Bessel Polynomials (Lecture Notes in Mathematics)

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

698 Emil Grosswald

Bessel Polynomials

Springer-Verlag Berlin Heidelberg New York 1978

Author Emil Grosswald Department of Mathematics Temple University Philadelphia, PA 19122/USA

1 6. JAN. 1979

AMS Subject Classifications (1970): primary: 33 A 70 secondary: 33 A 65, 33 A 40, 33 A 75, 33 A 45, 33-01,33-02, 33-03, 35 J 05, 41A10, 44A10, 30A22, 30A80, 30A84, 10F35, 12A20, 12D10, 60E05 ISBN ISBN

3-540-09104-1 0-387-09104-1

Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting. re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use. a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. (c-: by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

To

ELIZABETH

BLANCHE

and

VI VI AN

FOREWORD The present book consists of an Introduction, 15 Chapters, an Appendix, two Bibliographies and two Indexes.The chapters are numbered consecutively,

from 1 to

iS and are grouped into four parts, as follows: Part I

- A short historic sketch (i Chapter) followed by the basic theory

(5 Chapters); Part II

- Analytic properties

Part I I I Part IV

Algebraic properties

(5 Chapters); (4 Chapters);

- Applications and miscellanea

(2 Chapters).

According to its subject matter, the chapter on asymptotic properties would fit better into Part II; however, some of the proofs require results obtained only in Chapter 10 [properties of zeros) and, for that reason, the chapter has been incorporated into Part Ill. The Appendix contains a list of some 12 open problems. In the first bibliography are listed all papers, monographs, etc., that could be located and that discuss Bessel Polynomials.

It is quite likely that, despite

all efforts made, absolute completