum of values of the parameter X E [a,p] such that the map f x has a cycle of period m. Obviously, X(m’) < X(m”),if m’ 4 m”.

Theorem C . For any m such that ma 4 m 4 mp, m # (1) there exists an open interval A,

C

00,

[a,P] with the property: what-

ever X E A,, almost each solution of the BVP (*) is asymptotically m-periodic in t ; moreover, (a) i f m = 2i, i > 1, each nonconstant initial state generates WIT; (b) i f m # 2i, i = 0,1,2, ..., each nonconstant initial state generates I T ; (2) if m # 2i,i = 0,1,2, ..., there exists a set A$ c [ c Y , ~ of ] positive Lebesgue measure with the property: whatever X E A%, almost each solution of the B V P (*) is asymptotically m-periodic in t with respect to both the metric pa and the metric p# and generates S I T . Now consider the BVP (**)

wt - w, = 0, W t 12=1

z E [O, 11,

(18)

= g(w)wt L o ,

(19)

+

As far as the general solution of (18) is of the form w ( z , t ) = u(t x), we have u’(t 1) = g(u)u’(t),and instead of a single CDE, we obtain the one-parameter family of CDEs

+

u(t

+ 1) = f ( u ( t ) )+ A,

(20)

where f is an antiderivative of 9 , The initial condition w ( x , O ) = cp(z) for the BVP gives the initial condition u(t) = cp(t),t E [ 0 , 1 ] , for the CDE, which implies X = ~ ( 1-) f (~(0)).With this value X the equation (20) gives a continuous (piecewise smooth) solution

wlf(z, t ) = fx’t+f”’(cp({t

+x))),

= f (.)

+ A,

of the BVP (**). The continuous map y : C1([0,1],R ) + R given by Y[Pl = 4 1 ) - f(P(0))

t E R+,

629

induces a stratification of C1([O, 11,R) - the space of initial functions for the BVP (**). We assume that the function g in (19) satisfies the following “standard11 conditions:

g is a C2-smooth function with g’(u) < 0 f o r u E R and g(G) = 0 for some G E R, (u- U>g”(u- G) 2 0. The linear functions g ( u ) = -l(u - U ) with 1 > 0 are examples. Under these conditions on the function g, its antiderivative fx(u) = f(u) X is a convex function with negative Schwarzian. Moreover] there exists an interval r ( g ) such that for X E I’(g), the map fx has bounded nondegenerate invariant intervals] the largest of which we denote by Ix. The BVP (**) possesses bounded nonconstant solutions if and only if ‘p E B ( g ) ,where

+

Now it is easy to reformulate the theorem C for the BVP (**) and any family of initial data ‘ps from B(g). Let ‘pe depend continuously on the parameter B from some closed interval [a,b]. The map y induces on [a,b] the continuous map 7 : B -+ y[‘p~].Let a and p stand respectively for the minimum and the maximum of the values of 7(0) when 8 E [a,b]. We also use the notation m,, mp and X(m),that appeared in Theorem C.

Theorem D. For almost each family m,+m+mp, m # m ,

‘pe

and each m such that

(1) there exists at least one open interval 0, C [a,b] with the proper-

, the solution w v e ( x , t ) of the B V P (**) is ty: for each B € ,@ asymptotically m-periodic in t ; moreover,

(a) i f m = 2i1 i > 1, the initial state ‘ p ~ generates W I T ; (b) if m # 2i,a = 0,1,2, ..., the initial state ‘p(3 generates I T ; (2) if m # 2i1 = 0,1,2, ..., there exists a set 0%c [a,b] of positive Lebesgue measure with the property: whatever B E 0%,the solution wve (2,t ) of the B V P (**) is asymptotically m-periodic in t with respect to both the metric pa and the metric p# and generates S I T . Formulating the analogous results for equations of higher order, say, for the wave equation] creates no principal difficulties.

630

5. Visualization of ideal turbulence. Computer turbulence The visualization of I T is connected with certain difficulties in computing. These difficulties arise from the fact that IT is accompanied by cascade processes of birth of structures of infinitely decreasing scales.

f=O

t=8

t=12

Figure 3. Evolution of flow lines for the vector field (w’,w2) given by BVP w: = -wk - w;, wz = WE - 00 < x < +00, 0 5 y 5 1, and w2(z,0,t)= wl(x,O, t ) , w2(x, 1,t ) = f(w’(x, 1, t)) with f(u)= 2(1 - u 2 ) - 1.

+WE,

Figure 3 displays a typical evolution of the vector field given by twodimensional BVP which is accompanied by a cascade process of birth of coherent structures of decreasing scales [12, 27, 321. We see smooth integral curves of the vector field for the snapshots a t t = 0 , 8 , 1 0 , 1 2 . However, for t = 40, we observe a movement similar t o the Brownian one (Fig. 4) [54].This “phase transition” is due to just the incorrectness of calculation method stipulated by discreteness. The discreteness brought by calculation algorithm can be treated as adjustment taking into account finite sizes of real physical objects. When the sizes of structures in the vector field become less than the step of discreteness of calculations, the following situation is possible: the results of numerical investigation of models may be found more close (in certain

63 1

Figure 4. The smooth flow lines shown in Fig. 3 are replaced by “Brownian” movement, when t = 40; here is used Euler method with constant step of integration 0.001.

sense) to the reality than exact solutions. For example, when IT occurs in a BVP, the exact solutions of the BVP may give, nevertheless, laminar dynamics but just their space-time discretization results in turbulent dynamics. In such cases, we say that “computer turbulence” takes place [46, 50, 51, 53, 54, 551. Consider the following “gedanken experiment” based on the BVP Wt - C W , = 0,

2

E [O, 11,

Wlx4= f(wlx=o).

(21)

We assume that ‘‘the space” is not the interval [0,1] but a unit circle (i.e. the points z = 0 and 5 = 1 are glued). We can interprete w(z, t ) = u(z+ct) as a field of velocities on the circle, which evolves with time in accord to BVP (21) and assigns on the circle the movement of “particles”. Of interest is just the situation where w(x,t),as a function of z, oscillates about the constant c, that is, w(z, t ) := w(z, t ) - c is a function of alternating sign at every fixed t. In this case, for any t , 2 0, those zeros of the function w(z,t,) such that wk(z,t,)> 0 (these zeros are flowing unstable equilibrium positions) devide the circle into cells (with “cores”, by which is meant those zeros of w such that wk < 0, attracting exponentially the rest of the points of the corresponding cells). Theoretically, the movement of particles in the field of velocities is very simple during all time - the velocity of each particle is approximately equal to the velocity of the cells (which in turn equals to that of the field movement): every particle is captured by one of the cells, therewith the diameters of the cells tend to zero with time. But the numerical studies show that the discreteness brought in the model by a calculation algorithm results in a phenomenon, that we name computer turbulence. Namely, in numerical calculating, the particles movement becomes chaotic when the sizes of the cells become less than the size of the discreteness of calculations. Beginning with a certain moment, a particle moves disregarding the proper movement

632

0

80

I

so

0

(4

Figure 5. [50, 531 The movement of a particle in the velocity field w(z, t ) evolving in accord with (21); here c = .3 and f(w) = h-l(f(h(w)))where f(w) = 4w(l - w) and h ( w ) = w d ; d = .6 for (a) and d = .5 for (b); the step of integration A equals 0.01. The number of the complete rotations of the particle up to its breakdown moment- (this number = 7 for the figure) is equal approximately to - In A/ In X where X is for the Lyapunov multiplier of the map f (here X = 2). The averaged velocity c’ is equal to 0.1 in the case (a), and it equals 0 in the case (b). The graph (c) shows the “theoretical” particle movement (calculated by a computer too but with A = lo-’; for this A and t < 80, the particle can not yet leave the initial (moving) cell and moves together with the field); the movement does not depend on value of d, 0 < d < 1.

+

of the cells! We observe a “phase transition” and now, according t o wellknown G.D.Birkhoff’s theorem, the time-average velocity of the movement of almost each particle will be equal to the space-averaged (over the whole space) velocity, i.e. c* = J w dp(w), where p is an invariant measure on the phase space of our dynamical system. In the example represented by the Fig. 5, we use again the map w H 4 w ( l - w) having the invariant measure with the support [0,1]and the density l/ndfor the cases (a) and (b), the support is shifted to the left on d = .6 and d = .5, correspondingly. Therefore, c* = .1 for (a) and c* = 0 for (b). As a result, in the case (a), we can see a reverse movement of a particle, that is due to the fact that c* > 0. In the case (b), the “phase transition” leads to a Brownian-like movement of particles. Hence, one more reason for the chmticity of the

633

particle movement can, in numerical investigations, be the particle jumps from one cell into another, which are caused by the discreteness of calculation algorithm. In the reality, such a chaotic movement may be due to that the sizes of real particles are non-zero, which imply the particles collisions. In such case, we may also talk about an useful analogy between computer and real turbulence.

*

*

*

In conclusion, it should be noted that the creation of a perturbation theory, which will allow us t o analyse non-idealized models too, is both important and very difficult mathematical problem. The first steps in this direction were done still 20 years ago and are reflected partly in [12, 251.

References 1. A. N. Sharkovsky, in: Short Comm. of Intern. Congress Mathematicians, Warszawa, 1983, 10 (12), 4, 1983. 2. A. N. Sharkovsky, in: Proc. Conf. ‘(Nonlinear and Turbulent Processes in Physics” (ed. b y Sagdeev, R.Z.), Gordon and Breach, 3, 1621, 1984. 3. A. N. Sharkovsky, E. Yu. Romanenko, in: Encyclopedia of Nonlinear Science (ed. Alwyn Scott), Routledge, 955, 2005. 4. A. N. Sharkovsky, in: Proc. VIII ICNO (19?’8), Academia, Prague, 2 , 1073, 1979. 5. A. N. Sharkovsky, A. G. Sivak, in: Approximate and Qualitative Methods of the Theory of Differential and Functional-Differential Equations, Inst. Math. Ukrain. Acad. Sci., Kiev, 98, 1983 (in Russian). 6. A. N. Sharkovsky, in: I X Intern. Conf. on Nonlinear Oscillations, Kiev 2 , 430, 1984. 7. I. B. Krasnyuk, Yu. L. Maistrenko, A. N. Sharkovsky,Reports N A S Ukraine, ser. A, No 12, 27, 1984. 8. A. N. Sharkovsky, Yu. L. Maistrenko, Preprint 84.2, Inst. Math., Kiev, 1984. 9. A. N. Sharkovsky, in: Proceed. 2nd Intern. Workgroup “Nonlinear and turbulent processes”, Kiev, 2 , 114, 1985. 10. A. N. Sharkovsky, Yu. L. Maistrenko, Mathematical analysis, Teubner-Texte Math., 79, 220 (1985). 11. M. B. Vereikina, V. L. Maistrenko, E. Yu. Romanenko, A. N. Sharkovsky, Preprint 85.29, Inst. Math., Kiev, 1985. 12. A. N. Sharkovsky, Yu. L. Maistrenko and E. Yu. Romanenko, Difference Equations and Their Applications, Naukova dumka, Kiev, 1986 (in Russian) and English translation: Kluwer Academic Publisher, 1993. 13. Yu. L. Maistrenko, E. Yu. Romanenko, A. N. Sharkovsky, in: Mathematical mechanisms of turbulence, Kiev, Inst. Math., 75, 1986. 14. S. Ya. Aliyev, A. N. Sharkovsky, in: Dynamical systems and differential-

634

difference equations, Kiev, Inst. Math., 5, 1986. 15. Yu. L. Maistrenko, A. N. Sharkovsky, in: Proc. Intern. Workshop “Dynamical systems and environmental models” (Eisenach, 1986), Berlin, 40, 1987. 16. A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko, Ukrain. Math. J. 39, 123 (1987). 17. E. Yu. Romanenko, A. N. Sharkovsky, in: Differential-functional eguations and their applications to nonlinear boundary value problems, Kiev, Inst. Math., 74, 1987. 18. Yu. L. Maistrenko, E. Yu. Romanenko and A. N. Sharkovsky, in: Proc. Intern. Workshop “Plasma theory and nonlinear and turbulent processes in physics” (Kiev, 1987), World Sci. Publ., 520, 1988. 19. K. A. Lukin, Yu. L. Maistrenko, A. N. Sharkovsky, V. P. Shestopalov, Preprint 87.39, Inst. Math. of N A S of Ukraine, Kiev, 1987 (in Russian). 20. A. G. Bakhurskii, Yu. L. Maistrenko, Preprint 88.16, Inst. Math. of N A S of Ukraine, Kiev, 1988 (in Russian). 21. A. G. Bakhurskii, Yu. L. Maistrenko, Preprint 88.70, Inst. Math. of N A S of Ukraine, Kiev, 1988 (in Russian). 22. Yu. L. Maistrenko, in: Dynamical systems and ergodic theory (Warsaw, 1986), Banach Center Publ., 23,443 (1989). 23. E. Yu. Romanenko and A. N. Sharkovsky, in: Proc. IV Intern. Workshop “Nonlinear and turbulent processes in Physics” (Kiev, 1989), 2, 416, 1989. 24. K. A. Lukin, Yu. L. Maistrenko, A. N. Sharkovsky, V. P. Shestopalov, Reports of Acad. Sci. USSR 309,n2, 1989 (in Russian). 25. Yu. L. Maistrenko, Ukrain. Math. J . 41,779 (1989). 26. Yu. L. Maistrenko, I. I. Stukanyov, in: Dynamical systems and nonlinear phenomena, Akac. Sci. Ukrain. SSR, Inst. Mat., Kiev, 26, 1990 (in Russian). 27. A. N. Sharkovsky and E. Yu. Romanenko, Intern. J . Bifurcation and Chaos 2, 31 (1992). 28. A. N. Sharkovsky and E. Yu. Romanenko, in: Proceedings of ECIT-91, World Scientific, 241, 1992. 29. A. N. Sharkovsky, IEEE Transactions on Circuits and Systems.I. 40, 781 (1993). 30. A. N. Sharkovsky, Yu. L. Maistrenko, Ph. Deregel and L. 0. Chua, J . Circuits, Systems and Computers 3, 645 (1993). 31. Yu. L. Maistrenko, V. L. Maistrenko and L. 0. Chua, Intern. J . Bifurcation and Chaos 3, 1557 (1993). 32. E. Yu. Romanenko and M. B. Vereikina, Ukrain. Math. J. 45, 1398 (1993). 33. A. N. Sharkovsky, Intern. J . Bifurcation and Chaos 4, 303 (1994). 34. A. N. Sharkovsky and A. G. Sivak, Nonlinear Math. Physics 1, 147 (1994). 35. Yu. L. Maistrenko, V. L. Maistrenko, S. I. Vikul and L. 0. Chua, Intern. J . Bifurcation and Chaos 5 , 653 (1995). 36. E. Yu. Romanenko, A. N. Sharkovsky and M. B. Vereikina, Intern. J . Bifurcation and Chaos 5 , 145 (1995). 37. A. N. Sharkovsky, Ph. Deregel and L. 0. Chua, Intern. J . Bifurcation and Chaos 5 , 1283 (1995). 38. A. N. Sharkovsky, Intern. J . Bifurcation and Chaos 5 , 1419 (1995).

635 39. E. Yu. Romanenko, in: Dynamical Systems and Applications (Ser. Applicable Analysis), World Scientific, 4, 617, 1995. 40. Yu. L. Maistrenko, V. L. Maistrenko and S. I. Vikul, J. Tech. Phys. 37, 367 (1996). 41. E. Yu. Romanenko and A. N. Sharkovsky, Ukrain. Math. J . 48, 1817 (1996). 42. E. Yu. Romanenko, A. N. Sharkovsky and M. B. Vereikina, in: Proc. 1st Intern. Conf. “Control of Oscillations and Chaos”, St. Petersburg, 3, 492, 1997. 43. A. D. Fedorenko, V. V. Fedorenko, A. F. Ivanov, A. N. Sharkovsky, Bull. Austral. Math. SOC57, 37 (1998). 44. E. Yu. Romanenko, Computers Math. Appl. 36,No 10-12, 377 (1999). 45. 0. Yu. Romanenko, 0. M. Sharkovsky, Ukrain. Math. J. 51, 907 (1999). 46. A. N. Sharkovsky, S. A. Berezovsky, in: Proc. Intern. Conf. “Self-similar systems” (Dubna, Russia, 1998), 251, 1999. 47. A. N. Sharkovsky, E. Yu. Romanenko and M. B. Vereikina, in: Proc. Intern. Conf. “Self-similar systems” (Dubna, Russia, 1998), 237, 1999. 48. E. Yu. Romanenko, A. N. Sharkovsky and M. 8. Vereikina, Nonlinear Boundary Value Problems, Donetsk, 9 , 174 (1999). 49. A. N. Sharkovsky, Ann. Math. Silesianae 13, 243 (1999). 50. E. Yu. Romanenko, A. N. Sharkovsky, Intern. J. Bifurcation and Chaos 9, 1285 (1999). 51. A. N. Sharkovsky, S. A. Berezovsky, in: Proc. 2nd Intern. Conf. ‘(Control of Oscillations and Chaos” St. Petersburg 1, 6, 2000. 52. V. V. Fedorenko, Ukrainian Math. J. 54, 425 (2002) (in Russian). 53. 0. M. Sharkovsky, in: Proc. Ukrainian Math. Congress, Kiev, 12, 125, 2003 (in Ukrainian). 54. A. N. Sharkovsky, S. A. Berezovsky, Intern. J. Bifurcation and Chaos 13, 1811 (2003). 55. A. N. Sharkovsky, E. Yu. Romanenko, S. A. Berezovsky, in: Proc. Intern. Conf. “Physics and Control” (2003), Petersburg, Russia, 1, 23, 2003. 56. A. N. Sharkovsky, in: New Progress in Difference Equations, Proc. ICDEA’2001, Taylor & Francis, 3, 2004. 57. A. N. Sharkovsky, Grazer Math. Berichte 346, 403 (2004). 58. A. N. Sharkovsky, E. Yu. Romanenko, in: Proceed. Steklov Institute of Math244, 264 (2004). ematics, MOSCOW, 59. A. N. Sharkovsky, in: Proc. Intern. Conf. “Nonlinear Dynamics of Electronic Systems” (Evora, Portugal), 34, 2004. 60. E. Yu. Romanenko, Ukrainian Math. J. 57, 534 (2005) (in Russian).

636

FINE STRUCTURE OF THE ZEROS OF ORTHOGONAL POLYNOMIALS: A REVIEW

BARRY SIMON* Mathematics 253-37 Calafornia Institute of Technology Pasadena, CA 91125, USA E-mail: [email protected]

We review recent work on zeros of orthogonal polynomials.

1. Introduction Zeros of orthogonal polynomials have had a fascination at least since Gauss’ discovery that optimal quadrature for the Ftiemann integral on [-I, 11 involves the zeros of the Legendre polynomials. A special reason for recent interest concerns the fact that zeros are eigenvalues of cutoff finite difference matrices. Explicitly, if P,, p , are the monic orthogonal and orthonormal polynomials for OPRL (RL = real line) and a,, pn for OPUC (UC = unit circle), then

P,(z) = det(z - 7r,Mz7rn) @,(z) = det(z - 7r,Mz7rn)

where 7rn is the projection onto the n-dimensional space of polynomials of degree at most ( n - 1) and M x (resp. M z ) is multiplication by z (resp. z ) on Lz(R, dp) (resp. L 2 ( d ~d p, ) ) . By using the {pj}?:; basis, 7rnMX7rncan be replaced by a cutoff Jacobi matrix, and by using a cutoff CMV matrix, 7rnMz7rn can be replaced by a cutoff CMV basis (see Chapter 4 of [l]). In general, we follow the conventions of [l, 21 throughout this article. In particular, *Work partially supported by NSF grant DMS-0140592.

637

b,}r=l

{an, are the Jacobi parameters, {an},"=o the Verblunsky coefficients, and QE(z) = z" Q,(1/2) the Szego dual. We can also describe paraorthogonal polynomials (POPUC) (see [3]) in terms of (1.1)/(1.2). r n M z r n is norm-preserving on Ran7rn-1, the polynomials of degree at most n - 2, a space of codimension 1 in Ran 7rn. There is thus a one-parameter family, {C(p) I /? E of unitary modifications of 7rnMz7rlT, obtained by taking Rann, n [Ranrn-l]' (i.e., multiples of cp,-l) to vectors in Rann, n Ran(z7rn-1). One can show

am},

which defines the POPUC. In the past two years, I (partly jointly with Brian Davies and Yoram Last) have been looking at the fine structure of the zeros of orthogonal polynomials [4, 5, 6, 7, 81 as has my student, Mihai Stoiciu, in his thesis [9, lo]. It is this work that I want to review here. Earlier work (see Section 2) established that, in many cases, the bulk of the zeros of OP's approach a canonical density, often uniform density on a circle in the OPUC case and the measure for (first kind) Chebyshev polynomials in the OPFU case. When, motivated by earlier pictures of Saff [ll],I prepared pictures of zeros for my book [l,pp. 414-4231, I was struck by two kinds of regularity shown in Figures 1 and 2.

Figure 1.

638

Figure 1 (taken from [5]) shows the zeros of

when

a!%= ( ~ ) n + 1 ( l + 2 c o s ( ; ( n + 1 ) ) ) a somewhat complicated example t o show features I will discuss later. For At first sight, they now, I will focus on the eighteen zeros very near J z J= do seem t o be converging to a uniform distribution. In fact, the approach t o uniformity is strikingly strong - they seem to be equally spaced.

i.

.

-1

-0.5

.

’i

0.5

’L

‘I

-0.5

r.

5 9..

- 4

”

,“I

..’

Figure 2.

Figure 2, kindly prepared for me by Mihai Stoiciu, shows zeros for degree 70 with CQ, . . . , a 6 8 chosen randomly and independently, according t o a uniform distribution in { z I IzI < +}. The diagonal crosses show the zeros of OPUC with a 6 9 also having this distribution. The circles show the zeros for the POPUC where ,f3 is chosen uniformly in dD. In many cases, they appear as a cross upon a circle. Of course, a single choice is made using Mathematica’s random number generator, so this is a “typical” choice from a random ensemble. Theorems assert that again the bulk of zeros converge for either OPUC or POPUC to a uniform density on aD. At first sight, this seems questionable - look a t the clumping and gaps! But, in fact, this distribution is “regular”: 70 points placed a t random around the circle would show similar clumps and gaps! Thus, the main theme of the work I will describe is clock (strict equal spacing in the limit) behavior for one set of parameters and Poisson distribution for another.

oPuc/PoPuc of

639

Neither idea is totally new. Sixty-five years ago, Erdos-Tur6n [12] discussed O ( l / n ) upper and lower bounds and even proved clock behavior under some strong global hypotheses. In the 199O’s, VBrtesi [13, 14, 151 proved clock spacing for the zeros of Jacobi polynomials. As far as Poisson distributions of zeros, Molchanov [16] first proved an analog for certain random Schrodinger operators, and Minami [17] for some discrete models that overlap random OPRL. There are, however, new technical issues that need to be addressed to get the random results for POPUC [9, 101 and OPUC [8]. Before getting into the details, it is worth making a few remarks: 1. A well-known idea in quantum physics is that the levels repel each other as parameters are varied (see, e.g., [18, 19, 201). Equal spacing is an extreme form of eigenvalues trying to stay as far away from each other as possible. 2. The Poisson result can only hold in the random case because levels do not repel - this is connected to localization in the random model (see Section 12.6 of [2]). 3. There are still open questions especially for OPUC; see the discussion of open questions in [7]. 4. Fine structure of the zeros is intimately connected to suitable asymptotics for the OP’s. 5. There is a third distribution of eigenvalue spacing that occurs, but not, so far, for OP’s, namely those for random matrices [21]. It is intermediate to clock and Poisson. Indeed, GUE and GOE are distinct and both lie between. In fact, there is a one-parameter family (P-distributions) that all lie between. We discuss this further in Section 7. In Section 2, we provide some background on previous work that is related to the subjects we study here. We are not trying to be comprehensive in reviewing all aspects of previous work on zeros which is discussed in Saff [ll]and in Sections 1.7 and Chapter 8 of [I]. In Section 3, we discuss OPUC when the a, are a sum of competing exponentials and an error exponentially smaller. In Sections 4 and 5, we discuss some OPRL and POPUC in the Nevai class and perturbed periodic case, respectively. In Section 6, we discuss O ( l / n ) bounds. In Section 7, we discuss Stoiciu’s work on random POPUC, and in Section 8, the linear variational principle of Davies-Simon and the extension of Stoiciu’s results to OPUC.

640

2. Prior Work Here we mention earlier results that set the stage for later sections. The approximate density of zeros, dun, is the pure point probability measure giving weight k / n t o a zero of multiplicity k (for OPRL and POPUC, k = 1). If dun has a limit, we denote it d u and call it the density of zeros. It is certainly not automatic that a limit exists. Simon-Totik [22] have given examples of OPUC where the set of limit points of dv, is all measures on ID! But under mild regularity conditions, dun does converge:

Theorem 2.1. I n the OPRL case, i f bn to

+ 0,

a,

--t

1, then dv, converges

Theorem 2.2. Let {a,(w), bn(w)}r=l be given by an ergodic process. Then for a.e. w , dv,” converges to a limit d v that is w-independent.

+

Remarks. 1. If b, = 0 , a , = 1, then P, = c,sin((n l)O)/sin(O) where x = 2 cos 0 and d u ( z ) = dO/7r, “explaining” the form of (2.1) 2. Results of the form Theorem 2.1 go back to Erdos-TurAn [12]. It was Nevai [23] who realized all that was needed was b, --$ 0, a, .+ 1 (called the Nevai class). This OPRL work looks a t a , -+ rather than a, + 1. 3. Both theorems appear in the physics/Schrodinger operator literature; see [24, 25, 261. 4. The ideas behind the proofs are quite simple: By ( l . l ) ,J d d u , = ~ T I - ( J ; ;where ~ ) J,;F = 7rnMZr, realized as a cutoff Jacobi matrix. By uniform boundedness of the support of {dv,}, it suffices t o prove convergence of the moments, and that is immediate in the case of Theorem 2.1 and follows from the Birkhoff ergodic theorem in the case of Theorem 2.2. For OPUC and POPUC, the limit results are

Theorem 2.3. If limla,llln = R-l < - 1 and if ~ ~ ~ ~4 I 0, a then j I dv, for OPUC converges to d0/27r concentrated o n the circle of radius R-’. Theorem 2.4. If {a,(w)}r& is given by a n ergodic process and iE(logla,(w)l-’> < 00, then for a.e. w , du; for OPUC has a n w-independent limit supported on dD. Theorem 2.5. If o n dD.

$~

~

~

~ + l0 , a dunj

for l POPUC converges to d0/27r

641

Theorem 2.6. If { a , ( ~ ) } ~ is =given ~ by an ergodic process, du,” f o r POPUC converges for a.e. w to an w-independent limit du. I f also E(loglan(w)]-l) < cm,then this limit is the same f o r OPUC and POPUC.

Remarks. 1. Theorem 2.3 is due to Mhaskar-Saff [27], with a partially alternate proof using CMV matrices in Simon 113. Theorem 2.4 is Theorem 10.5.19 of [2]. Theorem 2.5 is implicit in Golinskii [28]. Theorem 2.6 is an easy consequence of the ergodic theorem (see Remark 3 below).

2. Golinskii [28] also discusses the case where C,”=i(oj - a( + 0 for some a E D. The density of zeros in this case is the equilibrium measure for an arc. Indeed, density of zeros are often equilibrium measures (see Stahl-Totik [29] and Chapter 11 of [2]). 3. As in the OPRL case, positive moments of dv, are given by traces of powers of cutoff CMV matrices. For POPUC, where measures live on r3D and positive moments determine the measure, this is all that is needed for convergence of dun. But for OPUC, one needs a separate argument that shows dun is asymptotically supported on a circle. In the ergodic case, this is easy. In the case of Theorem 2.3, one uses that the product of the zeros is 416, and Theorem 2.7 below.

4. The cutoff CMV matrices for POPUC and OPUC differ in only two rows, so moments of the dun for these two cases have the same limits. If OPUC has a limit supported on dD, POPUC has the same limit.

We mention two earlier results for OPUC that go beyond the bulk results of Theorems 2.1-2.6: Theorem 2.7. If limsupla,(lln = R-’ < 1, then the inverse of the Szego function D(z)-l has an analytic continuation to { z 1 ] z ]< R}. A point zo in { z I R-l < IzI < 1) is a limit point of zeros of @,(z) if and only i f D(z;l)-l = 0. Theorem 2.8. I f limn+m a,+l/o, = b and limn+mIonI = 0 , then f o r large n, @,(z) has no zeros in { z 1 ( z (< b - E } for each E > 0 .

Remarks. 1. Theorem 2.7 is due to Nevai-Totik [30] and the zeros asymptotically outside I z ( = R-’ are the Nevai-Totik zeros. Figure 1 shows three Nevai-Totik zeros. 2. Theorem 2.8 is due to Barrios-L6pez-Saff [31] who also treat cases where, for IbJ < 1, o,bPn approaches a periodic sequence.

642

3. OPUC With Competing Exponential Decay

Here, following [4, 51, we want to discuss zeros of OPUC where Verblunsky coefficients have the form L

an =

C Ctb? + O((bA)n)

(3.1)

e=i

where = b < 1 for f2 = 1 , . . . ,L and A < 1. It is known ([31]for L = 1, [l]for general L; see also [32, 33, 341) that (3.1) is equivalent to:

D ( z ) - l is meromorphic in { z I IzI < b-' with poles exactly at {bhl}&l

+E }

(34

We want to describe a complete asymptotic analysis of the zeros of

Q n ( z ) for n large. Related results were found independently by MartinezFinkelshtein, McLaughlin, and Saff (331. We already know the zeros outside IzI = b are the Nevai-Totik zeros which, by (3.2),are finite in number. Here, from [5], is what happens near the circle of radius b:

Theorem 3.1. If (3.1) holds, then for some 6 > 0, all zeros of Q n ( z ) in { z I b - 6 < IzI < b+6} lie in an annulus of width O(logn/n) about Iz = b, and for n large, they can be ordered via an increasing argument where INn - n1 is bounded. W e have

{.In I}Yz,

Moreover, if { 4n) z j }j=l N n + L are z y ) with the

inserted, and if

+

zN,+L+l -(n) = Zp) 27r, then arg

f o r k = 1,2,. . .,Nn

EZ,- arg zP)

=

n

(3.4)

+L.

Remarks. 1. The errors in Theorem 3.1 are global and can be strengthened away from those finite number of points on { z / JzI = b} where D ( l / z ) - ' = 0. If there are no zeros of D ( w ) - l on { w I IwI = b-'}, then the improvements are global. The improvements replace O(l / n log n) by O ( l / n 2 )and O(logn/n) by O ( l / n ) . 2. Basically, the zeros are equally spaced except for gaps a t z = C = 1 , 2 , . . . ,L. This is clearly seen in Figure 1.

be,

643

3. The proofs depend on detailed asymptotics near IzI = b which are obtained in [5] by carefully iterating the Szego recursion on the circle of radius IzI = b and IzI = b-l. For L = 1, [4] has a different argument analyzing second-order difference equations. As for zeros inside IzI = b, these are controlled by the degree L - 1 polynomials (where we = 6e/b),

which are almost periodic in n and when the we's are roots of unity, periodic in n.

Theorem 3.2. If (3.1) holds, for any E > 0 , there is any N so for n > N , the zeros of cpn(z) in { z I IzI < b - E } are precisely within E of the zeros of P,(z) in this region, where cpn(z) has k zeros within E of a zo with P,(zo) = 0 where k is the multiplicity of the zero of P,. Figure 1 has wl,w2,wg = l , i , - i so P, is periodic mod 4 and the one zero shown for Q.22 with IzI << will recur for Q26, Q30, Q34,. . . . The zero of P22 is at - 1 ) and the zero shown agrees numerically to Except for the results in Section 8, we will not discuss OPUC further. We note [6] has results on zeros of periodic OPUC. Among the open questions for OPUC, we mention:

;(a

Conjecture 3.1. If z,"==,(a,+l - a,I < 00 and lanll/n 4 1, then the zeros are mainly near dD and have clock behavior away f r o m z = 1. Conjecture 3.2. If a, O(n-7) near z = 1.

N

Cn-Y for y < 1, then the zeros have a gap of

4. Clock Behavior Within the Nevai Class

Anticipated results in the decaying random case (see Section 7) show that the Nevai condition b, -+ 0, a, -+ 1 is not expected to be sufficient for clock behavior, but a large subclass does have this behavior. Since the density of zeros is given by (2.1), clock behavior in this case cannot mean global equal spacing in the x scale but in a scale given by du/dx (i.e., the do of Remark 1 after Theorem 2.2). We have:

644

Theorem 4.1. [7] Suppose that the Jacobi parameters of an OPRL obey m

Ian - 11 + IbnI

+

C lan+l

0

-

an1

+ lbn+l

-

bnl < 00

(4.1)

n=l

Then one has clock behavior uniformly o n each interval [-2 the sense that n+m Jim

[{suplnlxl-

x1 -

+E, 2

-

E]

in

(~)-'l

x,x' are successive zeros of pn in [-2

+E, 2 - E]

11

(4.2) =

o

Remarks. 1. (4.1) holds if M

n=l

and also if bn

= n P u , an =

1

+ n-P

for any a , ,B > 0.

2. This result includes results of Vdrtesi [13, 14, 151 as well as results from [4]. The proof of this result has two elements. One is what Schrodinger operator experts would call Jost asymptotics, although in the context of Jacobi polynomials, they go back to Laplace, Heine, Darboux, and Stieltjes (see Szeg6 [35]). This oscillatory asymptotics guarantees that there are zeros with correct asymptotic spacing. The second issue is to ensure that there aren't additional zeros such as sin x- $ sin(n2x) has. In [4], this second issue is dealt with awkwardly. The point of [7] is that one can deal with it easily by proving a priori O(l/n) lower bounds as discussed in Section 6. We note that the control of oscillatory solutions only under (4.1) is subtle, using ideas of Kooman [36]; see also Golinskii-Nevai [37] and Section 12.1 of [2]. For POPUC, we have: T h e o r e m 4.2. [7] If the Verblunsky coeficients an of a POPUC obey M

n=O

645

then f o r any E

> 0 , the POPUC zeros are 2 r l n spaced in that

lim sup

n'oo

[{ [10' n

-

01

-

successive zeros

n

(4.4) of a POPUC of degree n; E

< 0 < 2r -E

5. Clock Behavior for Periodic OPUC It is well known (see, e.g., 138, 39, 40, 411) that if a set of Jacobi parameters, {aho),bho)},"==,, is periodic, that is, (0) = ( 0 ) an+p an

&owp = ($9

(5.1)

for some p and all n, then the essential spectrum of the corresponding Jacobi matrix, J(O, consists of p bands Up=1[aj,,4]where ,Bj <_ aj+l. Generically, ,Bj < aj+l and there are p - 1 gaps, but there can be some closed gaps. There is a polynomial A of degree p so AP1[-2, 21 = Uy=l [ a j ,,Bj] and the density of zeros is given by

A ( x )= 2 cos(pr(1- lc(x))) dlc = density of zeros

(5.2) (5.3)

Already in 1922, Faber [42] knew that dlc is the potential theory equilibrium measure for Up=,[aj,,Bj]. There is some overlap of the results below and work of Peherstorfer [43, 44, 451. As far as the strictly periodic case, [6] contains the following elementary but striking result:

Theorem 5.1. The zeros of Pmp-l consist of the zeros of Pp-l, which has exactly one point in each gap, and those points xc,(:), K = 1,.. . , p ; q = l , . . . , m - 1 , where

Remarks. 1 . Thus, we can precisely give the zeros of Pmp-l.The points in the gaps are related to "Dirichlet data" (see 161). Those in (5.4) are ( m - 1) points in each band. 2. [6] has two proofs of this result. One can also be based on results of Peherstorfer [43]. 3. [6] has additional results on the strictly periodic case and on the OPUC situation which is more subtle. As for perturbations of the periodic case, [7] has

646

Theorem 5.2. Let aL0),bLo) obey (5.1) and suppose that

00

C lan+p

-

an1

+ Ibn+p

-

bnl < 00

(5.6)

n=l

Then for any compact subset, S , of Uy=l(~,pj) (compact in the interior of the bands), we have lim [{suplnlx' - 21 -

n-+w

XI,

(~)-'l

x are successive zeros of p , in S

11

(5.7) =O

6. l / n Bounds

As noted in Section 4, one part of proving of Theorem 4.1 (and also Theorem 5.2) is proving a priori O ( l / n )lower bounds on eigenvalue spacing. Such upper and lower bounds are weaker than clock behavior but are known to hold in much greater generality and are of much greater antiquity, with techniques due to Erdos-Turfin [12], Szego [35], Nevai [23], and Golinskii [28]. [7] has a compendium of new techniques and refinements of old techniques on this problem. We will discuss one result here and refer the reader to [7] for others. The following is interesting because it only requires information at a single value of x. Let Tn(x) be the transfer matrix for solutions of the difference equation built out of the Jacobi parameters. Then Theorem 6.1. [7] Let z,'(z) be the zeros in [x,00) and x. Then

(-GO,%)

closest to

Remarks. 1. If supn llTn(x)II < 00,we get an O(l/n) bound. 2. [7] has an interesting application of this bound which shows that in ergodic cases, Poisson statistics for zeros implies zero Lyapunov exponents.

7. Zeros of Random POPUC In this section, we will discuss Stoiciu's results on zeros of random POPUC. Fix r E ( 0 , l ) once and for all. Our random space, a, will have a sequence

647 a o ( w ) , . . . ,a,(w),

. . . of numbers in D,independent identically distributed random variables (iidrv) with uniform density in { z I IzI 5 T } and a sequence in aD, P,(w), ,&(w), . . . of iidrv with density uniform on dD. The a’s and P’s are independent of each other. {Z;~)(U)};=~ is some listing of the zeros of the OPUC with Verblunsky coefficients {aj(~)}jOO=~ and {.$‘“’(~)}3n,~ of the zeros of the POPUC with { a j ( w ) }and ~ ~ ,B,-l(u). ~ We use #{j

I Z,!”’(w)

E S}

(74

to denote the number of zeros in a set (counting multiplicity) and P(.) to denote the probability of some event which we will list as a series of conditions. A final notation:

s(eo; a, b) = { z E I5 I z # 0 , arg z E (e, + a, e, + b ) }

(7.2)

We can now state Stoiciu’s main result [9, lo]: Theorem 7.1. For any r E ( 0 , l), any 00 E [0,27r), any . . . < be, and any kl,.. . , ke E {0,1,. . .}, we have that

a1

< bl 5

a2

<

This says the zeros are asymptotically Poisson distributed with the same asymptotic distribution as n uniformly randomly distributed points! It is the opposite of clock spacing. The proof uses the strategy of Minami [17] (and earlier, Molchanov [16]) with rather different tactics. One part is an a priori bound of the form Prob(#(j

I z;”)(w)

E S(&; a, b ) ) 2 2)

I C[n(b- .)I2

(7.4)

Minami gets this from a mysterious determinant cancellation, relying on rank one perturbations. Because the analogous POPUC perturbations are rank two, it is not clear how to extend his proof, so Stoiciu instead uses a Prufer angle argument. It is important to understand the other part of his argument since overcoming its limitations is key to handling the OPUC case. Using ideas of Aizenman-Molchanov [46], Aizenman [47],and del Rio et al. [48] and their

648

partial extension to OPUC by Simon [49],Stoiciu proves that the eigenfunctions of the unitary CMV matrices associated to POPUC are exponentially localized near centers. He then considers the unitary matrix obtained from an N x N CMV matrix by replacing the log N Verblunsky coefficients at spacing N / log N by P’s. Such a matrix breaks into log N blocks of size N / log N . A trial function argument and the exponential localization result shows that for all but O((1ogN ) 2 ) zeros of the Nth order POPUC, there are eigenvalues of the decoupled matrix very close to the zeros. Since (N-’N/ log N ) 2 . log N 4 0, for N large, by (7.4), the eigenvalues in S(&; come from different blocks which are independent, so one gets the classical situation where the Poisson distribution arises. The starting point of the next section will be the key step above, where a trial function argument is used. That completes what I have to say about Stoiciu’s work [9, 101. I want to complete this section with a brief mention of some work in preparation by Killip-Stoiciu [50]. As background, we recall some results of [2]. Let y E ( 0 ,l), C > 0, and

%,%)

&j(W)

= CY(n

+ C)-Yaj(w)

(7.5)

be random decaying Verblunsky coefficients. Then the spectral properties have a transition at y = with dependence on C at y = (a) (Theorem 12.7.1 of [2]) If y > the corresponding measures are purely a.c. for a.e. w . (b) (Theorem 12.7.5 of [2]) If y < the corresponding measures are pure point for a.e. w . (c) (Theorem 12.7.7 of [2]) If y = then there is pure point spectrum if iCr2 > 1, and otherwise purely singular spectrum with Hausdorff dimension 1 - i C r 2 . Killip-Stoiciu [50] have tentatively shown results on zeros with a similar structure: If y > there is clock behavior; if y < there is Poisson behavior; and at y = :, depending on C and r , there are P-distributions intermediate between clock and Poisson.

i

i,

i:

i, i,

i,

i,

8. Zeros of Random OPUC

As we noted, the control of random POPUC depends on trial functions, that is, a linear variational principle of the form dist(z,s~ec(A))I ll(A - z)(PII

(8.1)

649

if llpll = 1. This bound holds for normal operators, as can be seen by the spectral theorem. Zeros of POPUC are eigenvalues of a unitary matrix, so (8.1) applies. Since 7rnMz7r, is not normal, (8.1) does not apply to OPUC. Linear variational principles - even with a constant on the right side - do not hold for general non-normal matrices and z .

+

(:A)

Example 8.1. Let A = and pE = (l,&)'/(l E ~ ) ~ / Then ~ . dist(e,spec(A)) = E but [ / ( A- &)pll5 E'. In general, for n x n matrices, one can only hope for bounds if ll(A - z)pll is replaced by ll(A - ~ ) p l l ' / ~ which gains no smallness from exponential decay.

It was this lack that led Stoiciu to focus on POPUC. Davies-Simon [8] realized that by adding an additional condition valid in the case of OPUC, one can get a linear variational principle: Theorem 8.1. [8] If IzI 2 IlAll, A is an n x n mat7iq and llpll = 1, then

Remarks. 1. [8] proves (8.2) where 4n/7r is replaced by cot(7r/4n), which is shown to be the optimal constant. Using cot(z) 5 1/z, one gets (8.2). 2. The proof of (8.2) is not hard. By shifting to a Schur basis, one can suppose A is upper triangular, and by scaling, that z = 1 and IlAll 5 1. Since inf{llBpll I llpll = 1) = ~ ~ B - ' ~ (8.2) ~ - ' ,is equivalent to 4n 11(1- A)-'II 5 -dist(l,spec(A))-' 7r

+

Letting C = (1 - A)-' (1 - A*)-' - 1, and noting C 2 0, one has lCijl 5 ~Cii~1'2~Cjj~1~'. But for i < j , Cij = [(l- A)-l]ij since A is upper triangular. It follows that 2

i < j

(0

i > j

dist(l,spec(A))I[(l- A)-']ijl 5 This easily implies (8.3) with 2n rather than 4n/7r, and with some more work, cot(7r/4n). Once one has (8.2), one uses the localized states for test functions to get zeros of OPUC very close to POPUC and uses (7.4) to prove that with probability 1, for large n, the POPUC zeros are far enough apart that these zeros are distinct. The net result are the following three theorems of Davies-Simon [8]:

650

Theorem 8.2. For any r E ( 0 , l ) with probability 1,

for any k Theorem 8.3. For any r E ( O , l ) , any 00, a , b, and any have that, with probability 1, for large n, all .,'"'(w) in S(00;

8

> 0 , we

%,%) have

1zjn)(w)/ > 1 - exp(-n(l-E)). Theorem 8.4. For any r E (0, l), any 00 E [0,27r), any . . . < be, and any k l , . . . , k t E {0,1,. . .}, we have that

( -,-))27rbp

. Z ~ ) ( W )E S do;

27rap

a1

< bl I a2 <

=kpforp=l,

..., e

e (bP -

~

p=l

e-(bp-ap)

k,!

Note t h a t Theorems 8.4 and 7.1 are almost the same, but in Theorem 7.1, l.Zjn)l = 1, while in Theorem 8.4, 1- exp(-n(l-')) 5 Izj(")I< 1. The zeros of orthogonal polynomials continue t o provide beautiful mathematics.

Acknowledgments

It is a pleasure t o thank Andreas Ruffing for Herculean efforts in organizing what was a very successful conference.

References 1. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 1: Classical Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI (2005). 2. B. Simon, Orthogonal Polynomials on the Unit Circle, Part 2: Spectral Theory, AMS Colloquium Series, American Mathematical Society, Providence, RI (2005). 3. W. B. Jones, 0. Njktad, and W. J. Thron, Moment theory, orthogonal polynomials, quadrature, and continued fractions associated with the unit circle, Bull. London Math. SOC. 21,113-152 (1989). 4. B. Simon, Fine structure of the zeros of orthogonal polynomials, I. A tale of two pictures, preprint. 5. B. Simon, Fine structure of the zeros of orthogonal polynomials, 11. OPUC with competing exponential decay, to appear in J . Approx. Theory.

651 6. B. Simon, Fine structure of the zeros of orthogonal polynomials, 111. Periodic recursion coefficients, to appear in Comm. Pure Appl. Math. 7. Y . Last and B. Simon, Fine structure of the zeros of orthogonal polynomials, IV. A priori bounds and clock behavior, in preparation. 8. E. B. Davies and B. Simon, Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle, preprint. 9. M. Stoiciu, The statistical distribution of the zeros of random paramthogonal polynomials on the unit circle, to appear in J. Approx. Theory. 10. M. Stoiciu, Zeros of Random Orthogonal Polynomials on the Unit Circle, Ph.D. dissertation (2005). http://etd.caltech.edu/etd/available/etd-O5272005-110242/ 11. E.B. Saff, Orthogonal polynomials from a complex perspective, in Orthogonal Polynomials: Theory and Practice (Columbus, OH, 1989), pp. 363-393, Kluwer, Dordrecht (1990). 12. P.Erdos and P. TurBn, On interpolation. 111. Interpolatory theory of polynomials, Ann. of Math. (2) 41,510-553 (1940). 13. P.VBrtesi, On the zeros of Jacobi polynomials, Studia Sci. Math. Hungar. 25,401-405 (1990). 14. P. VBrtesi, On the zeros of generalized Jacobi polynomials. The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th birthday of T. J. Rivlin, Ann. Num. Math. 4,561-577 (1997). 15. P. Vbrtesi, Uniform asymptotics of derivatives of orthogonal polynomials based on generalized Jacobi weights, Acta Math. Hungar. 85,97-130 (1999). 16. S.A. Molchanov, The local structure of the spectrum of the one-dimensional Schrodinger operator, Comm. Math. Phys. 78,429-446 (1980/81). 17. N. Minami, Local fluctuation of the spectrum of a multidimensional Anderson tight binding model, Comm. Math. Phys. 177,709-725 (1996). 18. M. Affronte, A. Cornia, A. Lascialfari, F. Borsa, D. Gatteschi, J. Hinderer, M. HorvatiC, A. G. M. Jansen, and M.-H. Julien, Observation of magnetic level repulsion in Fe6:Li molecular antiferromagnetic rings, Phys. Rev. Lett. 88,167201 (2002). 19. N. Rosenzweig and C. E. Porter, “Repulsion of energy levels” in complex atomic spectra, Phys. Rev. 120, 1698-1714(1960). 20. B. Simon, Internal Lifschitz tails, J . Statist. Phys. 46,911-918 (1987). 21. M. Mehta, Random Matrices, 2nd ed., Academic Press, Boston (1991). 22. B. Simon and V. Totik, Limits of zeros of orthogonal polynomials on the circle, Math. Nachr. 278, 1615-1620 (2005). 23. P.Nevai, Orthogonal polynomials, Mem. Amer. Math. SOC.18,no. 213,185 pp. (1979). 24. L. A. Pastur, Spectra of random selfadjoint operators, Uspekhi Mat. Nauk 28,3-64 (1973). 25. J . Avron and B. Simon, Almost periodic Schrodinger operators, 11. The integrated density of states, Duke Math. J . 50,369-391 (1983). 26. W.Kirsch and F. Martinelli, On the density of states of Schrodinger operators with a random potential, J. Phys. A 15,2139-2156 (1982). 27. H.N. Mhaskar and E. B. Saff, On the distribution of zeros of polynomials

652

orthogonal on the unit circle, J. Approz. Theory 6 3 , 30-38 (1990). 28. L. Golinskii, Quadrature formula and zeros of para-orthogonal polynomials on the unit circle, Acta Math. Hungar. 96, 169-186 (2002). 29. H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge (1992). 30. P. Nevai and V. Totik, Orthogonal polynomials and their zeros, Acta Sci. Math. (Szeged) 53, 99-104 (1989). 31. D. Barrios Rolania, G. L6pez Lagomasino, and E. B. Saff, Asymptotics of orthogonal polynomials inside the unit circle and SzegGPadB approximants, J. Comput. Appl. Math. 133, 171-181 (2001). 32. B. Simon, Meromorphic Szego functions and asymptotic series for Verblunsky coefficients, preprint. 33. A. Martinez-Finkelshtein, K. McLaughlin, and E. B. SafT, Szego orthogonal polynomials with respect to an analytic weight in canonical representation and strong asymptotics, preprint. 34. P. Deift and J. Ostensson, A Riemann-Hilbert approach to some theorems on Toeplitz operators and orthogonal polynomials, preprint. 35. G. Szego, Orthogonal Polynomials, Amer. Math. SOC.Colloq. Publ., Vol. 23, American Mathematical Society, Providence, RI (1939); 3rd edition, (1967). 36. R. J. Kooman, Asymptotic behaviour of solutions of linear recurrences and sequences of Mobius-transformations, J. Approz. Theory 93, 1-58 (1998). 37. L. Golinskii and P. Nevai, Szego difference equations, transfer matrices and orthogonal polynomials on the unit circle, Comm. Math. Phys. 223, 223-259 (2001). 38. H. Hochstadt, On the theory of Hill’s matrices and related inverse spectral problems, Linear Algebra and Appl. 11, 41-52 (1975). 39. Y. Last, On the measure of gaps and spectra for discrete 1D Schrodinger operators, Comm. Math. Phys. 149, 347-360 (1992). 40. M. Toda, Theory of Nonlinear Lattices, 2nd edition, Springer Series in SolidState Sciences, 20, Springer, Berlin (1989). 41. P. van Moerbeke, The spectrum of Jacobi matrices, Innvent. Math. 37,45-81 ( 1976). 42. G. Faber, Uber nach Polynomen fortschreitende Reihen, Sztzungsberichte der Bayerischen Akademie der Wissenschaften, 157-178 (1922). 43. F. Peherstorfer, On Bernstein-Szego orthogonal polynomials on several intervals, 11. Orthogonal polynomials with periodic recurrence coeffcients, J . Approz. Theory 64, 123-161 (1991). 44. F. Peherstorfer, Zeros of polynomials orthogonal on several intervals, Int. Math. Res. Not., no. 7, 361-385 (2003). 45. F. Peherstorfer, On the zeros of orthogonal polynomials: The elliptic case, Constr. Approz. 20, 377-397 (2004). 46. M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: An elementary derivation, Comm. Math. Phys. 157, 245278 (1993). 47. M. Aizenman, Localization at weak disorder: Some elementary bounds, Rev. Math. Phys. 6, 1163-1182 (1994).

653 48. R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum. IV. Hausdorff dimensions, rank one perturbations, and localization, J. Anal. Math. 69,153-200 (1996). 49. B. Simon, Aizenman’s theorem for orthogonal polynomials on the unit circle, to appear in Const. Approx. 50. R. Killip and M. Stoiciu, in preparation.

654

ON THE SYMMETRIES OF INTEGRABLE PARTIAL DIFFERENCE EQUATIONS

ANASTASIOS TONGAS Department of Mathematics, University of Patras 26 500 Patras, Greece We investigate the Lie point and generalized symmetries of certain nonlinear integrable equations on quad-graphs. Applications of the symmetry methods to such equations in obtaining group invariant solutions, related to discrete versions of the Painlev6 differential equations, are also demonstrated.

1. Introduction The study of Sophus Lie in the late nineteenth century on the unification and extension of various solution methods for ordinary differential equations led him to introduce the notion of continuous groups of symmetry transformations. During the same period Backlund investigated possible extensions of Lie contact transformations, introducing an important class of surface transformations in ordinary space. A remarkable feature of Backlund transformations is that due t o a commutativity property repeated applications can be performed in a purely algebraic manner. This is known in classical geometry as the Bianchi permutability theorem and represents a nonlinear analogue of the superposition principle for linear homogeneous differential equations. The archetype is given by the equation

It relates a solution

u12

of the sine-Gordon equation u z y = sinu ,

(2)

with an arbitrary seed solution u and two solutions u1 and u2 obtained from u via the Backlund transformations specified by the parameters values p and q, respectively. On the other hand, equation (1) may be interpreted as a partial difference equation. This interpretation is obtained by simply identifying u1

655

and u2, respectively, with the values attained by the dependent variable u when the discrete independent variables n1 and 7x2 change by a unit step. Nonlinear partial differential equations (PDEs) possessing Backlund transformations and their compatible nonlinear partial difference equations (PAEs) arising from the associated superposition formulae have been the subject of intensive investigations during the past century, leading t o the modern theory of integrable systems. Such systems are characterized also by an extremely high degree of symmetry, and as a result, Lie symmetries and their generalizations have proven to be invaluable tools for generating solutions of difference and differential equations. Symmetry based techniques applied to difference equations have appeared quite recently in the literature starting from different philosophies, see e.g. [1]-[ll] and references therein. In the present work, we investigate the symmetries of certain partial difference equations living on elementary quadrilaterals. Our approach to this problem originates in the interplay between integrable quadrilateral equations and their compatible continuous PDEs, as this has been addressed recently in [5], [7]. We find Lie point and generalized symmetries for the celebrated discrete potential KdV equation. We show that appropriate linear combinations of Lie point and generalized symmetries can be used in obtaining group invariant solutions of the lattice equations. In particular we show that certain reductions of this type lead to discrete versions of the Painlev6 differential equations. 2. Symmetries of quadrilateral equations Central to our considerations are integrable discrete equations on quadgraphs, i.e. certain equations associated to planar graphs with elementary quadrilaterals faces. In particular, we consider equations where fields are assigned on the vertices and the lattice parameters on the edges of Z2. In the simplest case, one has complex fields f : Z2 -+ C assigned on the vertices a t sites ( n l , n2) which vary by unit steps only, and complex lattice parameters ( ~ 1 assigned ~ ~ 2 on the edges of an elementary square (Fig. 1). The basic building block of such equations consists of a relation of the form ~(f,f(l,0),f(0,1),f(l,l);al,Q2) = O7

(3)

which relates the values of four fields residing on the four vertices of an elementary quadrilateral. The forward shifted value of a field along nl and n2 will be denoted by f(o,l), f(o,l) respectively, i.e.

f(1,o) = f(.l + 1 7 n 2 ) 7

f(1,l) = f ( n l

+ 1,n2 + l).

(4)

656

Figure 1. An elementary quadrilateral.

f(O,-l)

f(-i,-i)

Figure 2.

f(1,-1)

A cross configuration.

A specific equation of the type (3) is given by the equation (1). Its linearized version is the following PAE

(P - Q)(f(l,l)- f) = (P+ W ( 0 , l ) - f ( 1 , O ) ) . (5) Let Q be a scalar function which depends on f and its shifted value forming the cross configuration of Fig. 2. We denote the first prolongation of a vector field X Q = Q by the vector field

a,,

(1)

XQ

- Qaf+Q(-l,o) %-1,o)

+Qw) afw) +Q(o,-1)

afw-1) +Q(o,l) afw.1).

(6) Similarly, the second prolongation of X Q is denoted by

X P =XQ ( l ) + Q(-l,-l)af(-l,-l) + Q(-l,l)af(-I,l) + Q(l,-l)af(i,-i) + Q(l,l)af(i,i) + .. .

(7)

Definition 2.1. We say that X Q = Q a f is a symmetry generator of the quadrilateral equation (3), if and only if

X(y(Z) =0 , (8) holds for all solutions of equation (3). Thus, in equation (8) we should take into account (3), and its consequences. It may be seen from the above definition that equation (8) is a linear functional relation for Q. Solutions of the latter equation provide symmetries admitted by equation (3). An indirect approach for determining solutions of the corresponding functional relation for Q, once a specific equation of the type (3) is given, is to derive first a compatible set of differential-difference and partial differential equations, by interchanging the role of the discrete variables ( n l , n2) with that of the continuous parameters ( p , q ) . The reasoning behind this construction is that one could

657

set up a natural framework for the description of the symmetries and reductions of discrete systems, by exploiting the notion of Lie-point symmetries and the infinitesimal methods for obtaining them, which are well known for the continuous PDEs. We next illustrate the relevant construction for the P A E (5). A particular solution of equation (5) is f =

(-y P-X P+X

(CJn2

(9)

X E C!. Differentiating f with respect to p, (respectively q ) and rearranging terms, we easily find that f also satisfies the differential-difference equations (DAEs)

where the minus sign denotes backward shift in the direction of the corresponding discrete variable. By interchanging completely the role of the lattice variables n1 , n2 with that of the continuous lattice parameters p , q, the aim now is to find a PDE which is compatible with equations (5) and (10). Such a PDE is the fourth order equation obtained from the Euler-Lagrange equation

apq(E)-ap(!+q(y)=o, fP9 for the variational problem associated with the Lagrangian density 1 L = z(P2 - q2)(fP9)2

2

+7 (n2fp - n1f9)(n2p2fp - n1q2f,). 9

(12)

Two of the divergence symmetries of Lagrangian L are the scaling transformations

Since every divergence symmetry of a variational problem is inherited as a Lie-point symmetry by the associated Euler-Lagrange equations, the transformations (13) are Lie-point symmetries of equations (11). They correspond to the characteristic symmetry generator

+

X Q = Q df , where Q = c1(pfP qfq) + cz f

,

c1 ,c2 E C! .

(14)

In view of the compatible DAEs (lo), the characteristic Q takes the form c1 Q = - 2( nl(f(-l,o) - f(1,O)) + ndf(o,-l) - f ( 0 , l ) ) ) + c 2 f . (15)

658

Equations (5), (10) and (11) form a compatible set of equations, in the sense that they share a non empty set of solutions. By virtue of this fact and since the symmetry generator XQ given by (14) maps solutions t o solutions of PDE (ll),XQ,with Q given by (15), should generate a symmetry of the discrete equation (5). In other words, Q given by (15) should satisfy

(P - ql(Q(i.1) - Q>= (P+ ~ ) ( Q ( o , i )- Q(i,o))

,

(16)

for all solutions f of (5). Taking into account equation (5) and its backward discrete consequences, we easily find that equation (16) holds. Thus, Q is indeed a symmetry characteristic of equation (5). The above considerations justify the reason why the symmetry characteristic Q of a general quadrilateral equation (3), depends initially on the values o f f assigned on the points which form the cross configuration of Fig. 2. In general, this dependence could be arbitrary. Indeed, as it is illustrated in the following sections, symmetries which correspond to extended cross configurations can be found from known ones.

3. Symmetries of equation ( 5 ) The symmetries of equation (5) are determined from the functional equation (16). TWOsimple solutions of the latter give the symmetry generators

x1 = (p+X(-l)n'+n2)af,

x 2

=fa,.

(17)

Symmetry characteristics corresponding t o the cross configuration of Fig. 2, and which can be found by exploiting the correspondence with the continuous PDE, are given by the vector fields

Yl =

(f(1,O)

- f(-l,O))

a,,

y2

=

(f(0,l) -

f(O,-l)) a,,

(18)

The latter serve t o construct an infinite number of symmetries. This follows from the fact that the commutator of two symmetry generators is again a symmetry generator. Let

be the characteristics of the vector fields

659

By induction we find that

holds V i , j E N \ (0). Repeated applications of the commutation relations (22), (23) produce new symmetries of equation (5), and thus the vector field 2 represents a master symmetry. The generated new symmetries correspond t o extended cross configurations.

4. Symmetries of the discrete potential KdV equation We next demonstrate how the above considerations can be applied equally well to a nonlinear discrete equation, namely the discrete Korteweg-de Vries (KdV) equation [12] ( f ( 1 , l )- f)(f(l,O) - f ( 0 , l ) )= P

(24)

- Q.

Exploiting the symmetries of the continuous compatible PDE and the interplay between the compatible set of differential and difference equations, we have found [13] the following symmetries for the discrete KdV equation x 1

=

a,,

x 2

Y1 =

2 1

=

z2=

( (

1 f(1,O) -

n1 f(1,O)

,Yl] =

df,

y2

+ f(0,l) -

=

1 f ( 0 , l )- f ( 0 , - 1 )

722

- f(-LO) nlP

21

722

+

f(1,O) - f(-l,O)

Taking the commutator of ator [Zl

f(-LO)

x, = ( - l ) n l + n z a f

,

= (-l)nl+n*faf

1

Q

f ( 0 , l ) - f(O,-l)

with

(f(1,O) - f ( - l , 0 ) l 2

f(O,-l)

Y1,

(f

) df, -

;f) a,,

7

a,,

(25) (26)

(27) (28)

one finds the new symmetry gener-

1 - f(2,O)

+ f ( - 2 , 0 ) - f ) a,

(29)

and a similar relation can be found for the commutator [Z,, Y2]. Further new symmetries are obtained by taking the commutator of 21 with the resulting new symmetries.

660

5. Symmetry reduction on the lattice

&a,

Let 3t = 0 be a quadrilateral equation of the form (3) and X Q = a symmetry generator. In analogy with the continuous PDEs, we adopt the following.

Definition 5.1. We say that a solution f of equation 3t = 0 is an invariant solution under X Q , if it satisfies in addition to 'H = 0, the compatible constraint X ( f ) = 0, or equivalently Q = 0. We next demonstrate the notion of invariant solutions of discrete quadrilateral equations by considering a specific symmetry reduction of the discrete KdV (24). For the symmetry constraint we choose a linear combination of Y 1 and Y 2 given by (26). The corresponding invariant solutions are obtained from the compatible system (f(1,l) -

f)(f(l,O)- f(0,l)) = P - 4

(30)

7

= f ( 0 , l ) - f(O,-l)

C(f(1,O) - & L O ) )

(31)

The aim is to eliminate from the above system, one direction of the lattice, i.e. to derive an ordinary difference equation. To this end, we define auxiliary variables z = f(1,l) - f

a = f(1,O) - f ( - L O )

7

9

(32)

.

(33)

Y = f(1,O) - f(0,l), b = f ( 0 , l ) - f(O,-l)

It follows from equations (32)-(33) that b(1,o)

= x - Y(o,-I) , b = x(o,-I) - Y 7

a(o,l)

=z

+ Y(-1,O)

7

a = 2(-1,0)

+ Y.

(34) (35)

With the help of the auxiliary variables (32)-(33),we arrive at the following

OAE (YW+P

=-

(36) yw+6' where w = x x ( - ~ , and ~ ) the parameters are given by cy = -6 = r c, ,B = r2(1+ c), y = 1 - c and r = p - q. Equation (36) is a discrete Riccati equation which can be solved explicitly, by using the symmetry generator W(1,O)

PI

x = ( y w 2 + (6 - Qow - @aw.

(37)

It should be noted that, when c = -1, the invariant solutions obtained above correspond to the periodic reduction f(-1,1) = f(1,-1).

661

6. Symmetry reductions to discrete Painlev6 equations In this section we show how discrete versions of the Painlev6 equations arise from invariant solutions of the discrete KdV. In particular, here we demonstrate two such symmetry reductions using the symmetry generators W1

= 21

+ AX1,

W2

= 2 1 +AX,,

respectively, where the symmetry generators XI, X2 and 21 are given by equations (25) and (27). Invariant solutions under W1 are solutions of discrete KdV, subject t o the compatible constraint W1 (f) = 0, i.e. n1

f(1,O)

- f(-LO)

+

122

f(0,l)

- f(O,-l)

+A=O,

(38)

which in terms of the auxiliary variables (32) reads 721

122

-+-+A=O.

(39) a b Using the auxiliary variables (32) and discrete KdV equation we have T

a = X(-l,O)

+ -X

(40)

7

r = p - q. On the other hand, a straightforward calculation shows that 1 -_ T l - 1 b(1,O)

x2 b

+ -x .

Eliminating the variable b between equations (39), (41) we get

Substituting a given by equation (40) into (42) and rearranging terms we arrive at the second order difference equation

(nl + 1)‘ XZ(1,O)

+ + T

nlr “Z(-l,O)

+T

= n1 + n 2

+ 1 - A-X + A x , T

(43)

known as alternate discrete Painlev6 I1 equation [14]. Invariant solutions of KdV under the symmetry Wz are obtained from the constraint nl n2 A(-l)nl+nz = 0 . (44)

f f(1,O)

-f

f(-l,O)

+f

f(0,l) -

f f(0,-1)

+

In a similar manner as previously, we arrive at the following coupled system of difference equations (Y(-l,O) - x)

(x - Y) = 7-x ,

(45)

662

A(-l)nl+nz(Y - Y(-l,O)) , (46) for the auxiliary variables (group invariants under the symmetry X2) = f f(l,O),

Y = f ( 1 , O ) f(1,l)* (47) The system of equations (45), (46) can be decoupled for the variable 2. Indeed, equation (46) can be integrated once to give 2

Due to a compatibility condition, it turns out that it is necessary to set y(n2) = ~ ( - 1 ) ~Solving ~ . equation (48) for y(-l,o) and inserting into equation (45) we obtain a second order difference equation for the variable 2, involving the parameters T , A, 722, c. The explicit connection of the latter equation with the various discrete Painlev6 equations known so far, cf. [15] and references therein, is under investigation.

7. Concluding remarks The main purpose of this work was to demonstrate that the notions of symmetry and invariance on the discrete level arise naturally from the interplay between PAEs and PDEs that share a common set of solutions. Moreover, certain symmetries which admit the aforementioned cross configuration can be used to derive invariant solutions, in exact analogy with the invariant solutions of the continuous PDEs. Recently in [16], the discrete multi-field Boussinesq system and the compatible PDEs were investigated. It was shown that scaling invariant solutions of the relevant PDEs are built from solutions of higher Painlev6 equations, which potentially lead to solutions in terms of new transcendental functions. We expect that appropriate reductions of the discrete Boussinesq system using the admitted symmetries, will lead to new discrete equations of Painlev6 type, [17]. A detailed study of all inequivalent reductions of this kind, for various integrable discrete equations on quad-graphs, will be given elsewhere.

Acknowledgements I would like t o thank F. Nijhoff, V. Papageorgiou, D. Tsoubelis and P. Xenitidis for useful discussions. This work was supported by the grant Pythagoras B-365-015 of the European Social Fund (ESF), Operational Program for Educational and Vocational Training I1 (EPEAEK 11).

663

References 1. S. Maeda. Canonical structure and symmetries for discrete systems. Math. Japon. 25 (1980), 405-420. 2. V.G. Papageorgiou, F.W. Nijhoff and H.W. Capel, Integrable mappings and nonlinear integrable lattice equations. Phys. Lett. A 147 (1990), 106-114. 3. F.W. Nijhoff and V.G. Papageorgiou, Similarity reductions of integrable lat-

4. 5.

6. 7.

8.

9.

10. 11. 12. 13.

14. 15.

16. 17.

tices and discrete analogues of the Painlevk I1 equation. Phys. Lett. A 153 (1991), 337-344. G.R.W. Quispel, R. Sahadevan, Lie symmetries and the integration of difference equations. Phys. Lett. A 184 (1993), 64-70. F.W. Nijhoff, Discrete Painlev6 equations and symmetry reduction on the lattice. in Discrete integrable geometry and physics (Vienna, 1996) ed. A. I. Bobenko and R. Seiler, Oxford Univ. Press, New York, 1999, 209-234, P.A. Clarkson and F.W. Nijhoff (ed), Symmetries and Integrability of Difference equations. Cambridge University Press, Cambridge, 1999. F.W. Nijhoff, A. Ramani, B. Grammaticos and Y. Ohta, On discrete Painlevk equations associated with the lattice KdV systems and the Painlevk VI equation. Stud. Appl. Math. 106 (2001), 261-314. V. Dorodnitsyn, R. Kozlov and P. Winternitz, Lie group classification of second order ordinary difference equations. J. Math. Phys. 41 (2000), 480504. D. Levi and P. Winternitz, Continuous symmetries of discrete equations. Phys. Lett. A 152 (1991), 335-338. D. Levi, S. Tremblay, P. Winternitz, Lie point symmetries of difference equations and lattices. J. Phys. A 33 (2000), 8507-8523. P.E. Hydon, Symmetries and first integrals of ordinary difference equations. R. SOC.Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), 2835-2855. F.W. Nijhoff and H. W. Capel, The discrete Korteweg-de Vries equation. Acta Appl. Math. (1995), 133-158. A. Tongas D. Tsoubelis and V. Papageorgiou. Symmetries and group invariant reductions of integrable partial difference equations. Proceedings of the 10th International Conference in Modern GRoup ANalysis, 24-31 October, 2004, Larnaca Cyprus, pp. 222-230. A S . Fokas, B. Grammaticos and A. Ramani. From continuous to discrete Painlevk equations. J. Math. Anal. Appl. 180 (1993), 342-360. B. Grammaticos, F.W. Nijhoff and A. Ramani. Discrete Painlevk equations. in The Painlevk property, CRM Ser. Math. Phys., Springer, New York, 1999, 41 3-5 16. A. Tongas and F. Nijhoff, The Boussinesq integrable system: Compatible lattice and continuum structures. Glasgow Math. J. 47A (2005), 205-219. A. Tongas and F. Nijhoff, A discrete Garnier type system from symmetry reduction on the lattice. J. Phys. A (at press).

664

HEUN FUNCTIONS VERSUS ELLIPTIC FUNCTIONS

GALLIANO VALENT * t: Laboratoire de Physique The'orique et des Hautes Energies CNRS Unite' associe'e U R A 280 2 Place Jussieu F-75251 Paris Cedex 05 fiance * Departement de Mathematiques Case 901 163 Avenue de Luminy 13288 Marseille Cedex 9 fiance We present some recent progresses on Heun functions, gathering results from classical analysis up to elliptic functions. We describe Picard's generalization of Floquet's theory for differential equations with doubly periodic coefficients and give the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integral representation which, at level one, is shown to be equivalent to their elliptic form.

1. Introduction

Heun functions [lo] are defined as a natural generalization of the hypergeometric function, to be the solutions of the Fuchsian differential equation

?+(:

_---- s

1-w

y+d+c =a+p

+ 1,

1-Pw €Ic2

dw ) dF+

(s

+ aPk2w)

w(l-w)(l-k2w)

F = 0, (1)

with four regular singularities 0, 1, 1/k2, 00. In [3] the fourth singularity l / k 2 and the auxiliary parameter s is taken as q - s / k 2 . We will is a follow the notation

Hn(k2,s;a7P,y,6;w),

k2 E [0,1].

(2)

In [3] one can find a lot of information on these functions and in the remaining chapters of [22] are gathered many results on their confluent limits. Due to limitation of space the former section misses several important items, particularly the relation between Heun functions and elliptic function theory. These last years this field has received new interesting developments from workers in condensed matter physics [4]and integrable systems [16].

665

Interesting accounts have been given by Smirnov [24], Brezhnev [5] and by Takemura [26], among many others. Our aim is to gather, for the community dedicated to the study of special functions and orthogonal polynomials, these new progresses relating Heun and elliptic functions. As a preliminary step we will give one further motivation for the study of Heun functions, coming from orthogonal polynomials and birth and death processes. 2. From orthogonal polynomials to Heun functions

Let us consider the three terms recurrence [l]

anPn

ZPn = bn-1 P n - 1 +

+ bn Pn+l,

n 2 1.

(3)

We will denote by P, and Q, two linearly independent solutions of this recurrence with initial conditions

PO(2) = 1,

PI(Z)=

x - a0

bo

'

bo a1 131 0

Qo(x) = 0,

1 Qi(x) = -. (4) b0

0 ...

0 bl a2 b2 0 ...

(5)

..................

If the b, > 0 and a , E R the P, (resp. the Q,) will be orthogonal with respect to a positive probabilistic measure II, (resp + ( I ) )

666

When considering applications to birth and death processes with killing [15], it is sufficient to consider the special case where

It is convenient to introduce new polynomials F,(z) by

The positivity constraints A,

> 0,

Pn+1

>0

n

L

0,

(9)

will be assumed. From (3) we deduce

-SF,

p,+lFn+l - ( A n

+ p n + y,)Fn + h - 1 F n - 1 , (10)

F-1(2)

= 0,

Fo(2) = 1.

The orthogonality relation (6) becomes

1

&Jb)= Tn&nn.

Fm(z)

(11)

The particular case where A,, p,, are quadratic is connected with Heun functions in the following way. Let us take

+

+ p),

A, = k2(n a ) ( n

a

p, = n(n

+ y - l),

> 0, p > 0, y > 0, k2 E (0, l),

y, = k"6n ( 12)

k r 2 = 1 - k2,

and consider the generating function

~ ( z ; w= )

C ~ , ( ~ ~ ~ , z ; a , pf, Hy n, (~k 2w, s~; a , b , y , 6 ; w )s, = x-apk2. n20

(13) This definition implies that F ( z ;w) is analytic in a neighbourhood of w = 0 and normalized by F(z;O) = 1. Routine computations show that F is indeed a solutiona of Heun's differential equation (1). Quite remarkably, the most general quadratic birth and death process with linear killing produces, for its generating function, the most general Fuchsian equation with four regular singularities. aIn [22] this function is called "local" Heun.

667

Let us conclude with the proof of

Proposition 2.1. The Hamburger (hence the Stieltjes) moment problem corresponding to the polynomials F,(x) with recurrence coeficients (12) is determinate. Proof. A theorem by Hamburger [l][p. 841 states that if the series c, = P,(O)’ is divergent then the Hamburger moment problem is determinate, i. e. the measure II, defined in (6) is unique. From relation (8) we can write

Now the generating function F ( z ;w),introduced in (13), is analytic for (wI < 1: hence its radius of convergence is one. It follows that, for n + 00 the previous ratio has for limit l / k 2 > 1 and the series { c n } diverges. 0 To have a clear view of the problems encountered in the construction of solutions of Heun’s equation, we will introduce some terminology: we will call “generic” solutions the solutions valid for arbitrary values of the auxiliary parameter s and “non-generic” solutions those which require particular values of s. Let us begin with some results on non-generic solutions. 3. Non-generic solutions 3.1. Special hypergeometric cases

The reason for the restriction k2 E ( 0 , l ) is that in the two limiting cases k2 = 0 (notice that the positivity conditions (9) require k2 > 0) and k2 = 1 the four singular points reduce to three and therefore Heun functions degenerate into hypergeometric functions. Indeed we have: (1) For k2 = 0 the parameters ( a ,p) become irrelevant and we have:

This case, which would correspond to a + 00, is missing in [3].

668

(2) For k2 = 1 we have the relation:

1

Hn(1, S ;a , P , y , 6; W) = (1- w)'

2Fl

(15)

p = z1 (y-(Y-p).

T=p-J-,

This case is considered in [3] but only with the extra constraint s = -a$, i. e. for T = 0. 3 . 2 . The "trivial" solution

This solution corresponds to the special values s = Heun's differential operator is factorized as

(YP = 0. In this case

)

(LD,,+MII D,F=O, with obvious L and M . This factorization leads to the explicit solutions

dw wr(1-

w)b(l-

k2w)E'

(16) Starting from these we can use the change of function

F

+F :

F ( w ) = w P ( 1 - w)"(l- k2w)'F(w)

(17)

which transforms, for special values of the parameters p, (T,T' a Heun function into another Heun function, up to changes in the parameters [3][p.18]. In such a way one generates 7 more Heun functions starting from (16). For all these solutions Heun's differential operator remains obviously factorized and, as shown in [23], this happens only for these cases. 3.3. Derivatives of Heun functions

In general the derivative of a Heun function cannot be expressed in terms of some Heun function with different parameters. However, as shown in [12], this happens in 4 cases. Correcting some misprints of this reference one can write

Hn'(k2,0;c4y'6;~)

+ 2,p + 2 ' 7 + 2'6 + 1;w), = y + 6 + 1 + (y + E + l ) k ?

= -$k% -s'

H n ( k 2 , s';

(Y

(18)

669

H n ‘ ( k 2 ,-apk2; a , p, y,6; w)

+ 2, p + 2, y + 1 , 6 + 2; w),

= @k2( Y 1 - w) H n ( k 2 , s’; a -s’ = y

(19)

+ 6 + 1 + (y + + ap)k? €

H n ’ ( k 2 ,-ap; a , P, 7,s;w)

+ 2, p + 2, y + 1 , 6 + 1;w),

= EYL? (1 - k2w) H n ( k 2 , s’; a

-s‘ = y + 6

(20)

+ ap + (y + + l)k? €

S

Hn‘(k2, s; 0 , p , y , 6; w) = -- H n ( k 2 , s’; 2, p s‘ = s - y - 6 - (y

+ 1 , y + 1,6 + 1;w), (21)

+ €)k2.

One can expand each member in powers of w and check the relevant recurrence relations. 3.4. Reduction to hypergeometric functions

For non-generic solutions it was realized some time ago the possibility for Heun functions t o reduce to hypergeometric functions. Some relations, using Weierstrass elliptic functions are given in [14]. Later on [17] has observed some reductions t o hypergeometric functions of some particular rational variable R ( w ) . This may happen only for polynomial R(w), with the following list :

Hn(-l,O;2~,2b,2c- 1 , 1 + a + b - c ; w )

+

Hn (1/2, -2ab; 2a, 2b, c, 1 2(u Hn (2, -4ab; 2a, 2b, c, c; w) =

+b

2F1

-

(

=

2F1

c ) ; w) =

(ut

7 2 )

7

(

Z F I u;b ;w(2 - 4) 7

).

; 4w(1- w)

(22) Notice that in all these cases R(w) is a second degree polynomial with 3 free parameters. These results have been completed recently by Maier [18], who discovered new cases with cubic and quartic polynomials R ( w ) and bWe have omitted the cases involving complex values of k2.

670

2 free parameters: Hn (1/4, -9ab/4; 3a, 3b, 1/2,2(a Hn (1/2, -8ab; 4a, 4b, a + b

+ b ) ;w) = 2F1

+ 1/2,2(a + b); w)

;4 4 2 - w ) ( l - w)'

(23) These results are not the whole story, since Heun functions may reduce to the product of some function f(w) by some hypergeometric function with variable R ( w ) ,not necessary polynomial. A result of this kind was obtained in [13] by Joyce : Hn (1/4, -1/8; 1/2,1/2,1,1/2; w) =

J

m

Z

(

s 'f2;1f2;~ ( w ) ),

(24)

with the non-trivial function

Let us turn ourselves to the generic solutions of Heun's equation. 4. Generic solutions

4.1. The I92 solutions of Heun's equation

As Heun himself observed [lo] there is a set of 24 substitutions of the variable w which produce a transformation of Heun's equation into another Heun's equation with different parameters. This leads to a complete list of 192 solutions. This list has been fully worked out recently by Maier in [19]. We will just quote the generalizations of the Euler transformation of the hypergeometric function: W k 2 ,s; Q , P; 7 ,6, E ; 'w) = (1 - w)l-6Hn(k2, -s - y(6 - 1); a - 6

+ 1, /3 - 6 + 1;y,2 - 6; w), (25)

and of the Pfaff transformation H n ( k 2 , s; a, p; y,6, E ; 20) = (1 - w)-" Hn(-k2/k'2,-k2/k'2(s+ay);a,a-6+

l,y,a-P+l;w/(w-

1)). (26)

671

4.2. A n integral transform

An integral transform was given in [28]:

H n ( k 2 ,s;a,p, 7,s; w ) =

valid for R e y > R e a > 0 and w E C\ [l,w[. Let us use orthogonal polynomials t o prove this relation [29]. To this end we define the monic polynomials M , by

Mo=Fo=l,

M n ( P ; z )=pl...pnFn(P;z)=n!(y),Fn(P;z), n > 1,

where P denotes the set of parameters k 2 , a , p, y,6. These monic polynomials satisfy the recurrence relation: (A,

+ p n + yn - .)Mn

= Mn+1

+ L 1 p n Mn-1,

n 2 0,

M - l ( z ) = 0, Mo(z) = 1. If we define the new set of parameters P' = ( k 2 , a' = y,p' = p, y' = a , 6' = s + y - a ) ,

z' = s

+ a'P'k2

it is easy to check the invariance relation M,(P';z') = M,(P;z). This induces 2

=s

+apk2,

For Re y > Re a

F,(P; z) = ( a ) , Fn(P';z'),

(r>n

> 0 we can write

2'

=s

+ a'p'k? (28)

then multiplying both sides of this relation by (28) and summing for n 2 0 gives (27) for 1wI < 1. Analytic continuation extends it t o w E C\ [l,w[.

Remarks : (1) This is not an integral equation, since the parameters of the Heun function are changed in the transformation. Notice that several integral equations are known [3].

672

(2) For Ic2

+ 0 we recover Bateman's

integral relation

valid for Re c > ReX > 0. Notice that now X is a free parameter, and this enables one to deduce Euler's integral representation. This does not work for Heun functions because the parameter X is no longer free. (3) For Ic2 = 1 we get the relation

with

F =D -

Jp2

- PY - s ,

p

1 2

= --(7 - a

+P).

This relation does not appear in the extensive list of hypergeometric integrals given in [8], so it could be new. 4.3. Carlitz solutions

In his analysis of some orthogonal polynomials of Stieltjes, Carlitz [6] discovered the following remarkable result: the linearly independent Heun functions with parameters ( k 2 ,s # 0; 0,1/2,1/2,1/2) are given by exp ( f 2i f i z ( w ) ) ,

z(w)=

IW

du

2 4 4 1 - U ) ( l - k2u)'

(31)

To check most simply this result we note that by the inversion theorem of elliptic functions we can write du 2 4 4 1 - u)(l - k2U)

w = s n 2 ( z ,k 2 ) .

673

This conformal transformation maps the singular points as W

0

1

4

4

4

4

-1

z

0

K

K+iK'

iK'

co

where K = K(lc2) and K' = K(lct2)are the complete elliptic integrals of first kind. Using z as new variable and setting F ( w ) = y(z), Heun's equation becomes dz2

1

cnzdnz snzdnz s n z c n z dy - (26 - 1) - ( 2 € - 1)Pdnz dz sn z cn z +4(s

+ ap k 2 sn2z)y = 0. (32)

At the very symmetric point where ap = 0 and y = 6 = E = 1/2, it reduces to

d2Y f 4 s y = 0, dz2

which proves (31). Using then the transformation (17) we can generate a full set of 8 generic solutions (see [28]) where the detailed list is given). So we realize that Heun's equation, in the elliptic functions setting, lies in the field of differential equations with doubly periodic coefficients. It happens that Floquet 's theory for differential equations with periodic coefficients does generalize to the case of doubly periodic coefficients and was derived by Picard. This will be discussed in the next subsection. 4.4. Second kind elliptic functions and Picard's theorem

Since this theorem is not easily available in the standard textbooks [2],[30], we will present some background material and give the detailed proof of Picard. Let us first recall the definition of an elliptic function:

Definition 4.1. A function @ ( z )is elliptic if it is meromorphic and it has two linearly independent periods

+

@ ( z 2w) = @ ( z ) ,

+

@ ( z 2w') = a(.).

674

These two periods define therefore a non-degenerate period parallelogram (usually w = K , w' = iK'). A parallelogram for which no pole lies on its boundary is called a cell. The principal properties of these functions are: Proposition 4.1. An elliptic function: (1) Has as m a n y poles as zeros in a cell, and at least either a multiplicity

2 pole. (2) Can be written @ ( z )= A

H ( z - b l ) . . . H ( z - bn) H ( z - a l ) . . . H ( z - a,) '

a1

+ ... + a ,

= bl

+ ... + b,, (33)

where H ( z ) is one of Jacobi's theta function defined as

(3) Let it be supposed that @ has poles of multiplicity n, at z = a; at can be expanded as:

- a ) + a1 D ,

( 5 )( z - a ) + ...

where the s u m is extended t o all the poles a in a cell. To state Picard's theorem we need another definition Definition 4.2. A function @ ( z )is a an elliptic function of second kind (or a function with constant multipliers) if it is meromorphic and that there exists two non-vanishing constants p and p' such that

+

@ ( z 2w) = ep @ ( z )

@ ( z+ 2w') = ep' @ ( z ) ,

where 2w and 2w' are two linearly independent periods. The constants and ep' are called the multipliers.

ep

Obviously for ep = ep' = 1 we recover elliptic functions. We will have to discuss separately the two cases: (1) Generic multipliers for which wp' - w ' p # 0, ( 2 ) Special multipliers for which wp' - w ' p = 0.

675

4.4.1. Theorems for generic multipliers

Let us consider the function

where a is not congruent to zero. One can check that it is elliptic of second kind with generic multipliers

(

p ' = i 2XK'-i7r-

p = 2XK,

"1 .

K

Proposition 4.2. An elliptic function of second kind with generic multipliers: (1) Admits as many zeros as poles in a cell. (2) Must have at least a simple pole in a cell, otherwise it vanishes. (3) Can be expanded, using f ( z ) defined in (35), as 9(z) =

c{

q)f

(z - a)

+ c1 f ' ( z - a ) + . .. + C*,-l

f (n"-l)(z - a ) } ,

a

(36) where the s u m extends to all poles z = a of multiplicity n, of 9.

Proof. Notice that for any elliptic function of second kind 9(z) with given generic multipliers (e", e"') it is always possible to find X and a in (35) such that f has the same multipliers. It follows that a(.)/ f ( z ) is elliptic. Using (33) we get the most general structure

which proves the first assertion. For the second assertion, let it be supposed that 9 has no pole: it cannot have any zero and therefore 9(z) reduces to A e X Z ,but in this case the multipliers are not generic, contradicting our hypothesis, hence A = 0. For the third assertion, let us consider a pole z = a of multiplicity n,. Near to z = a we have the Laurent expansion f

( k ) (2) =

Jk

( z - a)k+l

+ holomorphic,

so we can find coefficients {ck, k = 0 . . . n, - 1) such that

- ~ { q ) f ( z - a )+ c l f ' ( z - a ) + . . . + c , ~ - l f ( n . - l ) ( z - a ) } a

has no poles in a cell, hence it must vanish.

676

4.4.2. Theorems for special multipliers

This time let us define

H' f ( z ) = ex' z ( z ) .

(37)

We have now Proposition 4.3. Any elliptic function of second kind with special multipliers can be expanded as

@ ( z )= C e A z +

c{

cof(z- a)

+

Clf'(2 -

a)

+ ..

*

cna-l f ( n a - 1 ) ( 2

-u)}

,

a

(38) with the constraint

C e - A a ( c o+ c 1 x + . . . c n a - 1

p - 1 )

= 0,

where the sums extend to all poles a, of order na, of

a(.).

(39)

a

Proof. In this case w p' - w' p = 0. So we can find a value of X such that F ( z ) E @ ( z ) e V A is Z elliptic. Let us consider a pole z = a of order na of @. We have the Laurent series

co - c1 .. @ ( z )= z -a (z-a)2 hence we can write a0 a1 F ( z )= - ___ z -a ( z - a)2

where the new residue is a0

= e-Aa(co

+ xcl + . . . + ~ " 0 - 1c n a - l ) .

Using the expansion theorem for elliptic functions (34) we have

Inserting eAz into the right-hand side, and expressing the derivatives of H' - in terms of the derivatives of f gives (38). The constraint (39) comes H from the fact that the sum of the residues in a cell vanishes for an elliptic function. 0

677

4.4.3. Picard's theorem Now we can state Picard's theorem [20]

Proposition 4.4. Let u s consider the differential equation

dnF dn-lF dzn dzn--l with doubly periodic coeficients

-+ a1(z)-

a l ( z + 2w) = a l ( z ) ,

al(z

+...a n ( z ) F = O ,

+ 2w') = a l ( z ) ,

1 = 1 , 2 , . . . ,n,

A n y meromorphic solution F ( z ) i s a linear combination of elliptic functions of the second kind. Proof. To shorten, and in view of application t o Heun's case, we will consider a differential equation of second order:

with doubly periodic coefficients p and q. Let us consider a solution f ( z ) which is not elliptic of second kind: then the ratio f ( z 2 w ) / f ( z ) cannot be a constant. The periodicity of the coefficients implies that the functions f ( z 2w) and f ( z 4w) are also solutions, so we must have a linear relation of the form

+

+

+

f(z

+ 4w) = A f ( z )+ B f ( z + 2w).

Let us now consider the non-vanishing function 4 ( z ) = f ( z + 2 w ) + p f ( z ) . If we take for p a root of p 2 B p - A = 0 it is easy to check that 4 ( z 2w) =

+

+

( B+ P ) 4 ( Z ) . A similar argument works with respect t o the period 2w'. So we have proved that we can construct a first solution 4 ( z ) which is elliptic of the second kind. We have now to prove that a second linearly independent and meromorphic solution, which can be written as

P' = p , is also elliptic of second kind. The Wronskian

$4'

-

=Ce-p,

c # 0,

678

and the meromorphy of 4 and $ imply that e-p is meromorphic. Since p is doubly periodic, e-p is elliptic of second kind. So F is a meromorphic elliptic function of second kind. Let it be supposed first that its multipliers are generic. Using the expansion theorem (36) we can write

F(z)=

c[ca

f(z

- a)

+ . . . + cn-1

f'"-l'(.

- a)]

,

where the sum includes all the poles of multiplicity n of F ( z ) . Since the integral has t o be meromorphic all the coefficients co must vanish and we get

1

F ( z )dz =

c

[cl f ( z - a)

+ . + cn-1 f'n-2'(z

- a)] ,

and from (36) it is elliptic of second kind as well as $. Let it be supposed now that the multipliers are special. The expansion theorem (39) gives

with the constraint

If h # 0 we get

/

C F ( z ) d z = - ehz h

+

[CI

f(z - a)

+ . . + cn-1 f'""'(z '

- a)]

Since all the coefficients CQ vanish, the constraint (41) implies that F ( z ) dz is elliptic of second kind with special multipliers, hence $ is again elliptic of second kind. If h = 0 we have

G(z)

1

F ( z ) d z = Cz +

[CI

f(z - a )

+ . . . + cn-l f'n-2'(z

- a)] .

This implies

+

+

G(z 2w) = G ( z ) D , so, if (XI,

$(z

+

X2)

2W)

+

+

G(z 2 ~ ' = ) G ( z ) D',

are the multipliers of 4 we can write

= A1 $ ( z )

+ X1D 4 ( z ) ,

$(z

+

2W')

= A2 $ ( z )

+ XZD' 4 ( z ) ,

and we can find a suitable linear combination of $ and 4 which will be elliptic of second kind. 0

679

4.5. The rneromorphic solutions

So let us look for the necessary conditions on the parameters to get meromorphic solutions in the variable z. The computation of the exponents at the singularities of (32) is quite simple and gives

z=o

(sn z )2pl

(077- 1)

z = iK‘ (sn z ) p 2 P 3 ( a ,PI So the necessary conditions for meromorphy are 6 = l2 - m2, E = l - m 3, M=ml+mz+ms,

with the vector F = (mo,ml, m2, m3) E Z4. That these conditions are also sufficient is proved in [9]. From Picard’s Theorem the solutions will be elliptic functions of the second kind. The differential equation becomes -+2 d2y dz2

(-m1

cnzdnz snz

+ m2-s ncnz dzn z +(4s

dn z

+ N ( N - 2mo - 1)k2 sn2z)y = 0 ,

(44) and for F = (n, 0, 0, 0) we are back to Lamb’s equation. It is possible, extracting from y suitable factors of 20, to relate the negative and positive values of the parameters mi as summarized in [24][p. 2961, so from now on we will consider E N4. It is interesting to get rid of the derivative in (44) by the change of function d2Y - - - ( V ( Z) A)Y y(z) = (snz)ml (cnz)m2( d n ~Y)( z~) ~ ===+dz2

(45)

680

This equation has been considered by Darboux [7]. In this short article (3 pages) he claims: that the product of two solutions is a polynomial, which we denote by 9 , , ~ ( aw) ; of degree N in w = sn 2 z and of degree g in o = 4s. The knowledge of this polynomial is of paramount importance as we will see later. Let us quote Darboux: “Une fois le polyn6me 9 dhtermin6, l’int6gration s’achhve, comme on sait, sans aucune difficult&” that for half-integer values of m l , m2, m3 this equation can be integrated (for arbitrary A i. e. for what we call generic solutions). But neither detailed proofs nor the explicit forms of the solutions (even for the simpler case of integer mi) were given. We would like to point out that even for Lam& equation with halfinteger values of n only non-generic solutions, due to Halphen and Brioschi, are known, so that the claim of Darboux seems questionable. For integer n the general solution, due to Halphen and Hermite, is given in [3O][p. 5705751. However it is not so “explicit” since one has to solve a set of n linear equations, even if for low values of n everything can be made explicit. Interestingly enough, this differential equation has also appeared in the quite different field of integrable models, particularly KdV equation. Then relation (45) can be interpreted as a Schrodinger eigenvalue problem, with eigenvalue A (equivalent to the auxiliary parameter s) and potential V ( z ) given by (46). The solutions corresponding to mi E Z are called “finite-gap” solutions and sometimes V ( z ) is called a Treibich-Verdier potential [27], after their work on the subject. The name “finite-gap” refers to the appearance of a finite number of energy bands in the Bloch spectrum, a phenomenon discovered a long time ago by Ince [ll]. 4.6. Level one elliptic solutions

Proposition 4.5. The level one (N = 1) solutions, with o = 4s, are given by:

parameter mo = 1

solution y(z) ,mw)

H(z -W) @ (2)

constraint

*(o,

w)

dn2w = o - k2 o + k 2 w - 1 - k2

681

parameter m2

=1

m3

=1

solution y ( z ) ,ZZ(W)

0 1 ( z - w)

0 (2) ,ZZ(W)

( z - w) @(z)

constraint dn2w = (T

*(g, w)

+ 1- k 2

dn2w = (T

+

( ~ ( 1w) 1- k2 ( ~ ( 1 k- 2 w )- 1 + k 2

(47) 0’ w i t h Z ( z ) = - ( z ) . For generic values of (T a second linearly i n d e p e n d e n t 0 solution i s obtained by t h e substitution z -+ -2. T h e f u n c t i o n *(a, w) is proportional t o t h e product of t w o linearly i n d e p e n d e n t solutions. Proof. We will give the detailed proof for ml = 1, the other cases being analyzed similarly. So we start with d2y dz2

--

2cnzdnz d y -++y=o, snz d z

and we look for a solution y = epzH ( z - ’I. Taking derivatives we get

@(z)

Y’

- =P

+

H‘ +

- P) -

0’

@),

Y Y Let us define the auxiliary function 2 c n z d n z y’ 1 -x ( z ) ?z -2 snz y sn2(z-p)

(f)

1 sn2(z- p)

+ I C s~ n 2 z .

+ k2 sn2z+

(T,

so that proving that x vanishes identically is equivalent to proving that y is indeed a solution of (48). For z = -iK’one can check that the poles in x cancel automatically.

H’

Imposing that the pole at z = 0 is absent gives 1-1= - ( p ) . The absence H of the pole a t z = p is easily checked. So we know that x is bounded in a period parallelogram and since it is doubly periodic, it is bounded in all the complex plane: by Liouville theorem it is a constant. Imposing that x vanishes for z = K fixes the value of p by the equation sn 2 p = -l/u. Then we switch t o the new parameter w = p iK‘ and we express the solution in terms of w using the transformation theory of theta functions. We can now determine the spectral polynomial Q ( a ; z )defined, up to an overall constant factor, by the product of the two solutions of Heun’s equation. Let us do it first for mo = 1. The product of the two solutions

+

682

is computed, using the transformation theory of theta functions, to the identity

H(z -w)H(z 0 2

+

W)

( z )0 2 (w)0 2 ( 0 )

= IC2(sn2z- sn'w),

(49)

which is a first degree polynomial with respect to w = sn2z and to u = 1 k 2 - lc2sn2w. 0

+

Remarks: (1) Notice that w is uniquely defined up t o congruence. (2) The solution for mo = 1 is due to Hermite [3O][p.573] and the solution for m3 = 1 is due to Picard 1211. (3) As already mentioned the other linearly solution is obtained, for generic values of u, by the change z + -2. However, for special values of u, this may fail to give two linearly independent solutions. In this case factoring out one solution gives a second one via a quadrature. Let us consider some examples for the solution with mo = 1:

u=1+k2 w=O

A s n z + B s n z ( H'- ( z ) + ~ H

u=l

w=K

Acnz+Bcnz ($(z)

u=k2

w = K + i K ' Adnz+Bdnz

+

- IctZK (50)

(4) The general structure of the elliptic solutions, for arbitrary level, have been studied in [25], [9]. 4.7. Finite-gap solutions

Let us give a short account of the derivation of the finite-gap integral representations from [24], up to slight notational changes. We will write Heun's differential equation (1) as y"+py'+qy=O.

It is well known that the product Q of two solutions must satisfy Q'"

+ 3pQ" + (p' + 2p2 + 4q)Q' + 2(q' + 2pq)Q = 0.

(51)

683

The crucial point is to prove that there is a polynomial solution ! P g , ~ (w) a; of degree N in the original variable w and of degree g in the parameter a = 4s:

+ + LzN(U) - 1 ) y l - k 2 W ) m 3 ag + b1(w)

! P g , N ( U ,w)= @)(a) wN

= W-(w

+

a1 ( a )wN-l

* * *

xg-1

+ * . . + bg(w).

(52) The second line formula fixes the overall normalization of !#. Then, following a method due to Lindemann and Stieltjes [3O][p. 4201 adapted to Heun’s case, one looks for a solution of the form

Inserting this ansatz into Heun’s equation gives on the one hand

-N‘_ - - p

*

N

Wml(l

N(w)=

- w)rnZ(l- k 2 W ) m 3

Jw(1-

w ) ( l - k2w)

>

(54)

(recall that all the mi are positive) and on the other hand

2!PW‘ - ! P I 2 + 2p!Q9’ + 4 q P (55) N(w)2 The fact that v is a constant is checked by differentiating relation (54) and using the differential equation (51). From this relation it follows also that .’(a) is of degree 29 + 1 in 0. So we get the “finite-gap” solutions: u2(a) =

For the level N = 1 solutions (47) shows that we always have g = 1 and v 2 ( a )is:

mo = 1 m1

=1

m2

=1

m3

=1

( a - l ) ( a- P ) ( a- 1 - k 2 )

+ 1)(a+ k 2 ) a(a - k 2 ) ( a+ 1 - k 2 ) ~ ( -aI)(. - 1 + k 2 )

a(a

(57)

For a given level N one has first to determine the core ingredient, that is the spectral polynomial !P, and then .(a). Let us observe that there is no

684

explicit closed form for Q. This has been done partially up to level N = 5 in [24]. These results show that there do exist integral representations for Heun functions, but they are not “cheap”. Furthermore, as we will show now on a particular example, these integral representations are just a different dressing of the elliptic solutions: they are a kind of “algebraization” of the elliptic solutions, but are one and the same thing. 4.8. Finite-gap versus elliptic solutions

We will give the precise identification of the solutions on the example with rnl = 1, the other cases being discussed similarly. Certainly a general proof would be rather cumbersome! For this identification it is sufficient to consider w E [0,1]. We start from: O = - L 2 sn 2 w, Y~(~)=u(o.+I)(o.+L~),~ ( a ; w ) = 1 + o w . The factor appearing in the exponential of relation (53) is L2

f - sn w cn w dn w

2

J

w dw (1+ O. w),/w(i- w)(1 - Ic2 w).

Under the change of variable w = sn2z, with z E [O,K],it becomes an elliptic integral of the third kind, computed in [3O][p. 5231: J L2snwcnwdnwsn2zdz 1 - L 2 sn 2w sn2z

So, keeping the plus sign, we get for the exponential term in (53)

And we need to multiply this by the square root of

(

Q ~ l - - k ~ s n ~ w s n ~ z = ks 2nsn ~ 2w(w-i K ’) -lc2sn2z and upon use of the identity (49) we are first left with Q K

H(-w

+ 2 + iK‘)H(-w OZ(2)

-z

+ iK’) 7

and after use of the transformation theory for the theta functions we end up with O ( z + w) O ( z - w) Qloc 7 (59) W(z)

685

and gathering (58) and (59) we conclude that the finite-gap level 1 solution (53), with ml = 1, reduces to

which is nothing but the elliptic solution (47). 4.9. Conclusion

The finite-gap solutions or their equivalent elliptic counterparts solve Heun’s equation for all cases where these solutions are meromorphic functions of the variable z. It is quite difficult to say what new ideas will be required to cover the non-meromorphic solutions. These much more difficult problems are left for the future. References 1. N. I. Akhiezer, The classical moment problem, Oliver and Boyd, Edinburgh (1965). 2. N. I. Akhiezer, Elements of the Theory of Elliptic Functions, A. M. S. Tkanslations of Mathematical Monographs, Vol. 79 (1990). 3. F. M. Arscott, “Heun’s equation” in Ref. [22]. 4. E. D. Belokolos, V. Z. Enolskii and M. Salerno, “Wannier functions of elliptic one-gap potential”, arXiv: cond-mat/0401440. 5. Y. Brezhnev, “What does integrability of finite-gap/soliton potentials mean?”, arxiv: nlin. SI/O505003. 6. L. Carlitz, “Some orthogonal polynomials related to elliptic functions”, Duke Math. J.,27 (1960) 443-459. 7. G. Darboux, “Sur une kquation linkaire”, C. R. A. S. 94 (1882) 1645-1648. 8. A. Erdelyi et al, Tables of integral transforms, Vol. 2, McGraw-Hill, NewYork, 1954. 9. F. Gesztesy and R. Weikard, “Treibich-Verdier potentials and the stationary (m)KDV hierarchy”, Math. Z., 219 (1995) 451-476. 10. K. Heun, “Zur Theorie der Riemann’schen F‘unktionen zweiter Ordnung mit vier Verzweigungspunkten” , Math. Ann. 33 (1889) 161-179. 11. E. L. Ince, “Further investigations into the periodic Lame functions”, Proc. Roy. SOC. Edinburgh, 60 (1940) 83-99. 12. A. Ishkhanyan and K. A. Suominen, “New solutions of Heun’s general equation”, J . Phys. A, 36 (2003) L81-L85. 13. G. S. Joyce, “On the cubic lattice Green functions”, Proc. R. SOC. London A, 445 (1994) 463-477. 14. E. Kamke, Differentialgleichungen Losungsmethod und Losungen, Vol. 1, Chelsea, New-York (1971). 15. S. Karlin and S. Tavar6, “Linear birth and death processes with killing”, J. Appl. Prob., 19 (1982) 477-487.

686 16. I. M. Krichever, “Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles”, Funct. Anal. Appl., 14 (1980) 282290. 17. K. Kuiken, “Heun’s equation and the hypergeometric equation”, SIAM J. Math. Anal., 10 (1979) 655-657. 18. R. S. Maier, “On reducing the Heun equation to the hypergeometric equation”, J. Diff. Equations, 213 (2005) 171-203. 19. R. S. Maier, “The 192 solutions of the Heun equation”, arXiv:mathCA/0408317.

20. E. Picard, “Sur les Cquations diffhrentielles linkaires B coefficients doublement pkriodiques” , Journal de Crelle, 90 (1881) 281-302, reprinted in Oeuvres, Vol. 2, p. 61-82, Editions du CNRS, Paris, 1979. 21. E. Picard, “Sur une application de la thCorie des fonctions elliptiques”, C. R. A. S., 89 (1879) 74-76, reprinted in Oeuvres, Vol. 2, p. 11-13, Editions du CNRS, Paris, 1979. 22. A. Ronveaux (ed.), Heun’s Differential Equation, Oxford University Press, Oxford, 1995. 23. A. Ronveaux, “Factorization of the Heun’s differential operator”, Appl. Math. Comput., 141 (2003) 177-184. 24. A. 0. Smirnov, “Elliptic solitons and Heun’s equation”, in The Kowalevski Property (ed. V. B. Kuznetsov), CRM Proc. Lecture Notes, no 32, pp. 287305. American Mathematical Society, Providence, RI, 2002. 25. Sparre, “Sur 1’Cquation ...’’, Acta Math., 3 (1883) 105-140 and 289-321. 26. K. Takemura, “The Heun equation and the Calogero-Moser-Sutherland system IV: the Hermite-Krichever ansatz” , Comm. Math. Phys., 258 (2005) 367-403, and the many references quoted. 27. A. Treibich and J. L. Verdier, Elliptic Solitons, Prog. Math. 88, The Grothendieck Festschrift, Vol. 111, Birkhauser, Boston, 1990. 28. G. Valent, “An integral transform involving Heun functions and a related eigenvalue problem”, SIAM J. Math. Anal., 17 (1986) 688-703. 29. G. Valent, “Associated Stieltjes-Carlitz polynomials and a generalization of Heun’s differential equation”, J. Comput. App. Math., 57 (1995) 293-307. 30. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1965.

9+

687

Discrete Painlev6 equations for recurrence coefficients of orthogonal polynomials Walter VAN ASSCHE"

Katholieke Universiteit Leuven Department of Mathematics Celestijnenlaan 200B 8-3001 Leuven, Belgium * E-mail:[email protected]. be We give four examples of families of orthogonal polynomials for which the coefficients in the recurrence relation satisfy a discrete Painlev6 equation. The first example deals with Freud weights lelPexp(-1~1") on the real line, and we repeat Freud's derivation and analysis for the cases m = 2,4,6. The Freud equation for the recurrence coefficients when m = 4 corresponds to the discrete Painlev6 I equation. The second example deals with orthogonal polynomials on the unit circle for the weight exp(Xcos0). These orthogonal polynomials are important in the theory of random unitary matrices. Periwal and Shevitz have shown that the recurrence coefficients satisfy the discrete Painlev6 I1 equation. The third example deals with discrete orthogonal polynomials on the positive integers. We show that the recurrence coefficients of generalized Charlier polynomials can be obtained from a solution of the discrete Painlev6 I1 equation. The fourth example deals with orthogonal polynomials on {&q" : n E N}. We consider the discrete q-Hermite I polynomials and some discrete qFreud polynomials for which the recurrence ceofficients satisfy a q-deformation of discrete Painlev6 I.

Keywords: Orthogonal polynomials, recurrence coefficients, discrete Painlev6 equations

1. Introduction

Orthonormal polynomials on the real line are defined by the orthogonality conditions

where p is a positive measure on the real line for which all the moments exist and p , ( z ) = ~ ~ y n .z. .~ with positive leading coefficient T~ > 0. A family of orthonormal polynomials always satisfies a three-term recurrence

+

688

relation of the form

with

p-1 = 0

and

Comparing the leading coefficients in (2) gives a,+1=

"ln -> 0,

'Yn+1

(4)

and computing the Fourier coefficients of zp,(z) in (2) gives

(6)

The converse statement is also true and is known as the spectral theorem for orthogonal polynomials: if a family of polynomials satisfies a three-term recurrence relation of the form (2), with a, > 0 and b, E R and initial conditions po = 1 and p-1 = 0, then there exists a probability measure p on the real line such that these polynomials are orthonormal polynomials satisfying (1). This gives rise to two important problems:

Problem 1. Suppose the measure p is known. What can be said about the recurrence coefficients {a, : n = 1 , 2 , 3 , .. .} and {b, : n = 0,1,2,. . .}? This is known as the direct problem for orthogonal polynomials. Problem 2. Suppose the recurrence coefficients {a,+l, b, : n = 0,1,2,. . .} are known. What can be said about the orthogonality measure p? This is known as the inverse problem for orthogonal polynomials. The recurrence coefficients are usually collected in a tridiagonal matrix of the form

J=

(7)

689

which acts as an operator on (a subset of) &(N) and which is known as the Jacobi matrix or Jacobi operator. If J is self-adjoint, then the spectral measure for J is precisely the orthogonality measure p . Hence problem 1 corresponds t o the inverse problem for the Jacobi matrix J and problem 2 corresponds to the direct problem for J . In the present paper we will study problem 1 for a few special cases. In Section 2 we study measures on the real line with an exponential weight function of the form d p ( z ) = lzlp exp(-lzlm) d z , which are known as Freud weights, named after Gkza Freud who studied them in the 1970's. It will be shown that the recurrence coefficients a, satisfy a non-linear recurrence relation which corresponds to the discrete Painlev6 I equation and its hierarchy. In Section 3 we will study a family of orthogonal polynomials on the unit circle. We will first give some background on orthogonal polynomials on the unit circle and the corresponding recurrence relations. We will study the weight function w(0) = exp(XcosO), and it will be shown that the recurrence coefficients satisfy a non-linear recurrence relation which corresponds to discrete Painlev6 11. These orthogonal polynomials play an important role in random unitary matrices and combinatorial problems for random permutations. In Section 4 we will study certain discrete orthogonal polynomials related t o Charlier polynomials. The recurrence coefficients a, and b, are shown to satisfy a system of non-linear recurrence relations which are again related t o the discrete Painlev6 I1 equation. Finally, in Section 5 we consider certain q-orthogonal polynomials which are q-analogs of the Freud weight. We will show that the recurrence coefficients satisfy a qdeformed Painlev6 I equation. Most of the material in this paper is not new: the recurrence relations in Section 2 were already obtained by Freudg and are known as Freud equations in the field of orthogonal polynomials. The identification of the Freud equation with the discrete Painlev6 I equation apparently was first done by Fokas et al.7>8(see also Magnuslg). The recurrence relation in Section 3 was found by Periwal and Shevitz22 (see also Hisakado," Tracy and W i d ~ mBaikl ; ~ ~ used the Riemann-Hilbert approach to obtain the Painlev6 equation). The recurrence relations in Section 4 were obtained by Van Assche and F o ~ p o u a g n i g n iThe . ~ ~ results in Section 5 were first obtained by Nijhoff.21 We hope that bringing together these various orthogonal polynomials and the corresponding discrete Painlev6 equations will be illuminating and encourage researchers in the field of orthogonal polynomials and researchers in integrable difference equations to talk to each other, and that the interaction between both areas of mathematics will shed some extra light on either subject.

690

2. F’reud weights Freud weights are exponential weights on the real line

w,(z)

=

IzIPexp(-lxlm),

p

(-00,

co) of the form

> -1,m > 0.

They were considered by GQzaFreudg in his 1976 paper, where he gave a recurrence relation for the recurrence coefficients a, when m = 2,4,6. For these cases Freud found the asymptotic behavior of the recurrence coefficients a, and he formulated a conjecture for this asymptotic behavior for every m > 0. This conjecture really started the analysis of general orthogonal polynomials on unbounded sets of the real line, since before Freud’s work only very special cases such as the Hermite polynomials and the Laguerre polynomials were studied in detail. In this section we will repeat Freud’s analysis of the recurrence coefficients of Freud weights, make the connection with discrete PainlevQequations (which was not known to Freud but first pointed out by Fokas, Its, and Zhou7 and Fokas, Its, and Kitaev,8 see also Magnuslg), and point out what has been done after Freud. Observe that Freud weights are symmetric, i.e., wp(-x) = wp(z), which implies that b, = 0 for n 2 0. 2.1. Generalized Herrnite polynomials The case m = 2 corresponds to generalized Hermite polynomials (and p = 0 are the Hermite polynomials). Generalized Hermite polynomials were already investigated by Chihara3 (see also Chihara’s book4). The weight wo(z) = exp(-x2) satisfies the first order differential equation

[wo(z)]l= -2zwo(z)

(8)

which is the Pearson equation for the Hermite weight. In general a weight w satisfying a Pearson equation of the form [a(z)w(z)]’ = T ( ~ ) w ( z with ), (T a polynomial of degree at most two and r a polynomial of degree one, is called a classical weight. The weight functions for Hermite polynomials, Laguerre polynomials, and Jacobi polynomials are the classical weights for (T of degree zero, one and two, respectively. Bessel polynomials appear for a(.) = z2, but they are not orthogonal on the real line with respect to a positive measure. Freud’s idea was to compute the integral

L 03

(P,(z)Pn-l(z)14p)’ wo(z) dx

(9)

in two different ways, The first way is simply working out the derivative in the integrand and to use the orthogonality to evaluate the resulting terms.

69 1

This gives

For the right hand side in (10) we use the fact that

+ lower order terms = n-pn-l(z) "In + lower degree terms. "In- 1

pk(x) = nynxn-l

This gives

L 00

"In

p ~ ( ~ ) p n - l ( ~ ) l ~ l fdx ~ o= ( ~n--. )

"In- 1

For the integral in (11) we see that P ~ - ~ ( Xis)a polynomial of degree n - 2 and hence by orthogonality the integral vanishes. For the integral in (12) we use the fact that the weight function w,, is even, i.e., w p ( - X ) = w,,(x), which implies that p n ( - z ) = (-l)npn(x). This means that ~,(X)/X is a polynomial of degree n - 1 when n is odd and pn-l(x)/z is a polynomial of degree n - 2 when n is even. Hence when n is even the integral in (12) vanishes, and when n is odd we have

so that

where

0 if n is even, 1 if n is odd.

692

Observe that this holds whenever w is a symmetric weight on the real line. A second way to compute the integral in (9) is t o use integration by parts, combined with Pearson’s equation (8) for the weight. This gives W

00

(Pn(Z)Pn-l(.)

l ~ l )’” wo(x) dx = -

1,

Pn(x)P,-l(.)

I.I”w1,(x)

.d

00

xP,(z)p,-l(z)lxL:”Wo(x)dx

=2 J--00

= 2a,.

Combining (13) and (14) then gives a: =

so that a, = d one has a, =

n

+ PA, 2

m / f i . For p = 0, which are the Hermite polynomials,

m.For generalized Hermite polynomials one has

2.2. Freud weight r n = 4 Let us now consider the weight w p ( x )= lxlPexp(-x4), where p Pearson equation for the weight wo(x)= exp(-x4) is

> -1. The

[wo(z)]’ = -4x3wo(z), which is a first order linear differential equation with polynomial coefficients, but since the polynomial coefficient in front of wo is now of degree 3, the weight wo is no longer a classical weight but a semi-classical one. The equation (13) remains valid for this Freud weight, but integration by parts give a different result. Indeed 00

L

(P,(S)Pn-l(.)

”1.1 1’ wo(x) dx 00

-

-

-1,

P, (Z)Pn-l(.)

I.l”WXZ)

dx

W

= 4 1 0 0 ~ 3 ~ , ( 3 . ) ~ , - l ( ~ ) 1 ~ 1 dx. pw~(5)

(17)

This integral can be computed by applying the three term recurrence (2), with b, = 0, repeatedly. Indeed

+ a,xPn-l(.) 2 2 = an+lan+2Pn+2(x) + (a,+, + a,)Pn(.)

x2Pn(x) = an+lxP,+l(x)

+ anan-lPn-z(x),

693

J -03

This result holds for n 2 1 if we define a0 = 0. Observe that one can obtain this also by using the calculus of the Jacobi operator, since

1, M

(J3)m,,

=

Z 3 P m ( 5 ) P n ( ~ ) 4 2dX1 )

and the quantity of interest is (J3)n,n-1. This computation is quite simple since it amounts to some simple matrix multiplications. Combining (13) and (17)-(18) then gives 2

2

4a,(a,+,

+ a: +

2

=n

+ PA,.

(19)

This time we do not get a, explicitly, but instead we get a second order non-linear recurrence relation for the recurrence coefficients. The initial conditions are a0 = 0 and for a1 we require that pl(x) = xpo(x)/al has norm one, which means M

x2wp(x)dx = 1, and together with (3) th'is means

If we put x, = 2 4 then clearly xo = 0 , x,

xn(x,+l

+ x, + x,-l)

> 0 for n > 0 and =n

+ PA,.

(20)

This recurrence relation is the discrete Painleve' equation d-PI

with a = 1, p = p / 2 , y = - p / 2 and 6 = 0, since we can write A , = (1 - (-1),)/2. The observation that Freud's equation (19) is a discrete Painlev6 equation was not known to Freud but was pointed out much later by Fokas, Its,

694

and Zhou7 and Fokas, Its, and Kitaev8. This means that the equation has the discrete Painlev6 property, which is known as singularity confinement lo:

Definition 2.1 (discrete Painlev6 property). I f x , is such that it results in a singularity for x,+1, then there exists a p E N such that this singularity is confined to x,+l,.. . ,x , + ~ . Furthermore X,+~+I depends only on xn-1,x,-2,.. .. The usual Painlev6 property for differential equations is that the only movable singularities (singularities which depend on the initial conditions) of solutions of a Painlevi: equation are poles. Poles are isolated singularities, hence a discrete version of poles as singularities is t o require that singularities of a discrete equation are confined. This is the case for discrete Painlev6 I. Consider for instance d-PI in the form xn(xn+1+ xn

+ xn-1)

= ni

then

and if x , = 0 then we have a singularity for x,+1 which becomes f m . For x,+2 we then find ~m and for xn+3 we have the indeterminate form (k~) (rfm). A more careful analysis is t o put x , = E and t o expand x,+k in a Laurent series in E . This gives

+

2, = E

n

- xn-l - E

%+I

=-

x,+3

= --

n+3 E n

+ O(E2)

So we see that as E + 0 the indeterminate form for x,+3 becomes 0, but it does not give a new singularity for x,+4. The singularity is confined to Xn+1, 2n+21Xn+3.

The solution of d-PI can not be obtained in a closed form, but one can say a few things about the behavior of the solution. Freud obtained the asymptotic behavior of the solution of (20) in the following way. Since x, 2 0 for n 2 0 we have X:

I Xn(xn+l+ xn + xn-1)

=n

+pan,

695

so that x n / f i is a bounded and positive sequence. Define A and B as the smallest and largest accumulation points X n

X n

0 5 A = liminf - < limsup - = B < m.

6-

n-00

12-00

Choose a subsequence n' such that xni/& we have 1 5 A(2B

J5i

4

A, then as n'

-+

co in (20)

+ A).

In a similar way we can choose a subsequence n" such that x , i / / f l j B , and as n' -+ 00 in (20) we then find B(2A+ B ) 5 1. Together this gives

B(2A

+ B ) I A(2B + A ) ,

so that B 2 I A2. But since we already know that A 5 B , this implies that A = B and hence X n

lim - = A = = . n-00

If we take the limit in the recurrence relation ( 2 0 ) then one finds 3A2 = 1 so that A = l/& Recall that xn = 2a:, hence as a result we have

The recurrence relation ( 2 0 ) , with initial conditions

is very unstable for computing the recurrence coefficients. Lew and Quarled5 showed that there is a unique positive solution of the recurrence relation ( 2 0 ) with zo = 0. Hence a small error in x1 eventually destroys the positivity of 2,. In Figure 1 we plotted the values of xn obtained from the recurrence relation (with p = 0) by using 30 digits accuracy. The xn are following the asymptotic behavior quite well until n 50 and then large deviations from the true solution appear. Lew and Quarles proved the unicity by showing that there is an operator F acting on a Banach space of infinite sequences with xo = 0, such that the positive solution x = ( Q , x ~x2,. , . .) of ( 2 0 ) is a fixed point: F ( z ) = x . The operator F is then shown to be a contraction, so that the fixed point is unique.

-

Fig. 1. The result of computing x, from d-PI using 30 significant digits

Observe that if we take a weight function of the form w P ( z ) = exp(-z4 Ax2), then the Pearson equation becomes

+

121P

[wo(z)]’ = (-4x3

+ 2Xz)wo(z),

and a slight modification of the previous computations gives the recurrence 4a:(u,+l+ 2

U:

+

- 2Xu; = n + p a n .

If we put z, = 2 4 , then the z, satisfy the discrete Painlev6 equation d-PI with o = 1, ,h’ = p / 2 , y = - p / 2 and 6 = A. 2.3. Freud weight m = 6

For the Freud weight w,(z) = IzlPexp(-z6) one can proceed in a very similar way. The Pearson equation now becomes [WO(X)]’ = -~z~wo(z),

so that ~ (P,.(~)P,-~(~)I~I~)‘~~(~) da: = 6

5 ~ ~ ( ~ ) ~ , - l ( ~dz. ) ~ ~ ~ P ~

697

The integral on the right is

and this is 2

2

(J~),,,-~= a,(~,-~a,-~

2 2 + a,-l + 2a,-,4 + a,-la:+l + a: + 2a:a:+, + a;+1+ 4

2

an+1ai+2).

Together with (13) this gives 2

2

2 + a,-l+ 2an-1a: + ~:-laX+1 + a: + 2a:a:+, + + 0:+1a;+2) 4

2

6an(an-2an-l

+:I.

= TZ

+ PA,.

(22)

This is a fourth order non-linear recurrence relation for the recurrence coefficients. It is within the hierarchy of discrete Painlev6 I (Cresswell and Joshi'). Freud was also able to obtain the asymptotic behavior for the a, in this case. Obviously 2

6 4 6 6a:(an-2a:-1 hence a,/nl/'

+ an-l + 2a:-1~2 + ~:-1a:+1 4

+ a: + 2a:a;+1+

a:+i+ a:+la:+2) = n

+ pan,

is a positive and bounded sequence. If we define

a: < limsup a: = B 0 6 A = liminf n+co nU3 n1l3

< m,

then by taking a subsequence that converges to A we find

6A(5B2+ 4AB

+ A) 2 1

and by taking a subsequence that converges to B we find

6B(5A2+ 4AB + B2)6 1. Combining both inequalities gives

6B(5A2+ 4AB so that A2B

+ B2)5 6A(5B2+ 4AB + A )

+ B3 6 AB2 + A3. This is equivalent with ( B - A)(A2+ AB + B2)5 ( B - A)AB. +

If we assume that A < B , then this would imply that A2 B2 6 0 , or A = B = 0, which is impossible (since A < B ) . Hence our assumption is false and A = B. Taking the limit in (22) gives 60A3 = 1, hence

698

2.4. bud's conjecture

On the basis of (16)’ (21) and (23), Freud made the conjecture that for every m > 0 and p > -1 one has

Furthermore Freud showed that if the limit exists for even m, then it is equal to the expression in (24). Freud could not prove the existence of the limit for even m L 8 because the central coefficient a,, in the non-linear recurrence relation does not occur sufficiently often and more non central coefficients a n f k with k # 0 appear, making the recurrence no longer ‘diagonally dominant’. The simple trick using limsup and liminf then no longer suffices to show that the limit exists. The proof of Freud’s conjecture for every even integer m was given by Alphonse Magnus17>18.His proof still consists of obtaining a non-linear recurrence relation for the a, (the Freud equation, which is within the hierarchy of discrete Painlev6 I), but a more subtle argument is used to prove the existence of the limit. Freud’s conjecture for general m > 0 was finally proved by Lubinsky, Mhaskar and Saff 16. The proof for general m > 0 no longer uses a recurrence relation for the recurrence coefficients but relies on the Mhaskar-Rakhmanov-Saff number and results of weighted polynomial approximation and an equilibrium problem of logarithmic potential theory with external field. For m an even positive integer, MBt6, Nevai and Zaslavsky2’ obtained an asymptotic expansion of the form

where Q is the constant in Freud’s conjecture (24), but the other coefficients C k with k > 0 are not explicitly known. Their analysis is again based on the non-linear recurrence relation for the recurrence coefficients.

3. Orthogonal polynomials on the unit circle In this section we will consider orthogonal polynomials on the unit circle T = {z E C : = 1). A very good source for the general theory is the recent set of books by Barry Simon24. The sequence of polynomials {cp,, n = 0 , 1 , 2 , .. .} is orthonormal on the unit circle with respect to a weight w if

(zI

z = ei0.

699

These polynomials are unique if we agree to make the leading coefficient positive: cp,(z) = K,Z,

+ .. *

,

K,

> 0.

The monic polynomials are usually denoted by a n ( z ) = cp,(z)/~,. An important property, which replaces the three term recurrence relation for orthogonal polynomials on the real line, is the Szego recurrence = %+l(Z)

Z@n(Z)

+(.,a;(.),

(26)

where @t(z) = z"S,(l/z) is the reversed polynomial and 5, is the polynomial @, but with complex conjugated coefficients. In Simon24 the recurrence coefficients a, (n = 0,1,2,3, . . .) are called Verblunsky coefficients. They are given by a, = -@,+1(0) and they satisfy la,( < 1 for n 2 0 and a-1 = -1. An important relation between K, and a, was found by Szego: n

K:

=

19k(o)12, k=O

from which it follows that

so that

4

- - - 1 -\a,[ 2

.

G+1

3.1. Modified Bessel polynomials We will take a closer look at the orthogonal polynomials on the unit circle for the weight

w(8) = exp(Xcos8),

8E

[-T,T].

observe that w ( 4 ) = w(8), which implies that the Verblunsky coefficients are real. Ismail [13, pp. 236-2391 calls the resulting orthogonal polynomials the modified Bessel polynomials since the trigonometric moments of this weight are in terms of the modified Bessel function

These polynomials appear in the analysis of unitary random and play an important role in the asymptotic distribution of the length of the longest increasing subsequence of random permutations2.

700

Periwal and Shevitz22 found a non-linear recurrence relation for the Verblunsky coefficients of these orthogonal polynomials (see also Hisakado’l and Tracy and W i d ~ m ~If~z) = . eis then

and this weight satisfies a Pearson equation of the form x 1 W’(Z) = -(1 - -)G(z). 2 22 Consider the integral

(28)

then by means of Pearson’s equation ( 2 8 ) we find

The first integral on the right is zero because of orthogonality. For the second integral we use the recurrence ( 2 6 ) for the orthonormal polynomials

to find

If we use (30) for n

+ 1 and orthogonality, then

because

+

d + I ( Z ) = ( P n + l ( o ) ~ n + l ( t ) lower degree terms =-

+

C X ~ ( P ~ + ~ ( lower ~ )

degree terms.

701

In a similar way we have

zcp;(z) = --an-l-

Kn

pn+1(z)

Kn+1

+ lower degree terms

so that zcp~(Z)cpn+l(Z)W(B) dB =

x Kn -pG--. Kn+1

Combining all these results gives dz

A

K,

- = --(an+la,+ z

(31)

an=).

2&+1

We can compute this integral also using integration by parts, to find dz

dz.

-= Z

We have to be a little bit careful because zcpn(z) is not analytic in the complex plane, but on the unit circle T we have cpn(z) = z-"cp;(z) so that

dz dz

dz

If we use the recurrence relation (30) then

by orthogonality we find

= 0,

and if we use

~p;+~(z)= (n+ l)-cpn(z) Kn+ 1 Kn

+ lower degree terms,

702

then

Kn+ 1

= ( n -t1)-.

Kn

These computations give

Now we can combine (31) and (32) t o find

which, together with (27) gives

x

+

--(1 - Ia,(2)(a,+la,+a,=) = ( n l)la,12. 2 Recall that w(-19) = w(0) implies that the a, are real. Hence when a, then

+

#0

+

- - x( 1 - a,)(a,+1 2 an-1) = ( n l)a,. (33) 2 This non-linear recurrence relation corresponds to the discrete PainlevB equation d-PII

with a,

= x,,

a = P = -2/X and y = 0. The initial values are

4. Discrete orthogonal polynomials In this section we will study certain discrete orthogonal polynomials on the integers N. The orthonormality now becomes ca

x p n ( k ) p m ( k ) w k = &,m,

n, m 2 0.

(34)

k=O

Instead of the differential operator we will now be using difference operators, namely the forward difference A and the backward difference V for which

Af(x) = f(x + 1) - f(x),

Vf(4

=

f(.)

-

f(.

-

1).

703

We now have two sequences { a , : n = 1 , 2 , .. .} and {b, : n = 0 , 1,2,. . .} of recurrence coefficients, and we need two recurrence relations to determine all a, and b,.

4.1. Charlier polynomials Charlier polynomials are the orthonormal polynomials for the Poisson distribution

Observe that

k

wk-1 = - wk

a

(35)

which is the (discrete) Pearson equation for the Poisson distribution. It can also be written as aVwk = ( a - k)wk. The Pearson equation gives the following structure relation for Charlier polynomials.

Lemma 4.1. For the orthonormal Charlier polynomials one has an Pn(Z 1) = P,(Z) -Pn-l(z), a where a, are the coeficients an the recurrence relation (2).

+

Proof. If we expand p,(z

+

+ 1) into a Fourier series, then n

P,(Z

+ 1) = C A , , j P j ( X ) , j=O

and if we compare the leading coefficients then A,,, = 1. The other Fourier coefficients are given by

and if we use (35) then this gives

+

The polynomial z p j (z - 1) has degree j 1, hence by orthogonality A,,j = 0 whenever j < n - 1. For j = n - 1 we have Yn-1

XPn-l(z - 1) = -p,(z) Yn

+ lower degree terms

704

so that (4) gives the desired result.

0

Note that we can write (36) also as

If we compare the leading coefficient in the latter, then ny, = a,y,-l/a, so that

a; = an. We will now compute the sum

in two different ways. If we use the Pearson equation ( 3 5 ) then 00

c p i ( k

M

+ 1)Wk = x p i ( k ) w k - l

1 ”

-

a

kpi(k)wk =

bn

k=l k=O On the other hand, the structure relation (36) gives k=O

Combining both computations gives

b n = a + - a: ==+a.

a

These simple computations show that the recurrence coefficients for Charlier polynomials are given by an =

bn = n + a. 6,

4.2. Generalized Charlier polynomials

If we take the weights wk

ak (k!)N’

-

k E N , a>0,

with N E { 1 , 2 , 3 , . . .}, then for N = 1 one finds the Charlier polynomials and for N >_ 2 the generalized Charlier polynomials. These were introduced by Hounkonnou et a1.12. The Pearson equation is

705

which can also be written as a V w k = ( a - k N ) W k . For N 2 2 the factor a - k N is a polynomial in k of degree greater than one, and hence the weight is no longer classical but semi-classical. We will investigate the case N = 2 in more detail.

Lemma 4.2. For N structure relation

=

2 the generalized Charlier polynomials satisfy the

where an are the recurrence coeficients in the three-term recurrence relation (2). Proof. If we expand p n ( z

+ 1 ) into a Fourier series, then n

pn(z

+ 1) = C A n , j p j ( z ) . j=O

Comparing coefficients of zn gives An,n = 1 , and comparing coefficients of 1 gives An,n-l = n / a n . The remaining Fourier coefficients are given by 00

00

=c

pn(k k=O

&,j

+ l ) p j ( k ) w k= c p n ( k ) p j ( k

-

1)wk-1.

k=l

If we use the Pearson equation (37) then

k=O

+

The polynomial z 2 p j ( z - 1 ) is of degree j 2 and hence by orthogonality An,j = 0 for j < n - 2. For j = n - 2 we have 2

7n-2

z pn-2(z - 1 ) = -pn(z)

"ln

+ lower degree terms

so that

where we used (4), which gives the required result. The structure relation (38) can also be written as

706

If we compare coefficients of xn-', where we use

then we find

If X I , , < ~ 2 , n< . . . < xn,n are the zeros of p,, then by ViBte's symmetric formulas we have n

dn = -CXb,". "In

k=l

The zeros of p , are equal to the eigenvalues of the truncated Jacobi matrix

(ao

ai a1 bi a2

I

...

and the sum of all eigenvalues is the trace of the matrix, hence n-1

-=-Cbj. dn "In

j=O

If we use this in (39), then

(1)

n- 1

- nbn-l+

C bj = a i aai P 1 ~

j=O

In order t o get rid of the non-homogeneous terms, we put b, = n the relation becomes

+ in, and

Differencing both sides gives *

-na(bn

I

- bn-1)

2

2

2

= an(an+l- an-l),

(41)

which may be considered as the first Freud equation for the recurrence coefficients.

707

Next, we will compute 00

+

x P n ( k + l)Pn-l(k 1)wk k=O in two different ways. First we use the Pearson equation (37) to find 00

00

xPn(k k=O

+ l)Pn-l(k + 1)wk = xPn(k)Pn-l(k)wk-l k=l

The entry (J2)n,n-1can be computed easily by repeatedly using the recurrence relation (2) and is equal t o (J)&-l = an(bn bn-1)7 so that

+

On the other hand, we can use the structure relation (38) t o find 00

k=O

n (n-l)a, -7 (43) an U where the last equality follows from the orthonormality (34). Combining (42) and (43), and recalling that b, = n in7then gives the second Freud equation

- +

+

A

+ b,-l+

.

.

n ) = nu.

(44)

If we eliminate nu from the two equations (41) and (44), then .

-(bn

.

+ b,-l+ A

2 n ) ( L - 6 - 1 1 = a:+1 - an-l.

Summing both sides of this equation gives n- 1

4:

+ C i& - nbn = a,+, + a: 2

k=O

- a.

(45)

708

Summing both sides of (41) gives

-)

bk - nb,

= a,a,+l. 2 2

Combining (45) and (46) then gives -

2 a(a,+,

+ a i j + a2 = a i i ,

which is equivalent to

aii

- a ) ( a2, - a ) .

=

This means that a: - a and we must conclude that a; - a

(47)

- a have the same sign, and since a0 = 0

< 0 for n 2 0. We may therefore write

a: = u ( 1 - c,), 2

(48)

with q,= 1, and then (47) becomes in =

(49)

J;lcn&+l.

The second Freud equation (44) becomes

(1 - ci)J;l(cn+l+ cn-1)

n&*

1

(50)

If we compute the coefficients c, from the recurrence relation (50), then we obtain the recurrence coefficients b, = n 6, from (49) and the a, from (48). The non-linear recurrence relation (50) corresponds to the discrete Painlev6 I1 equation

+

L+1+

2,-1

=

+ +Y

xn(an P ) 1 -xi

with c, = x, and a = I/@, P = y = 0. We need to find the solution with Q = 1 and cf = 1 - .:/a. Observe that if we require that pl(x) = (x - bo)po/al is orthogonal to PO,then 00

k=O so that

k=O

709

where

is the modified Bessel function. From (49) we then see that c1 =

I1( 2 4 -

IO(24)' The non-linear recurrence relation (50) with initial conditions co = 1 and c1 = I1(2&)/10(2&) is again very unstable for computing all the c, recursively. One can show that the discrete Painlev6 equation with y = 0 and Q = f l has only one solution for which -1 < c, < 1 for all n 2 1 (Van Assche26), and this is the solution that we need since u i = a(1- c i ) needs to be positive. Hence a slight deviation from the actual initial value c1 will destroy the positivity of the eventually. In Figure 2 we have plotted the c, obtained from the recurrence relation with an accuracy of 30 digits. The c, converge quickly to zero, but for n near 40 we see that the c, deviate quite a lot from zero.

The discrete Painlev6 I1 equation also arose in Section 3 for the Verblunsky coefficients of certain orthogonal polynomials on the unit circle.

710

Verblunsky coefficients always have the property that [ant< 1 for n 2 0, hence in that case one also requires the unique solution with a-1 = -1 for which -1 < a, < 1 for n 2 0. Observe that there is a shift in the index since we are using Verblunsky coefficients, in which case the recurrence starts with a-1 = -1. Obviously the equation (50) satisfies the discrete Painlev6 property. Indeed, we have . -

hence a singularity will appear in %+I gives that for c, near 1

whenever c, = f l . A careful analysis

&=1+E

n 1 n c"+l=-G;-V%-1 + O(E) c"+2

= -1

n+2 +O ( 2 ) +n E

and near -1

n

Cn+1

= -2&

%+2

=1

%+3

=-

1

n

;+ 4&

- %-l+

O(E)

n+2 + O(E2) +n E

n n+l -&-1 &(n+2) n+2

+ O(E),

so that in both cases the singularity is confined to %+I and &+2. Observe that the critical value 1 for c, results in the critical value -1 for &+2, and that the critical value -1 for c, results in the critical value 1 for &+z.

5. q-Orthogonal polynomials Here we consider orthogonal polynomials on the exponential lattice {fq,,n E N}, where 0 < q < 1. The orthogonality is of the form

711

where the q-integral is defined by ,.l

00

We will only consider even weights for which w(-x) = w ( x ) , in which case the orthogonal polynomials have the symmetry property p n ( - x ) = (-l)npn(x), i.e., the polynomials are even when n is even and odd when n is odd. The recurrence relation will then be of the form

xcpn(.)

= an+lPn+l(z)

+ anPn-l(x),

(53)

with p-1 = 0. The results in this section were obtained for the first time by Nijhoff 2 1 , but we take a slightly different approach.

5.1. Discrete q-Hermite I polynomials The orthonormal discrete q-Hermite I polynomials [14, 83.281 are given by

where

n 00

(2; 4 ) m

=

(1 - Wk).

k=O

Observe that

so that the weight can be defined as w ( x ) = ( x 2 q 2 ; q 2 ) , . In terms of the q-exponential function E q ( z )= ( - ~ ; q we ) ~have w(x) = Eq2(-x2q2),and since E q ( ( l - q ) z ) -+ exp(z) when q --+ 1 it follows that w(4-x) -+ exp(-x2) when q + 1, which shows that this weight is a q-analog of the Hermite weight. One easily finds that (1 - .”w(.)

= w(z/q),

(54)

which is the Pearson equation for this weight on the q-lattice. The structure relation for the corresponding orthogonal polynomials is in terms of the qdifference operator D, for which

712

Lemma 5.1. The discrete q-Hermite I polynomials satisfy

Proof. Clearly D q p n ( z )is a polynomial of degree n - 1 and Dqpn(-z) = (-l)n-lDqpn(z). If we expand this polynomial into a Fourier series, then n-1 ~

C aj,npj

p (%>~= n j=O

(21,

with 1

aj,n = S _ , 4 I P n ( 4 P j ( 4 W ( 4 dqz. The symmetry shows that aj,n = 0 whenever n - j is even. When n - j is odd then

00

k=O 00

00

k=O

k=O

Both sums are finite since either pn or p j is an odd polynomial. Using the Pearson equation (54), and a shift in the summation index in the first sum, gives 00

00

The first integral on the right is zero because of orthogonality. The second integral only gives a contribution when j = n - 1, in which case

an--l,n

=

s'

1 - q -1

zPn(z)Pn-l(z/q)W(z)dqz.

The recurrence relation (53) gives w n - 1 ( z / q )= q a n p n ( z / q )

+ qan-lpn-z(z/q),

713

and since

+ lower degree terms

p , ( z / q ) = q-np71(z)

this gives

which gives the desired structure relation.

0

If we compare the leading coefficients on both sides of (55),then 1 - qn an - 771- 1 qn- 1

7 7 1

(1 - 4 ) ’

1-q

so that we find

a; = q y 1 - q n ) , which are indeed the recurrence coefficients as given in [14, 83.281. So for these orthogonal polynomials the recurrence coefficients can be found immediately from the structure relation (55). Observe that the a: tend to zero exponentially fast and that

and

2

-

n

lim -- q - + l 1 - q2 2’ and the latter are the recurrence coefficients (15) for the Hermite polynomials ( p = 0). 5.2. Discrete q-Freud polynomials

A q-analog of the Freud polynomials with weight exp( -x4) can be obtained by taking the weight ~ ( z= )(q4z4; q4)m = E44(-q4z4) on the exponential lattice. Observe that w ( q m z ) 4 exp(-z4) as q 4 1. This weight satisfies (1 - .4)w(.>

=w(z/q),

and the structure relation for these semi-classical polynomials is:

Lemma 5.2. The orthonormal polynomials for which

(56)

714

satisfy

with

An =

anan-1%-2 q - 3

(58) (59)

Proof. Expanding Dqpn into a Fourier series gives

with

Again aj,n = 0 whenever n - j is even. When n - j is odd then, as in the proof of Lemma 5.1, we have

where we have now used the Pearson equation (56). Again the first integral on the right vanishes because of orthogonality. The second integral only gives a contribution when j = n - 1 or j = n - 3. For j = n - 3 we have

1’

an-3,n = -

1-Y

x3Pn(z)Pn-3(x)w(x>dqx,

-1

and since 7n-3

x 3 p n - 3 ( z / y ) = -4

7n

we easily find

-n+3

pn(z)

+ lower degree terms

715

which gives (58). For j

=n

-

1 we have

and if we write

Z3Pn-i(z/q) = An+zpn+2(2)

+ B n p n ( z ) + lower degree terms,

(60)

then the orthonormality gives

Bn

an-1,n = 1-q'

If we compare coefficients of zn in (60) then 6 n - 1 ~ - ~= + An+26n+2 ~

+

+ Bnyn,

+ . . ., so that using (58) gives

where p n ( z ) = 7nzn hnxn-'

If we compare coefficients of

9 - l

in the recurrence relation (53) then

6n = an+lhn+l+ anTn-1, from which one easily finds dn+1

6n -

'Yn+l

7n

2

-an

7

which gives n- 1

and using this in the formula for Bn gives the desired expression (59).

0

If we compare coefficients of znP1in the structure relation (57) then ~ n ( 1 - qn) = Tn-lBn.

Comparing coefficients of znp3in (57) gives dn(1-

qn-2)

which together with (62) gives

= hn-lBn

+ 7n--~An,

(62)

716

Together with (58) and (61) this gives

On the other hand, if we compare (62) with (59) then we find

which can be written as $-'(I

- qn) =

ai+1

+ a: + ai-1 + (1- q 2 )

j=1

If we take (63) with the index n raised by one, then we can find

c

n-2

(1 - q 2 )

=p + 3 2

2 2

2 an+lanan-l - (1- q2 )un-l - (1- q-n+l)a:,

j=1

and if we insert this in (64) then we find the second order non-linear equation qn-l(l-

qn) = a: (an+l +

P + la,

+ q an-l+ q-2n+3an+la:d-l).

(65) We claim that this equation is a q-deformation of the discrete Painlev6 I equation. Indeed, if we take xn = then

ai/dm

= xn

which for q

(xn+1+ 4

-+

-n+l

xn

+ q2xn-1 + (1- q4)q-2n+3xn+1xnxn-1)

1 converges to n

- = zn(xn+1+

271

+ xn-l),

which is the discrete Painlev6 I equation (19) for Freud polynomials (with p = 0). If we put a: = qn-'gn then (65) can be rewritten as n

4 (Yn+lYn

+ l)(Yn-lYn + 1) = 1 - Y2.

(66)

This could therefore be called a q-discrete Painlev6 I equation (q-PI). We can easily find the asymptotic behavior as n .+ 00. First observe that from (65) we find the upper bound q-n+1a;

5 qn-l(l

- qn),

717

so that a t 5 qZn-'(1 - qn), and a, tends t o zero as n -+ 00. Let A = limsup,,, a2/qn-', then if we take a such that ai/qn-' converges t o A, equation (65) gives A2 = 1. A similar reasoning also shows that B = liminf,,, a2/qn-' is such that B2 = 1. Hence we may conclude that

The equation (66) has the singularity confinement property. Indeed, a singularity occurs for yn+l whenever yn = 0. So if we put yn = E , then some straightforward calculus gives

Hence the singularity is confined to yn+l, yn+2, yn+3. Again the recurrence relation (65) or (66) is very unstable for computing the recurrence coefficients recursively. One can show (Van AsscheZ6)that there is again a unique solution of (66) with yo = 0 which is positive for all n > 0, and this is the solution for which yn = ai/qn-'. This solution is such that y, 4 1 and

where the integrals can be computed using the q-binomial theorem. In Figure 3 we have computed log lynl recursively for q = 0.9 with 50 significant digits. 5.3. Another discrete q-Freud case

If we take the weight ~ ( z=) (z2q2;q2)m(cz2q2;q2)m, with c 5 1, then w is positive on the q-lattice and it satisfies the Pearson equation (1 - 2 ) ( 1- cz2)w(z)

+

qmz)

= w(z/q).

(67)

If c = -1 a d then w( + exp(-z4 - 2 a z 2 ) so that this gives us a q-deformation of the Freud weight exp( -z4 - 2az2). Observe that

r

50 digits accuracy

I

41

3-

2-

1-

0: 10

~ ~

.&.

o 0

Q

~O**"~O"

"

40

50

g o o * Q o o

-1-

0

c = -1 gives the discrete q-Freud polynomials considered in the previous section and c = 0 gives the discrete q-Hermite I polynomials.

Lemma 5.3. The structure relation for the orthonormal q-polynomials with weight W(Z) = (22q2;q2)m(cx2q2;q2)00 on the q-lattice { f q n , n E N} 1:s

where

Proof. The proof is a straightforward copy of the proof of Lemma 5.2, except that one uses the Pearson equation (67). The Pearson equation contains the quartic polynomial (1 - x2)(1- cx2)= 1 - (1 c)z2 x4 so that

+

+

719

one ends up with integrals of the form

11[(1 + 1

2, =

c)z - cz3]p,(z)p,-3(z)w(z) d,z = -cA,

where A, and B, are given by (58)-(59).

I7

Reasoning in the same way as in the previous section, i.e., comparing the coefficients of 9 - l and xn-3 in (57), one arrives at

-

-ca:

(u:+l

+ q - n f l a,

If we put u: = qn-'y,,

+q

u,-1-

l+c

-C

then this can be rewritten as

(1 - y,)(l - cy,)

= qn(cy,+lyn - l)(CY,-lY,

- 11,

(72)

which is a more general form of the q-discrete Painlev6 I equation in (66). Acknowledgments

This research was supported by INTAS Research Network NeCCA (0351-6637)' FWO projects G.0184.02 and G.0455.04, and OT/04/21 of the Research Counsel of K.U.Leuven. Appendix A. Appendix

Several discrete Painlev6 equations have appeared in the literature, and the list is certainly longer than the six Painlev6 differential equations. Sakai23 made a classification in terms of rational surfaces associated with affine root systems and the most general (elliptic) discrete Painlev6 equation is related with the affine Weyl group symmetry of type Es. Sakai's classification does not give explicit expressions for the discrete Painlev6 equations. A few important discrete Painlev6 equations are listed below. The list was compiled by Peter Clarkson and I thank him for his permission to present it in this paper.

720

Appendix A . l . Discrete Painleve' equations

where z,

= an

+ 0 and a, 0,y,6,

K,

p are constants.

Appendix A.2. q-discrete Painleve' equations

where A,

= Aoqn

and a , P, y and 6 are constants.

72 1

Appendix A.3. Asymmetric discrete Painleve' equations zn+1

=

fl(Yn) f3(Yn)

Yn+Yn-l

Yn

+ Yn-1

with

+z,

=

+ %f2(Yn)

Yn-1

+ znf4(Yn)

=

s+

91(Zn) 9 3 ( 4

2n+1/2

2(zn&+1/2 -

=

4

1 -x;

a+b+c+d=O

+Y

+ Yng2(%)

+ YnS4(Yn)

722

where zn

= an

+ P and a, b, c, d , p , q , r , s, a, P, y and S are constants.

Appendix A.4. Alternative discrete Painleve' equations

z,+1+

2n-1 =

zn

Xn

zn

%+l%

+1+

+

- y

zn- 1 2,2,-1+

1

= -2,

1 ++ zn + y Xn

723

+-11 +-xxnn [ i z n+

where zn = an

+ ,8 and a , ,8, y and 6 are constants.

Appendix A.5. Other discrete Painleve‘ equations

where zn = an+P and a , b, c, d , p , q, r , s, a , p, y,6, K. and p are constants.

724

References 1. J. Baik, Riemann-Hilbert problems f o r last passage percolation, in “Recent Developments in Integrable Systems and Riemann-Hilbert Problems” (K. McLaughlin, X. Zhou, eds.), Contemporary Mathematics 326 (2003), pp. 121. 2. J. Baik, P. Deift, K. Johansson, O n the distribution of the length of the longest increasing subsequence of random permutations, J . Amer. Math. SOC. 12 (1999), 1119-1178. 3. T. S. Chihara, On quasi-orthogonal polynomials, Proc. Amer. Math. SOC.8 (1957), 765-767. 4. T. S.Chihara, A n Introduction t o Orthogonal Polynomials, Mathematics and its Applications 13,Gordon and Breach, New York, 1978. 5. R. Conte (editor), T h e Painleve‘ Property. One Century Later, CRM Series in Mathematical Physics, Springer-Verlag, New York, 1999. 6. C. Cresswell, N. Joshi, The discrete Painleve‘ I hierarchy, in ‘Symmetries and Integrability of Difference Equations’ (Canterbury, UK, July 1-5, 1996), Lond. Math. SOC.Lect. Notes Ser. 255, (P.A. Clarkson et al., eds.), Cambridge University Press (1999), pp. 197-205. 7. A.S. Fokas, A.R. Its, X. Zhou, Continuous and discrete Painleve‘ equations, in ‘Painlev6 Transcendents’ (Sainte-AdBle, PQ, 1990), NATO Adv. Sci. Inst. Ser. B Phys. 278,Plenum, New York, 1992, pp. 33-47. 8. A S . Fokas, A.R. Its, A.V. Kitaev, Discrete Painleve‘ equations and their appearance in quantum gravity, Comm. Math. Phys. 142 (1991), 313-343. 9. G. Freud, O n the coeficients in the recursion formulae of orthogonal polynomials, Proc. Roy. Irish Acad. Sect. A 76 (1976), no. 1, 1-6. 10. B. Grammaticos, F. W. Nijhoff, A. Ramani, Discrete Painleve‘ equations, in,5 pp. 413-516. 11. M. Hisakado, Unitary matrix models and Painleve‘ III, Mod. Phys. Letters All (1996), 3001-3010. 12. M. N. Hounkonnou, C. Hounga, A. Ronveaux, Discrete semi-classical orthogonal polynomials: generalized Charlier, J . Comput. Appl. Math. 114 (2000), 361-366. 13. M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications 98,Cambridge University Press, 2005. 14. R. Koekoek, R. F. Swarttouw, T h e Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Reports of the faculty of Technical Mathematics and Informatics no. 98-17, Delft University of Technology, 1998. 15. J. S. Lew, D. A. Quarles, Nonnegative solutions of a nonlinear recurrence, J . Approx. Theory 38 (1983), no. 4, 357-379. 16. D. S. Lubinsky, H. N. Mhaskar, E. B. Saff, A proof of Freud’s conjecture f o r exponential weights, Constr. Approx. 4 (1988), no. 1, 65-83. 17. A. P. Magnus, A proof of Freud’s conjecture about the orthogonal polynomials related t o lxlPexp(--52m), f o r integer m, in ‘Orthogonal Polynomials and Applications’ (Bar-le-Duc, 1984), Lecture Notes in Mathematics 1171, Springer, Berlin, 1985, pp. 362-372.

725 18. A. P. Magnus, O n Fkeud's equations for exponential weights, J. Approx. Theory 46 (1986), no. 1, 65-99. 19. A. P. Magnus, Fkeud's equations for orthogonal polynomials as discrete Painleue' equations, in 'Symmetries and Integrability of Difference Equations' (Canterbury, 1996), London Math. SOC.Lecture Note Series 255, Cambridge University Press, Cambridge, 1999, pp. 228-243. 20. A. MBt6, P. Nevai, T. Zaslavsky, Asymptotic expansions of ratios of coeficients of orthogonal polynomials with exponential weights Trans. Amer. Math. SOC. 287 (1985), no. 2, 495-505. 21. F. W. Nijhoff, O n a q-deformation of the discrete Painleue' I equation and q-orthogonal polynomials, Lett. Math. Phys. 30 (1994), 327-336. 22. V. Periwal, D. Shevitz, Unitary-matrix models as exactly solvable string theories, Phys. Rev. Letters 64 (1990), 1326-1329. 23. H. Sakai, Rational surfaces associated with a f i n e root systems and geometry of the Painleue' equations, Commun. Math. Phys. 220 (2001), no. 1, 165-229. 24. B. Simon, Orthogonal Polynomials on the Unit Circle, Amer. Math. SOC. Colloq. Publ. 54, Part 1 and Part 2, Amer. Math. SOC.,2005. 25. C. A. Tracy, H. Widom, Random unitary matrices, permutations and P a i n l e d , Commun. Math. Phys. 207 (1999), 665-685. 26. W. Van Assche, Unicity of certain solutions of some discrete Painleue' equation, manuscript 27. W. Van Assche, M. Foupouagnigni, Analysis of non-linear recurrence relations for the recurrence coeficients of generalized Charlier polynomials, J. Nonlinear Math. Phys. 10 (2003), suppl. 2, 231-237.

726

ORTHOGONAL POLYNOMIALS ON R+ AND BIRTH-DEATH PROCESSES WITH KILLING

PAULINE COOLEN-SCHRIJNER* AND ERIK A. VAN DOOR"

* Department of Mathematical Sciences Durham University Durham DH1 3LE, United Kingdom E-mail: Pauline.SchrijnerOdurham.ac.uk

t Department of Applied Mathematics University of Twente

P.O. Box 217, 7500 AE Enschede, T h e Netherlands E-mail: [email protected] The purpose of this paper is to extend some results of Karlin and McGregor's and Chihara's concerning the three-terms recurrence relation for polynomials orthogonal with respect to a measure on the nonnegative real axis. Our findings are relevant for the analysis of a type of Markov chains known as birth-death processes with killing.

Key words and phrases. Favard's theorem, Stieltjes moment problem, three-terms recurrence relation, birth-death process. 2000 Subject Classification: Primary 42C05, Secondary 60J80.

1. Introduction and main results Our point of departure will be the familiar three-terms recurrence relation for orthogonal polynomials. That is, we consider a sequence of monic

727

polynomials { P,(Z)}~=~ satisfying the recurrence relation = .( - cn)P,-1(2)

Pn(.)

- dnP,-2(.),

n

> 1, (1)

Po(.) = 1, PI(.) where c, is real and d,

= 5 - c1,

> 0. Hence, by Favard’s theorem

(see, for example,

Chihara [3]), there exists a measure $ on R with respect to which the polynomials {P,(z)} are orthogonal, that is,

L co

P,(.)Pm(.)$(d.)

with Ic,

> 0.

= lC,6,m,

n, m 2 0,

(Here and in what follows the term measure stands for Bore1

probability measure with infinite support.) It is well known that, conversely, any sequence of polynomials {P,(.)}

orthogonal with respect to a measure

on R satisfies a recurrence relation of the type (1) with c, real and d,

> 0.

Clearly, whether or not the orthogonalizing measure $ is uniquely determined by the recurrence relation (l),the moments

L M

m, :=

zn$(dz), n 2 0,

are uniquely determined by (1), and may be expressed recursively in terms of the parameters c, and d,. We will therefore refer to the problem of finding an orthogonalizing measure for the polynomials {P,(x)}as the Hamburger

moment problem (Hmp) associated wzth (1). Favard’s theorem thus tells US that the Hmp associated with (1) has a solution. The Stieltjes m o m e n t problem (Smp) associated with (1) is the problem of finding an orthogonalizing measure for {P,(x)}on

R+ := {X E R :

5

2

0}, if any exist. A necessary and sufficient condition for the Smp associated

with (1) t o have a solution is given in the next theorem, which is due to Karlin and McGregor [8] and Chihara [l](see also [3, Theorem 1.9.1]), but can be partly traced back to the work of Stieltjes [lo].

728

Theorem 1.1.

(Karlin and McGregor [d], Chihara [I]) For the sequence

of polynomials {P,(x)} satisfying the recurrence relation ( I ) t o be orthogonal with respect to a measure on R+, it is necessary and suficient that there exist numbers a,

> 0 and P n + 1 > 0 , n 2 1, such that

It has been observed by Karlin and McGregor [8] and Chihara [l](see also the Corollary to [3, Theorem 1.9.11) that (2) may actually be relaxed to

But we may weaken condition (2) even further, as appears from the next result.

Theorem 1.2.

For the sequence of polynomials {P,(x)} satisfying the

recurrence relation ( I ) to be orthogonal with respect to a measure on R+,

it is necessary and suficient that there exist numbers a, > 0 and

,&+I

>

0 , n 2 1, such that

Proof.

The necessity of the condition follows immediately from Theorem

1.1. To prove the sufficiency, suppose there are positive numbers a, and &+I,

n 2 1, such that (4) is satisfied. We can then recursively define the

quantities

and note, by induction, that

o < p, 5 p,,

G,,

2 c, - P, 2 a, > 0, n > I.

729

So we have constructed positive quantities 5, and ,&+I, P,(X)

n 2 1, such that

= ( x - 6, - &)P,-~(x) - ~ n - ~ j n P n - ~ ( xn) > , I,

(6)

Po(.) = 1, PI(X) = x - 51. Theorem 1.1 now implies that the Smp associated with (1) has a solution. 0

Remark The condition in Theorem 1 may be formulated alternatively using the concept of a chain sequence (cf. [3, Theorem 1.9.21). T.S. Chihara (personal communication) has observed that this alternative formulation, together with Wall's comparison test for chain sequences (see [3, Theorem 111.5.71) immediately leads to the result of Theorem 2. We know from [8] and [l]that there exists an orthogonalizing measure $ on R+ for {P,(x)} with a finite moment of order -1, that is,

if and only if there exist positive numbers a, and

Pn+l,

n

2 1, satisfying

(3) with the inequality being strict. Also this condition can be relaxed, as shown in our second main result. Theorem 1.3.

There exists an orthogonalizing measure on R+ for

{P,(x)} with a finite moment of order -1 if and only if there exist positive numbers a, and

Pn+l,

n 2 1, satisfying

(4) with at

least one of the

inequalities being strict. The necessity of the condition in this theorem follows from the result preceding the theorem. The sufficiency of the condition will be proven in Section 3, where also some remarks are made about how much freedom we have in choosing a, and

Pn

successively when c, and d , are given.

730

The remainder of this paper is organized as follows. Section 2 contains preliminary results, and in Section 4 we discuss criteria for determinacy, that is, for the existence of a unique solution, of the Smp and Hmp associated with (1) when the parameters satisfy (4). In Section 5 we briefly discuss the relevance of orthogonal polynomials defined by the recurrence relation (1) with parameters satisfying (4) for the analysis of a type of Markov chain known as birth-death process with killing. Some of the results mentioned in this paper are quoted from the recent papers [5], [6] and [7], which deal with birth-death processes with killing.

It is the purpose of this note to collect these results, and to elaborate on them from the perspective of orthogonal polynomials.

2. Preliminaries

In this section we will assume that the parameters c, and d, in the recurrence relation (1) satisfy the condition of Theorem 1.2. Letting

we then have 7, 2 0, and the recurrence relation (1) can be written as

731

in which we recognize the type of recurrence relation introduced in [6]in the setting of birth-death processes with killing (about which more in Section 5 )*

Letting

it follows readily with induction from (10) (see also [ 5 ] ) that n

a n p n ( Q n (~ Qn-1(z)) ) = C ( y j -z)pjQj-l(X),

n

> 0.

(12)

j=1

Hence we have

and in particular

So Qn(0) 2 1, and Qn(0) is non-decreasing, while Qn(0) = 1 for all n 2 1

e yn = 0 for all n 2 1.

Moreover, we know from [7, Lemma 11 that

Another useful observation from (12) is given in the next lemma.

If yn > 0 for some n 2 1, then

Lemma 2.1.

00

n= 1

Proof. Suppose ^lm n

1 < CQ. anpnQn-l(0)Qn(O)

> 0. Then

(12) implies that we can write, at least for

L m, 1 anPnQn-i(o)Qn(O)

-

1

Cj”=,r j p j Q j - l ( O )

732

and hence

It follows that

Next, let Cin and

Bn be as in (5). We define Qn(z)and f i n

by analogy with

(9) and (ll),respectively, and note that the polynomials Q n ( z ) ,n 2 0 , satisfy the recurrence relation C n Q n ( 4 = (6, &I&)

= 1,

+ Sn - Z ) Q n - l ( Z )

Ci1Q1(s)

- SnQn-2(4,

n

> 1, (15)

= Ci1 - 2.

With (5) it is easy to establish the relations

and

Moreover, it is evident from (15) that Qn(0) = 1 for all n 2 0, so (16) and (17) imply

The result of Lemma 1 can therefore be stated as

We have now collected sufficient information about the sequences

{ Q n ( z ) }and

{On(.)}

to continue with the proof of Theorem 1.3 and dis-

cuss some related issues.

733

3. Proof of Theorem 1.3 and related issues It remains to be proven that the condition in Theorem 1.3 is sufficient. So we suppose that the parameters c, and d, satisfy the condition of Theorem 1.2 (as in the previous section), and, in addition, that at least one of the

inequalities in (4)is strict. That is, defining 7, as in (7), we have

We define the parameters Fin and

bn as before.

Since the condition in Theorem 1.2 is fulfilled there exists an orthogcnalizing measure $ for {P,(x)}on R+. In what follows $ will always denote either the unique solution of the Smp associated with ( l ) , or, if a unique solution does not exist, the one known as minimal solution (in the terminology of Karlin and McGregor [S]) or natural solution (in the terminology of Chihara [2]). The latter may be characterised as the solution that is supported by the limit points of the zeros of the polynomials P,(x),n 2 1. Ft-om [8, Lemma 6 on p. 5271 we know that $({O})

= 0 and

if and only if the sum in (21) is finite. By (19) the latter indeed holds under the prevailing condition (20), so $ has a finite moment of order -1. This concludes the proof of Theorem 1.3. The following results will be needed in the second part of this section. Maintaining assumption (20), it follows by induction from the recurrence relation (15) and (21) that

734

and hence, in view of (16),

Moreover, it is shown in [6, Lemma 31 (under the implicit assumption (20)) that

Since, by (13), 1 5 Qm(0) 5 00, it thus follows that

We will now describe a procedure for successively choosing the parameters a, and

Pn

of (4), given the parameters c, and d, of the recurrence

relation (1). First, by letting

and checking whether 6,

>0

for all n 2 1, we can verify whether the

polynomials {P,(z)} are orthogonal with respect to a measure on R+. If this is the case we must check whether the right-hand side of (21) is finite.

If not, the (minimal) orthogonalizing measure

$J

does not have a finite

moment of order -1, and the only permissible choice for the parameters a, and

Pn

x

is a, := 6, and

Pn+l := Pn+l (and hence 7,

= 0) for n

2 1. If,

however, the right-hand side of (21) is finite, so that )I does have a finite moment of order -1, we may proceed as follows. First, choose y1 2 0 such that

735

and let

a1 := c1 -

71. By [8, Lemma 1 3 we know that 0

<

a1

5

c1.

Subsequently, in step n = 2 , 3 , . . . , let

choose yn 2 0 such that

(recall (23)) and let an := c , - bn - yn. Evidently, if equality prevails in (24) in step n = m, say, then we must have yn = 0 for all n

> m. It remains

to be shown that the values of an obtained by this procedure are positive. This, however, is a consequence of the following lemma. Lemma 3.1.

Let c,"==,(GnPn)-l

<

00,

and suppose the polynomials

Pn(x), n 2 0, satisfy the recurrence relation (8) with parameters an

0,

Pn+l >

>

0 and yn 2 0, such that yn = 0 for n 2 m 2 1. Then the

polynomials Pn(x), n 2 0, also satisfy a recurrence relation of the type (8) with parameters a; > 0, /3L+l Pn+l

and

= yn for n

> 0 and 7; 2 0, such that a;

< m, 7 ; > 0, and

= 0 for n

= an,

@L+l

=

> m, i f and only if

Proof. Adapting the proof of [8, Lemma 11 to the present setting leads

to the conclusion that a necessary and sufficient condition for { P n ( x ) }to satisfy the requirements is given by

But, since yj = 0 for j 2 m, the conditions (25) and (26) are equivalent, in view of (14) and (23).

0

736

4. Determinacy of the Smp and Hmp

We will assume again that the parameters cn and d , of the recurrence relation (1) satisfy the condition of Theorem 2, so, defining hn and

Pn

as

before, the polynomials Pn(z), n 2 0, satisfy the recurrence relation ( 6 ) . From Karlin and McGregor [8, Theorem 141 we know that the Smp associated with ( 6 ) ,and hence with (l),is determined, that is, has a unique solution, if and only if

while, by Karlin and McGregor [9, p. 3911 and Chihara [4], the Hmp associated with ( 6 ) , and hence with (l),is determined if and only if

Now suppose the parametrization of the recurrence coefficients c, and

dn is given by (8) rather than ( 6 ) . The preceding results together with (18) and (19) then tell us the following.

Theorem 4.1.

If yn > 0 for some n 2 1, then the H m p and S m p

associated with (8) are both determined o r are both indeterminate according as

00

pn+lQi(0) diverges or converges.

Remark An alternative proof for this theorem may be based on the observation that, by a theorem of Wall's mentioned just before Theorem 1 of [2], convergence of

Cp=,(Gnpn)-'

is equivalent to the condition in Theorem

2(A) of [2]. Considering the recursive expression (13) for Qn(0), it does not seem possible t o give a necessary and sufficient condition for convergence of

737 00 Cn=opn+lQ:(0) explicitly in terms of the parameters an, pn, and yn.

Since Qn(0) 2 1, we can, however, formulate the following sufficient condition, which does not require assumption (20).

Corollary The Hmp (and hence the Smp) associated with (8) is determined if

"

k

1

(29)

k=l

n=l

We finally note that a determined Hmp (associated with (8)) remains determined when some yn is increased in value, since, by (13), C:=, pn+lQ;(O) is an increasing function of each individual -yn. This observation can be extended t o the boundary case yn = 0 for all n. Indeed, in this case (28) reduces to 00

/ n

.

\ 2

so the Hmp associated with (8) is determined if and only if (30) is satisfied.

But if yn

> 0 for some n 2 1, then (30) implies (29). It follows in particular

that a sufficient condition for determinacy of the Hmp associated with (8) is determinacy of the Hmp that results after replacing all yn by 0. A similar

statement cannot be made for the Smp associated with (8). 5. Birth-death processes with killing

A birth-death process with killing is a Markov chain X := { X ( t ) , t 2 0) taking values in S := { 0 , 1 , .. .} with q-matrix Q := ( q i j , i , j E S ) given by 4i,i+l

qij

= Xi,

= 0,

Qi+l,i

= pi+1,

li -jl > 1,

qii

= -(Xi + p i

+ Vi),

(31)

738

where

Xi

> 0 and

ui 2 0 for i 2 0, Pi

> 0 for i > 0, and PO = 0. The

parameters X i and pi are the birth and death rates in state i, while ui may be regarded as the rate of absorption, or killing, into a fictitious state 8, say. It has been shown in [6] that the transition probabilities p i j ( t ) := Pr{X(t) = j I X(O) = i},

t 2 0, i , j E S,

of the process X can be represented in the form

where nn, n 2 0 , are constants given by TO := 1

and xn :=

XOXl

. . .X n - l 7

P1P2-..Pn

n>0,

Rn(x), n 2 0 , are polynomials defined by the recurrence relation XnRn+l(z)= ( A n XORl(5) = A0

+YO

+ P n + ~n- x)Rn(z)- PnRn-l(Z), - 5,

Ro(5) =

n 2 1,

(33)

1,

and $J is a measure with respect to which the polynomials { R n ( z ) }are orthogonal. Making the identification

we see that, for all n 2 0, the constant xn can be identified with the constant pn+l of ( l l ) ,and the polynomial Rn(z)with the polynomial Q n ( z )of (10).

It has been shown in [6, Theorem 41 that the transition probabilities (if they are to satisfy both backward and forward Kolmogorov differential equations) are uniquely determined by the parameters (31) if and only if at least one of the conditions limn+m Rn(0)= oo or m

c -)

n=O

1 (rn + Xnrn

=oo

(35)

739

be satisfied. If we apply the identification (34) we see that, by (18), condition (35) is equivalent to (27), and hence determinacy of the associated Smp, if limn+mRn(0)

<

00.

In view of (14) and (22), we therefore con-

clude that the process is uniquely determined by its rates if and only if the associated Smp is determined or

This generalizes [8, Theorem 151.

Acknowledgement This paper was conceived and largely written while the second author visited Durham University holding a Grey College Fellowship. He thanks Grey College and the Department of Mathematical Sciences of Durham University for the hospitality extended during this visit.

References 1. Chihara, T.S. (1962). Chain sequences and orthogonal polynomials. Trans.

Amer. Math. SOC.104 (1962) 1-16. 2. Chihara, T.S. (1968). On indeterminate Hamburger moment problems. Pa-

cific J. Math. 27 (1968) 475-484. 3. Chihara, T.S. (1978). A n Introduction to Orthogonal Polynomials. Gordon and Breach, New York. 4. Chihara, T.S. (1982). Indeterminate symmetric moment problems. J. Math.

Anal. Appl. 85 (1982) 331-346. 5. Coolen-Schrijner, P. and van Doorn, E.A. (2006). Quasi-stationary distributions for birth-death processes with killing. J. Appl. Math. Stoch. Anal., to appear.

740 6. van Doorn, E.A. and Zeifman, A.I. (2005). Birth-death processes with killing.

Statist. Probab. Lett. 72, 33-42. 7. van Doorn, E.A. and Zeifman, A.I. (2005). Extinction probability in a birthdeath process with killing. J. Appl. Probab. 42, 185-198. 8. Karlin, S. and McGregor, J.L. (1957). The differential equations of birthand-death processes, and the Stieltjes moment problem. B a n s . Amer. Math.

SOC.85, 489-546. 9. Karlin, S. and McGregor, J.L. (1957). The classification of birth and death processes. Trans. Amer. Math. SOC.86, 366-400.

10. Stieltjes, T.J. (1894, 1895). Recherches sur les fractions continues. Ann. Fac. Sci. Toulouse 8, 51-5122, and 9, Al-A47. (pp. 406-570 in: Oeuvres Complktes

- Collected Papers, Vol. 2. G. van Dijk (ed.), Springer-Verlag, Berlin, 1993).

741

COMPUTING TOPOLOGICAL INVARIANTS IN BOUNDARY VALUE PROBLEMS REDUCIBLE TO DIFFERENCE EQUATIONS

RICARDO SEVERINO~, ALEXANDER SHARKOVSKY~, J. SOUSA RAMOS3 AND SANDRA VINAGRE4 'Department of Mathematics, Universidade do Minho Campus de Gualtar, 4710-057 Braga, Portugal e-mail: [email protected] Institute of Mathematics, National Academy of Sciences of Ukraine Tereshchenkivska str., 3, 01601 Kiev, Ukraine e-mail: asharkovQimath. kiev.ua Department of Mathematics, Instituto Superior Te'cnico Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal e-mail: [email protected] Department of Mathematics, Universidade de Evora, Rua Romiio Ramalho, 59, 7000-671 Evora, Portugal e-mail: [email protected]

Among boundary values problems (BVP) for partial differential equations there are certain classes of problems reducible to difference equations. Effective study of such problems has became possible in the last 20-30 years owing to appreciable advances done also in the theory of difference equations with discrete time, specifically given by one-dimensional maps. Here we apply how this reduction method may be used in simple nonlinear BVP, determined by a bimodal map. We consider two-dimensional linear hyperbolic system with constant coefficients, with nonlinear boundary conditions and usual initial conditions. The objective is to characterize the dependence of the motions of the vortice solutions with the topological invariants of the bimodal map.

1. Introduction In recent years, some attempts have been made to develop a method of investigation of boundary value problems (BVP) for partial differential equations (PDE), in particular, for hyperbolic systems with one spatial variable

742

which is based on the reduction of the original problem to a difference equation or to an equation of another type with one independent variable (see, for example, Romanenko- Sharkovsky', Sharkovskyg, Sharkovskylo, Sharkovsky" and Sharkovsky et ~ 1 . l ~This ) . method can be also applied to many-dimensional hyperbolic systems. In this paper we consider the case of two spatial dimensions proposed by Sharkovsky et al.12. The reduction to a difference equation with continuous argument followed by the employment of the properties of the one-dimensional map associated with the difference equation allows to give an insight into the properties of such chaotic s o htions. The contents in this paper are organized as follows: in the next section we present the problem following Sharkovsky et a l l 2 . In the section 3, we give a brief description of the kneading theory and we present the definition of the growth number of vortices and a characterization of the growth number of vortices of a BVP from the topological entropy (see Severino et ~ 1 . ~ Finally, ). in the section 4,we apply symbolic dynamical techniques to study how the motions of the vortex solutions depend on the topological invariants of the bimodal map associated with the difference equation and the solutions of difference equations. 2. Preliminaries

Following Sharkovsky et u1.12, we consider a linear hyperbolic system with constant coefficients

du -

at

_ -au --

du

ax ay' -0a<x<+0a,

t20,

O l y I 1 ,

av -dv dv -dt dx d y + - 1

with the nonlinear boundary conditions

and with the initial conditions ult=0

= 210

(x,y) ,

Vlt=O

= vo ( X I y)

,

-Oa

< x < +0a,

0 I y 5 1. (3)

We assume that the boundary and initial conditions are consistent, that is, uo (XI 0) = vo

( 2 ,01,

210

(x11) = f (uo (x,1))*

743

The general solution of (1) is of the form

u (z, Y,t ) = cp (z - t , Y - t ) ,

+ t, Y + t )

(z, Y,t ) = ?b .(

(4) (5)

7

where cp and .1cI are arbitrary C1-functions. Using the boundary conditions (2), we obtain, for -oo < z < +oo, 0 I y 5 1, t 2 0, cp (z - t , 4 )= $J(z

+

11, (z t , 1+ t ) = f

+ t ,t ) ,

(6)

.

( ' p ( a : - t , 1- t ) )

(7)

Excluding cp from these relations and introducing the new variables

z=z+t-2

and

~ = t - l ,

(8)

we obtain the difference equation with continuous argument W (2

+ 2 , T + 2) = f (W (2,7)),

--oO

< Z < +W,

-1 5 7

I +oO,

(9)

with the initial condition WO (277-1 =

uo ( 2 - 27, -7)

{

uo ( 2 1 7 )

if if

< z < +a,-1 5 7 5 0, -co < z < +oo, 0 5 T < 1.

--o~

(10) Then it follows from (4),(5) and (9)-(10) that the solution u ( z , y , t ) , u (z,y, t ) of the boundary value problem (1)-(3) is given by

b , Y , t ) = w .( - t - 2 (Y - t ) 2/ (z, y, t ) = W (z t, y t ) .

'u.

+ +

7

-

(Y - 4)

7

(11) (12)

When the boundary value problem (1)-(3) is nonlinear, its solution cannot be explicitly expressed in terms of elementary functions. However, the boundary value problem (1)-(3) is equivalent to the initial boundary value problem (9)-(10) which solution can be written, by the iteration of equation (9), as follows

( z ,7) = f " (200 (2 - 2n, 7- - 2n)) , -oo
(13)

The reduction of the boundary value problem (1)-(3) to the difference equation (9) with the initial condition (10) enable us to characterize the behavior of the solution u (z,y, t ) , u (2, y, t ) completely. The properties of the solution u (z, y, t ) , u (z,y, t ) depend on the map f : I I and on the initial conditions uo (z,y) and wo (z, y) . At every fixed time t = t o , the solution defines a vector field (u and u are projections of this field onto the axes z and y, respectively) in the

-

744

region % = {-cc < z < +cc, 0 5 y 5 1). The streamlines of this field characterize a “state” of the original partial differential system at t = t o . Using the formulas ( l l ) , (12), we have the solution u (z, y, t ) , Y (2, y, t ) of the boundary value problem (1)-(3), a t times t = 2n, n = 0,1, . . . , given by

that is, the formulas of the solution u (z, y, t ) , v (2, y, t ) a t times t = 2n, n = 0 , 1 , . . . , become significantly simpler: a solution can be found by successive iterations of the initial functions uo (5, y) and 210 (z, y) , see Sharkovsky et al.I2. In what follows, we render concrete the initial conditions in the region

{

%a = z 2 + ( y - 1 / 2 )

2

1

<1

=a2/4, a

and, in order to obtain a initial “vortex” , we assume that

uo (z, y) = 2y - 1 and

YO

(z, y) = -2x

(z, y) E Ra.

for

(15)

According to the initial conditions, there is a “vortex” in the region % a t t = 0: the functions uo (2, y) , YO (z, y) define streamlines which form circles; moreover, a modulus of the vector field along each circle is equal to the doubled radius of this circle, see Figure 1.

A

YA 1

u=2a

v=o

’

u =o V=-W

V

> X

Figure 1. Representation of the streamlines of the vector field of the solution (I, y, t ) in the region %.

745

Let us investigate how the motions of the vortex solutions depend on the topological invariants of the bimodal map f. Let us take the family of cubic maps f a $ given by fa$ (W)

= aW3

4-bw2 + (1- a )

-

b,

( a ,b) E 0,

E [-I, 11,

W

where 0 C R2 is the set of parameters such that the map f a , b has two distinct critical points and it is a map from the interval [-1,1] into itself, see Figure 2 and see Martins et aL3. 3. Symbolic dynamics

Given a bimodal map f a , b in a interval I = [-1,1],with the critical points c1 and c2, we assign the symbols L , M and R to each point x of the subintervals of monotonicity [-I, c1) , (c1,c2) and (cp,11 , respectively, and the symbols A and B to their critical points c1 and cp. It is called the address of x and it is denoted ad (x). By doing this, we get a correspondence between orbits of points and symbolic sequences of the alphabet da = { L ,A , M, B , R}, the itinerary by the map f a , b ,

The kneading invariant of the map of each critical point, (fa,b)

=

(itfa,b

fa$

is the pair of itineraries of the image

( f a $ ( c l ) )7 Ztfa,b ( f a & ( c 2 ) ) )

.

Given a finite symbolic sequence S of d ! , we denote by n M the frequency of the symbol M in S and we define the M-parity of this sequence, p(S), according t o whether nM is even or odd, that is, in the first case we have p(S) = +1 and in the second p(S) = -1. From the order relation L 4 M 4 R, inherited from the order of the interval, we introduce an order relation between sequences as follows: given any distinct sequences P and Q of d ! , admitting that they have a common initial subsequence, i.e., there is a k > 0 such that P I . . .Pk = & I . . . Q k and Pk+1 # Qk+l, we will say that P 4 Q if and only if Pk+l + &k+l and and p ( p 1 . . .Pk) = -1. p(P1.. .%) = +I, O r Qk+l 4 Let V be a vector space of three dimension defined over the rationals having as a basis the formal symbols { L ,M, R } , then to each sequence of symbols S = SoSlS2.. . Sj . . . we can associate a sequence 0 = 0 0 . . . 6 j . . . of vectors from V , setting O3 = E ( S ~ )with S ~ j > 0, 60 = SO,when i = 0, and E ( L )= -E(M) = E ( R )= 1, where to the symbols corresponding to the critical points c1 and cp we associate the vectors ( L Ad)/2 and

niIi

+

746

+

( M R) /2. Thus € ( A )= E ( B )= 0. Choosing then a linear order in the vector space V in such way that the base vectors satisfy L 4 M 4 R we are able to order the sequence 8 lexicographically, that is, 8 4 8 iff 8 0 = 80, . . . ,8,-1 = 8,-1 and 8, 4 8, for some integer 3 2 0. Finally, introducing t as an undetermined variable and taking 8, as the coefficients of a formal power series 8 (invariant coordinate) we obtain 8 = O0 + t + . . . = 8, ’t In Milnor-Thurston*, the authors introduced basic invariants called kneading increments, kneading matrix and kneading determinant. The kneading increments are formal power series that measure the discontinuity evaluated at the critical points. For the case of a bimodal map we have two kneading increments defined by v, ( t )= OC; ( t )- 8,; ( t ), i = 1 , 2 , where 0 is the invariant coordinate defined previously and 8 5 ( t )= limx+c; 8, ( t ). cz After separating the terms associated with the different symbols in v, ( t ), we get v, ( t ) = Nzl ( t )L N,2 ( t )M Nz3 ( t )R, with i = 1 , 2 , and from these we can define the kneading matrix by N ( t ) = (N,, ( t ) ) ,i = 1 , 2 , 3 = 1 , 2 , 3 . Closely related is the kneading determinant which is defined from the kneading matrix according to the following formula D ( t )= D1 ( t )/ (1 - t ) = - 0 2 ( t )/(1 t ) = 0 3 ( t )/(1 - t ) , where D , ( t ) is the determinant obtained by eliminating the i - th column of the kneading matrix . Let fa$ be a bimodal map in the interval with kneading invariant x ( f a , b ) and growth number s (fa$) > 1. Then, s (fa&) is given by the inverse of the smallest zero of the kneading determinant associated to x ( f a , b ) and the entropy is given by the logarithm of s (fa$), (see Milnor-Thurston4, Misiurewicz-Szlenk5). Given a family of bimodal maps on the interval, we need to consider a second topological invariant, other than the usual topological entropy, in order to classify it. This invariant is introduced using the results of Milnor and Thurston about a topological semiconjugate X of fa$ to a piecewise linear map Fs,T with slope f s (fa$) everywhere, see Almeida et al.’ and Milnor-Thurston4. There exists one and only one map Fs,T: [0,1] + [0,1] SO that Fs,T (A (x)) = X (fa$ (z)) for every x E [0,1]. The map Fs,Tis piecewise linear with slope f s (fa$,) everywhere defined by

I,”=, ’

+

+

+

+e

sz

+ 1- s

if if if

0 2 x 5 X (c1), X (c1) 5 z 5 X (c2), X (c2) 5 z 5 1,

747

+

where e = ( 4 r + l + s ) / 2 , X(c1) = e / ( 2 s ) and X(c2) = e / ( 2 s ) ( s - 2 ) / ( 2 s ) . Note that s = s ( f a $ ) and r = T ( f a , b ) are determined by the kneading data and therefore depend on f a & According to the kneading theory, we have that X(c1) = limt.+l,s L ( J ,t )/ L ( I ,t ) , where J = [0,cl] and I = [0,1], with the formal power series L ( J ,t ) =

c:==, e

( f t , b ) tn-1

and e

0 f:,b

is the lap number of lln

f & . The growth number s ( f a $ ) is given by the limit (f2b) = l / t * ,which verifies D (t*)= 0 , where D ( t )is the kneading determinant. In a family of two parameters bimodal maps there exist dynamical behaviors which we can name by not factorizable and by factorizable, see Lampreia et aL2. Now we associate two topological invariants to not factorizable kneading data ( P A ,QB) or PBQA. One of them is the growth number s ( f a $ ) and the other is the invariant given in the following theorem.

Theorem 3.1. Let f a , b be the bimodal map with a not factorizable kneading data. Then, the pair ( s ( f a , b ) , r ( f a $ ) ) is a complete topological invariant f o r the map f a , b , where r ( f a , b ) as given by

(c1) - 1 - S (f a , b ) 2 Remark 3.1. The second topological invariant can be interpreted as a measure of the difference between the growth number in the third interval of monotonicity and the first interval of monotonicity. (fa,b) =

(fa,b)

Following Severino et al.', we have the definition of the growth number of vortices and a characterization of the growth number of vortices of a BVP from the topological entropy of that map.

Definition 3.1. For the boundary value problem (BVP) given by (1)-(3) and (15), we define the growth number of vortices as

p(BVP)= lim (Nk(BVP)lIk k-+w

where

Nk

( B V P ) is the number of vortices obtained in the iterate k.

Now we can characterize the growth number of vortices of a BVP from the topological entropy, as the following result shows.

Theorem 3.2. The growth number of vortices p ( B V P ) > 1 if and only i f the topological entropy of the map f a , b is positive. Proof. See Severino et aL8.

0

748

4. Results

As said before, our purpose is to study how the motions of the vortex solutions depend on the topological invariants of the bimodal map f a , b . Denote by A2 the subset of the parameter space s1 for which the corresponding maps f a $ have growth number 2, that is, let A2 be the topological entropy level set for ht ( f a , b ) = In 2, see Figure 2 and Martins et d 3 .

b

Figure 2.

The parameter space 0,with the isentropic A z .

+

+

Example 4.1. Consider f a $ (w) = aw3 bw2 (1 - a ) w [-1,1] , and fix a = 3.598.. . and b = 0. See Figure 3.

-

b, with w E

Figure 3. u) Graph of the map fa,b with a = 3598.. . and b = 0. b) Diagram of the streamlines for the boundary value problem (1)-(3) and (15), with fa,(, (w) = uw3 -Ibw2 + (1 - U ) w - b, w € [-1,1], u = 3.598.. . , b = 0.

749

+

+

Example 4.2. Consider fa,b (w) = aw3 bw2 (1- a ) w - b, with [-1,1], and fix a = 3.236. . . and b = 0.361 . . . . See Figure 4.

20 E

0.75 0.751 0.5.

-1

-0.5

0

0.5

1

Figure 4. a) Graph of the map fa,b with a = 3.236.. . and b = 0.361.. . . b) Diagram of the streamlines for the boundary value problem (1)-(3) and (15), with fa,b (w) = .w3 bW2 (1 - a ) -~ b, 2~ E [-I, 11, a = 3.236.. . , b = 0.361.. . .

+

+

+

+

Example 4.3. Consider f a , b (w) = aw3 bw2 (1- a ) w - b, with w E [-1,1], and fix a = 3.236.. . and b = -0.361.. . . See Figure 5. Theorem 4.1. For a bimodal boundary value problem, given by (1)-(3) and (15), the growth number of vortices is equal to the growth number of laps of the corresponding bimodal map.

Proof. Since the vortices of the vector field ( u , ~ )at, time t = 2n, are related with the zero crossings of the map f & ( z ) , we are going to identify those crossings by studying the sequences a d ( z )a d ( f a , b ( z ) ).

*

ad(fcb(z)),

z E [-I, 11.

We will say that such a sequence is of type 1, if it has no symbols A , and B ; of type 2, if it has a unique symbol A , or B , at its end; and of type 3, otherwise. Since the number of sequences of type 3, T3(n),gives us the number of zeros of the derivative, the number of laps, !(n),of f t , b ( z ) is equal to T3(n)minus all the possible multiple zeros, which can also be described symbolically. The difficult part is to characterize symbolically

750

Figure 5. a) Graph of the map fa,b with a = 3 . 2 3 6 . . . and b = - 0 . 3 6 1 . . . . b) Diagram of the streamlines for the boundary value problem (1)-(3) and (15), with f a , b ( w ) = a w 3 + h 2 + ( l - a ) w - b , w E [ - 1 , 1 ] , a = 3 . 2 3 6 ..., b = - 0 . 3 6 1 . . . .

the zero crossings of the map f z , b ( % ) , and that has to be done considering all the possible situations. Here, we will present only the simplest one. Let f a $ be such that both its kneading sequences have no symbols A , M , and B. Thus, all the zeros of fr,b(z) are simple and then the number of laps of f&(z) is equal to T3(n) 1. Moreover, the number of zero crossings is one half the number of type 2 sequences, Tz(n),since all type 3 sequences ends either with a symbol L or R and, therefore, every time f E b ( % ) goes by the first critical point, it has to go also by the second critical in the same lap, and vice versa. Our result is a consequence of the relation TS(n+l)= T3(n)+T~(n),i.e., denoting by co(n) the number of zero crossings of the map fcb(s), we have then a relation between the number of laps of the map and its zero crossings, co(n) = !(n 1)- !(n).

+

+

Theorem 4.2. The second topological invariant of the bimodal map fa$ measures the difference of the distribution of vortices relative to the symmetric distribution.

Proof. From its definition, we have that the second topological invariant is related with the growth number of laps of fa$ in the first subinterval of monotonicity. Thus, if we do the same kind of analysis as done for Theorem 4.1, but now for the first subinterval of monotonicity, we conclude that the growth number of laps wanted is just the growth number of zero crossings of f:,b(%), in the same subinterval. Therefore, the second topological in-

751 variant is a measure of how t h e vortices are distributed along t h e region [-1/2,1/2] x [0,1], with the zero of the second invariant corresponding to the symmetric situation, in respect with the anti-diagonal of the square. Then, we conclude t h a t t h e second topological invariant of fa,b measures t h e difference of the distribution of vortices relative to t h e symmetric distribution. 0

References 1. P. Almeida, J. P. Lampreia and J. Sousa Ramos, Topological invariants for bimodal maps, Proceedings of ECIT92, eds: W. Forg-Rob, D. Gronau, C. Mira, N. Netzer, G. Targonski, World Scientific, (1996), 1-8. 2. J. P. Lampreia, R. Severino and J. Soma Ramos, Symbolic product of bimodal kneading invariants. Iteration theory (ECIT '02), 245-254, Grazer Math. Ber., 346, Karl-F'ranzens-Univ. Graz, Graz, 2004. 3. N. Martins, R. Severino and J. Sousa Ramos, Isentropic real cubic maps, Int. J. Bifurcation and Chaos, 13 (2003), n7, 1701-1709. 4. J. Milnor and W. Thurston, On Iterated Maps of the Interval. In: J. C. Alexander (ed.) Proceedings Univ. Maryland 1986-1987. Lect. Notes in Math. 1342, 465-563, Springer- Verlag, Berlin-New Yorlc, 1988. 5. M. Misiurewicz and W. Szlenk, Entropy of piecewise monotone mappings. Studia Math., 67, (1980), 45-63. 6 . E. Yu. Romanenko and A. N. Sharkovsky, Fkom boundary value problems to difference equations: a method of investigation of chaotic vibrations, Int. J. Bifurcation and Chaos 9 (1999), n7, 1285-1306. 7. R. Severino, A. N. Sharkovsky, J. Sousa Ramos and S. Vinagre, Symbolic dynamics in boundary value problems, Iteration theory (ECIT '02), 393-402., Grazer Math. Ber., 346, Karl-F'ranzens-Univ. Graz, Graz, 2004. 8. R. Severino, A. N. Sharkovsky, J. Sousa Ramos and S. Vinagre, Symbolic dynamics in boundary value problem f o r systems with two spatial variables, submitted to Grazer Math. Ber. 9. A. N. Sharkovsky, Universal phenomena in some infinite-dimensional dynamical systems, Int. J. Bifurcation and Cham, 5 (1995), n5, 1419-1425. 10. A. N. Sharkovsky, Iteration of continuous functions and dynamics of solutions f o r some boundary value problems, Ann. Math. Sil., 13 (1999), 243-255. 11. A. N. Sharkovsky, Difference equations and boundary value problems, in "New Progress in Difference Equations", Proceedings of the ICDEA'2001, Taylor & F'rancis (2003), 3-22. 12. A. N. Sharkovsky, Yu. L. Maistrenko and E. Yu. Romanenko, Diflerence Equations and Their Applications, Kluwer Academic Publishers, 1993.

752

ABEL'S METHOD ON SUMMATION BY PARTS AND BILATERAL WELL-POISED 3+3-SERIES IDENTITIES

CHU WENCHANG College of Advanced Science and Technology Dalian University of Technology Dalian 116024, P. R. China E-mail: chu. wenchang Ounile. it By means of Abel's lemma on summation by parts, two functional equations with an extra integer parameter m will be established for Bailey's bilateral well-poised s$s-series. Their limiting cases lead us directly t o Bailey's non-terminating bilateral well-poised g$g-series identities.

8000 Subject Classification: Primary 33D15, Secondary 05A30

For two complex x and q , the shifted-factorial of x with base q is defined by (x;q), = 1 and

When IqI

(z;q), = ( l - x ) ( l - x q ) . . . ( l - z q n - l )

for n E N .

(11

< 1, we have two well-defined infinite products Ca

(x; q),

= n(1-qkx) and

(2; 41,

= (x; q),

/ (xqn; d,

,

(2)

k=O

In particular, the shifted factorial with negative integer order can be written explicitly from the last fraction as

The product and ratio of shifted factorials are abbreviated respectively to

753

Following Gasper-Rahman [12] and Slater [17], the bilateral basic hypergeometric series is defined by

where the base q will be restricted to IqI < 1 for non-terminating q-series. One of the most important and useful identities in the theory of basic hypergeometric series is Bailey’s summation formula [2] for a nonterminating bilateral very-well-poised Gqs-series. It has recently been provided a completely new and simple proof by Chu [lo] through Abel’s lemma on summation by parts. By employing this approach further, this paper will present a new and elementary proof for Bailey’s non-terminating bilateral well-poised basic hypergeometric 3$3-series identities, whose finite forms are also called q-Dixon formulae. Abel’s lemma on summation by parts is very useful and important in classical analysis. For an arbitrary complex sequence { r k } , define the backward and forward difference operators V and A, respectively, by

VTk = 7 k - r k - 1

= r k - 7k+1

and

(7)

where A is adopted for convenience in the present paper, which differs from the usual operator A only in the minus sign. Then Abel’s lemma on summation by parts may be reformulated as

k=-m

k=-m

provided that the series on both sides are convergent or terminating. The proof of Abel’s lemma is an easy exercise and can be done by any student following a calculus class. In fact, according to the definition of the backward difference, we have +m

+cc

+m

C

A ~ V B= ~

k=-m

C

+W

C

C

A ~ { B ~ - B = ~ - ~ } A ~ -B ~

k=-m

k=-m

A ~ B ~ - ~ .

k=-m

+

Replacing lc by 1 lc for the last sum, we get the following expression: +m

+m

+m

k=-m

k=-m

k=-m

which is the formula stated in Abel’s lemma.

754

By means of Abel’s lemma on summation by parts, we shall show that Bailey’s two non-terminating bilateral well-poised basic hypergeometric 3+3-series satisfy two crossing recursions and two independent recurrence relations. Iterating these relations, we get two functional equations with an extra integer parameter m. Letting m -+ 00 and using Jacobi’s triple product identity, we derive Bailey’s bilateral well-poised 3+3-series identities. Compared with Bailey’s original proofs based on Rogers’ q-Dougall 645series identity, the author believes that the proof presented in this paper is more elementary and accessible to readers. 1. Bilateral Well-Poised s?,!~a-SeriesIdentities

In this section, we review two non-terminating bilateral well-poised series identities due to Bailey [4] and their implications.

3$~~-

1.1.

Bailey [4, Eq 2.21 discovered the following well-poised bilateral series identity (cf. [lgl Eq5.51 also):

Its reversal, also due to Bailey [4, Eq 1.21, reads as

Jackson [16, Eq 21 (see also Carlitz [7, Eq 1.21 and Bressoud [6, Eq 21) was the first person to get the finite form c = q-n of (9). Similarly, the finite form for identity (10) has been given by Bailey [3, Eq2], Guo 113, Eq 1.11 and Verma and Joshi [20, Eq 3.101. In view of the fact that (4% Q ) k 1 - wqk 1 W -(w; q ) k 1-w 1-w 1 - w qk the linear combination of (9) and (10) leads to a bilateral identity with an extra w-parameter:

This identity has explicitly appeared in Chu [9, Eq3.16~1. Bressoud [6, Eq5] found one of its special cases. Other special cases and terminating

755

forms have been obtained by Jain and Verma respectively in [19, Eqs 5.3 & 5.41 and [19, Eqs4.12 & 4.191.

1.2. In the same paper [4, Eq2.31, Bailey (1950) found another bilateral wellpoised series identity

It is not hard to get its reversal as follows:

When c = q P n , the finite form of (12) was obtained by Carlitz [7, Eq 2.121 and Verma and Joshi [20, Eq 3.131. Guo [13, Eq 2.61 has recently given, among others, an inductive proof for the case c = q-n of (13). Reversing the summation index by lc -lc i and then shifting it by lc -+ 1 Ic, we can verify without difficulty that

+

By means of the three terms relation

(qu; d k (4u; 4 ) k (u; d

(v; Q ) k

-

1 - uqk 1 - uqk -

1-u 1-u 1 - qk(u+4 q2”u (1 - u)(1- u) (1 - u)(1 - u) (1- u ) ( l - u) k

+

we can establish from the combination of (12), (13) and (14) the following general identity [9, Eq 3.16a-b] with two extra free parameters:

Two special terminating cases have appeared in Bressoud [6, Eqs 4 & 101.

756

2. New Proofs via Abel’s Method on Summation by Parts

In order to save writing space, denote two well-poised 3+3-functions by

According to (10) and (13), their reversals can be restated respectively as

2.1.

757

Recalling (16b) and (17b), we see that the last sum reduces to 2D(qb, c, qd). This leads to the following crossing relation between two well-poised functions: !B(b,c,d) = D(qb' "'dl

(1 - l / b c d ) (1- q / c )(1 - 1/ b d ) '

2.2.

Similarly, for two sequences defined by

we can check the following differences

Combining (16b) with (17b), we can manipulate, by means of Abel's lemma on summation by parts, the s&-series as follows:

The last sum is equal to 2B(b, c / q , d ) thanks to (16a) and (17a). This leads to another crossing relation between two pJ3-functions:

758

The combination of (20) and (23) leads us to two independent relations:

2.3.

Inverting the last two relations, we can reformulate them as follows:

Their duplication under the parameter exchange c e d yields respectively two important relations:

Observe that the latter relations result from the shifting parameter b by q 2 . Iterating them by shifting c and then d by q2, we establish the following two identities:

Iterating these relations further m-times, we finally find two relations

with an extra natural number parameter m as follows:

Recall the Jacobi triple product identity (cf. [12, 11-28]) +00

k=-cc

According to the Weierstrass M-test on uniformly convergent series (cf. Stromberg [18, P 141]), we may compute the following limits

k

= [ q3, Q , q2;q3100 = ( q ;Q ) 0 0 .

Letting m + 0;) in (28a-28b) and (29a-29b), we recover the bilateral series identities (9) and (12), respectively. This completes the proof of Bailey’s two bilateral well-poised sqs-series identities (9) and (12). It is interesting to remark that the present proof ties two nonterminating bilateral well-poised series (16a-16b) together with the crossing recursions (20) and (23), which lead t o the independent recurrence relations (24a-24b) and then t o both evaluations displayed in (9) and (12).

3. q-Analogue of Dixon’s Theorem For the alternating cubic-sums of binomial coefficients, there is Dixon’s well-known theorem, which states that

760

Its terminating q-analogue was first found by Jackson [16] and subsequently generalized by Bailey [4]. Define the q-Gaussian binomial coefficient by l+s-n.

Taking c = qVn in (9), we may reformulate the result as

which is equivalent to the formula discovered by Jackson [16, Eq 21. Letting b = d = q-n further, we find the identity due t o Bailey [4, Eq 3.31:

which is the q-analogue of Dixon’s well-known theorem (31) for 6 = 0. Similarly, taking c = q-n in (12), we may restate the result as

=[

2n + 1 (Q;q)n+l(q2/bd;q)n n ( q 2 / b ;q ) n ( q 2 / d ;q)n

1

which is equivalent t o the formula discovered by Carlitz [8, Eq 2.121. Letting b = d = q P n , we find another identity due to Bailey [4, Eq3.41:

which is the q-analogue of Dixon’s well-known theorem (31) for 6 = 1.

Acknowledgements This work has been carried out during the summer visit t o Dalian University of Technology (2005).

761

References 1. W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935. 2. W. N. Bailey, Series of hypergeometric type which are infinite in both directions, Quart. J. Math. (Oxford) 7 (1936), 105-115. 3. W. N. Bailey, A note o n certain q-identities, Quart. J. Math. Oxford 12 (1941), 173-175. 4. W. N. Bailey, O n the analogue of Dixon’s theorem f o r bilateral basic hypergeometric series, Quart. J. Math. Oxford [Series 21 1 (1950), 318-320. 5. W. N. Bailey, O n the s u m of a terminating &(l), Quart. J. Math. Oxford [Series 21 4 (1953), 237-240. 6. D. M. Bressoud, Almost poised basic hypergeometric series, Proc. Indian Acad. Sci. (Math. Sci.) 97:l-3 (1987), 61-66. 7. L. Carlitz, S o m e formulas of F . H . Jackson, Monatshhefte fur Math.73 (1969), 193-198. 8. L. Carlitz, S o m e applications of Saalschutz’s theorem, Rend. Sem. Mat. (Padova) 44 (1970), 91-95. 9. W. Chu, Basic Almost Poised Hypergeometric Series, Memoirs of American Mathematical Society. 135:642 (1998), pp 99+iv; MR 99b333035 10. W. Chu, Bailey’s very well poised a$,j-series identity, To appear in Journal of Combinatorial Theory. series A 11. G. Gasper, Elementary derivations of summation and transformation formulas f o r q-series, Fields Institute Communications 14 (1997), 55-70. 12. G. Gasper and M. Rahman, Basic Hypergeometric Series (2nd ed.), Cambridge University Press, Cambridge, 2004. 13. V. J. W. Guo, Elementary proofs of some q-identities of Jackson and Andrews- Jain, Discrete Math. 295 (2005), 63-74. 14. F. H. Jackson, Summation of q-hypergeometric series, Messenger of Math. 50 (1921), 101-112. 15. F. H. Jackson, T h e qe equations whose solutions are products of qe equations of lower order, Quart. J . Math.(Oxford) 11 (1940), 1-17. 16. F. H- Jackson, Certain q-identities, Quart. J. Math. Oxford 12 (1941), 167172. 17. L. 3. Slater, Generalized Hypergeoemtric .Functions, Cambridge University Press, Cambridge, 1966. 18. Karl R. Stromberg, An Introduction t o Classical Real Analysis, Wadsworth, INC. Belmont, California, 1981. 19. A. Verma and V. K. Jain, Certain summation formulae f o r q-series, J . Indian Math. SOC. 47 (1983), 71-85. 20. A. Verma and C. M. Joshi, S o m e remarks on summation of basic hypergeometric series, Houston J. Math. 5:2 (1979), 277-294.

762

2+1 DIMENSIONAL LATTICE HIERARCHIES DERIVED

FROM DISCRETE OPERATOR ZERO CURVATURE EQUATION

ZUO-NONG ZHU DEPARTMENT OF MATHEMATICS, SHANGHAI JIAO TONG UNIVERSITY, 800 DONGCHUAN ROAD, SHANGHAI, 200240, P.R. CHINA In this article, several 2+1 dimensional lattice hierarchies are constructed by using discrete operator zero curvature equation.

1. Introduction

In recent years there has been a wide interest in the study of nonlinear integrable lattice systems. It is well known that discrete lattice systems not only possess rich mathematical structures, but also have many applications in science, such as mathematical physics, numerical analysis, statistical physics, quantum physics, etc. Much of the research has been focused on 1 1 dimensional lattice systems. The most famous and well studied 1+1 dimensional integrable lattice systems are the Toda lattice and the Volterra lattice. It has been shown that they possess all the integrable On the other hand, 2 1 dimensional integrable lattice systems have also been studied, for example, the 2 1 dimensional Toda lattice Recently, by means of the Lie algebraic approach, Blaszak and Szum constructed some 2+1 dimensional continuous and discrete integrable hierarchies which have the Hamiltonian structures". The discrete spectral equation considered by them is

+

+

+

'l1O.

LlCln = XlCln,

(1)

where

L =p

f a

+ Urn+a-lE m+a-1 + ... + U , E ~ - a,,

02a

> -m,

(2)

where the field function uj is function of n, t , and y (here n is discrete, and t and y are continuous), where the operator E is defined by EZU(72)

= (EiU(72))EZ = u(n

+

i)Ei,

2

E 2,

763

E being the shift in n operator. By choosing a different operator L , Blaszak and Szumllconstructed 2+1 dimensional generalizations of some known lattice systems such as the 2+1 dimensional Toda lattice as well as some new lattice-field systems proposed by Blaszak and Marciniak12. It is known that the derivation of new integrable lattice hierarchies, specially the 2+1 lattice hierarchies, is an interesting and important topic. In this paper, our aim is to construct the 2+1 integrable lattice hierarchies. This we do by making use of the discrete zero curvature operator equation. 2. 2+1 dimensional lattice systems derived from discrete operator zero curvature equation It is well known that 1+1 dimensional integrable lattice hierarchy can be derived from the following discrete isospectral problem:

dlCt(n,t7

= V(u(n,t ) ,A)$(n,t , A).

dt

(4)

where u(n,t ) = (u1(n,t),u2(n,t ) ,...,u p ( nt))T , is a field function, $ is a eigenfunction, and A is a spectral parameter. The integrability condition of the spectral problem (3-4) is

at

= (E V )U - UV.

(5)

For a given discrete isospectral equation, one chooses the proper time evolution equation such that integrable lattice systems can be derived from discrete zero curvature equation (5). Usually one supposes that the element u i j of matrix V has the form13>14 m 1 =o

or

where a1 and bl are functions of the field function u(n,t ) , then one determines a1 and bl and derives the 1+1 integrable lattice hierarchy. In the current paper, we will generalize the method to 2+1 dimensional lattice systems. Let us consider the following 2+1 dimensional discrete isospectral

764

problem:

-wJ(n, t,Y, d$(n,t,Y,A) dt

= U('1L(n,t,Y),

+ a,)$(?

= V ( u ( nt, , Y),

+ a,)$(%t,Y,4.

t , Y, A),

(8) (9)

The integrability condition of the spectral problem leads to the following discrete operator zero curvature equation:

aU

-4 at

= [(EV)U- U V ] 4 .

(10)

Here 4 is an arbitrary function. For the 2+1 discrete isospectral problem (8), our aim is to seek the proper evolution equation of the eigenfunction $ such that this discrete operator zero curvature equation yields the 2+1 nonlinear lattice hierarchy. We first set the element Kj of matrix V in the following form:

, ~that the 2+1 nonlinear and then, we determine the functions a ~ such lattice hierarchy are constructed by means of Eq (10). In the following, we illustrate in detail how to get 2+1 dimensional lattice hierarchies by the presented approach. For simplicity, we set fn = f(n,t,y), f n , y = (1) 2+1 dimensional Benjamin-Ono lattice hierarchy. Take

%.

and

From the discrete operator zero curvature equation (lo), we have

m

1=0

765

In order t o get the 2+1 lattice hierarchy, aj,m(j = 0,1, ....,m - 1) must satisfy:

ao,m = 1,

+(E

2 a3,m = m(E - 1)- un,yy

-

m(m - 1) %yy 2

1)-1

........

+ munu,,, + ( m - l)un,y

We thus obtain the following 2+1 dimensional B-0 lattice hierarchy:

Eq (16) is trivial for m = 0 and m = 1. The equations for m = 2 and m = 3 are, respectively,

+ + Hun,,,, + + u,,,(34 + 221, + 3 ( E - 1)-1u,,,) 1)-'%+1,yy + ( E 1)-1(3(~,u,,y)y + 2~,,,,) %,tz

%,t3

= %,yyy

+3%(E

-

= %L,y

2U,%,y

(17)

'Il,,yy

-

+3(E - 1)-2%+l,yyy

(18)

+

where the operator H = ( E 1)(E- 1)-l. The term V associated with Eqs (17) and (18) is, respectively,

+ a,y + x + a, + 2(E 1)-1u,,,, v 3 = (A + a,)3 + (A + a,)2 + 3 ( E 1)-1un,,(A + a,) +3(E - 1)-2%+1,yy + ( E 1)-1(3u,~,,y + 2~,,,) v 2 =

(A

-

(19)

-

-

(20) It is interesting to note that Eq (17) leads to the dicrete K P equation15-17

( E - l)(qn,t + qn,,

-

2%&,,)

=(E

+

where we have set q, = u, 1. (2) 2+1 dimensional Toda lattice hierarchy. Let us consider the spectral problem (8) with

u=

(0 -21,

x -p,

+ a,

>-

+ l)q,,,,

(21)

766

Set

j=O

j=O

Substituting them to Eq (lo), we have

m v22

= C(Ebj,,)(A j=O

and

where

+ dy)m-j+l

767

Therefore, we find that aj,m and bj,m should satisfy the following equations:

j=O,l,

..., m - 1 .

Here we define b,+l,, results:

=

0. Solving Eqs (30-32), we have the following

......... And we get the 2+1 dimensional Toda lattice hierarchy:

vn,t,

= Fm,

pn,tm

= Gm.

(35)

We write here the first equations of the 2+1 Toda lattice hierarchy:

vn,to = vn(Pn - P n - 1 ) , Pn,to = vn+1 2

-

vn,ti = v n b n Pn,ti =

vn

+P n , y ;

2 -Pn-1

+vn+1

(36) -vn-l

( E - l)vn(Pn-l+ P n )

+Pn,y +pn-l,y),

+ vn,y + vn+l,y + Hpn,yy + 2pnpn,y(37)

768

The matrix V associated with Eqs (36) and (37) reads, respectively, as follows,

(38) 2 vn-1 + P n - 1 +

H~n-1,y

+ a y + Pn)

-vn(x

- vn,y

(A

A + a,

+

)

+Pn-l

+ +2 21,

( -~~ ) - ' p ~ ,(39) ~

(3) 2+1 dimensional Blaszak-Szum three-field lattice hierarchy. Consider the 2+1 spectral problem (8) with 0

1 +ay -pn-l 0

-un-l

(40)

0

We set the matrix V as the following: m

v 3 1

=a =

C

Uj,m(A

+

m

ay)"-j,

v 3 2 =

b=

j=O

C bj,,(A

+ aY)"-j,

j=O

then we have

and .irn =

(A

+ a, -

vn)EV11 -

+

( E 2 h 2 ) ( A 8, - vn)

-E2V23 - pnE&l - unEV31 Pn

= (E2V22)Pn- ~

-PnEV22 ?in =

+

' ~ 2 1(A+

(41) a y - vn)EV12

(42)

- ~nEV32

+ + a, -

( E 2 h 2 ) ~ n(A

v,)EVi3

- PnEV23 - ~nEV33

(43)

We can specify ~ j ,bj,m ~ ,and cj,m such that Eqs (41-43) yields a 2+1 lattice hierarchy. Here we only give the lattice equations corresponding t o m = 0.

769

Solving them, we get two solutions:

Thus the following two lattice systems are given:

The corresponding matrix V of the two equations is, respectively,

v=

(x+ay pn-2

0 X+ay

1

-u,-1 0 vn-2

(52)

(4) 2+1 dimensional Blaszak-Szum four-field lattice hierarchy. Let us construct another 2+1 dimensional Blaszak-Szum four-field lattice hierarchy. For the 2+1 discrete spectral problem (8), take

u=

(

0 0 - un-2 1

+

ay

1 0 -vn-2 0

0 1 -pn-2

Then we set the matrix V as the following:

0

-qn-2

0

(53)

770

It follows from Eq (10)that

and

we find that Eqs (54-57) are quite complex. Here we give lattice systems which correspond t o the case where a , b, c, and d are only dependent on the field functions. Under this case, we have

771

in which

Here the operator A is defined by

772

The matrix V associated with Eqs (62)-(64) is, respectively,

v=

(

0 1

APn-2

1

0

APn-1

- 21,-2

-%-2

-A(Pn-l+

0

0

+ a,

1

Pn-2) -47n-2 APn-3

where

v=

(

x+ay

0

-qn-1

x+ay

0 0

pqn-2

0

-qn

x+ay

0

vn-3

Pn-3

1

Un-3

In this paper, we have constructed several 2+1 lattice hierarchies by means of the discrete operator zero curvature equation (10). The method is a generalization of the one used in deriving 1+1 lattice hierarchies. It should be remarked that the 2+1 lattice hierarchies constructed in the current paper have been derived by Blaszak and Szuml'by means of the Lie algebraic approach. However, to our opinion, it is interesting to construct 2+1 lattice hierarchies by means of the discrete operator zero curvature equation. We believe that many more 2+1 lattice hierarchies can be constructed in this way.

Acknowledgments The work is supported by the National Natural Science Foundation of China under Grant No. 10671125 and also sponsored by Shanghai Pujiang Program.

773

References M. Toda, J . Phys. Soc. Japan22, 431 (1967). M. Toda, Theory of nonlinear lattice, (Springer, Berlin, 1981). H. Flaschka, Phys. Rev. B 9 , 1924 (1974). H. Flaschka, Prog. Theor. Phys. 51,703 (1974). S. V. Manakov, JETP 40, 269 (1975). S. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons. The Inverse Scattering Method (Consultants Bureau, New York, 1984). 7. R. Hirota and J. Satsuma, J. Phys. Soc. Jpn. 40, 891 (1976). 8. M. Wadati, Suppl. Prog. Theor. Phys. 59, 36 (1976). 9. K. Kajiwara and J. Satsuma, J . Math. Phys. 32, 506 (1991). 10. J. Villarroel and M. Ablowitz, Physica D 6 5 , 48 (1993). 11. M. Blaszak and Szum, J. Math. Phys. 42, 225 (2001). 12. M. Blaszak and K. Marciniak, J . Math. Phys. 35, 4661 (1994). 13. 0. Ragnisco, P. M. Santini, Inverse Probl. 6 , 441 (1990). 14. G. Z. Tu, J. Phys. A 23, 3903 (1990). 15. E. Date, M. Jimbo, and T. Miwa, J . Phys. SOC.Jpn.51, 4116, 4125 (1982); 52, 388, 761, 766 (1983). 16. T. Tamizhmani, S. Kanaga Vel, and K. M. Tamizhmani, J. Phys. A 31, 7627 (1998). 17. H. W. Tam, X. B. Hu, and X. M. Qian, J . Math. Phys. 43, 1008 (2002).

1. 2. 3. 4. 5. 6.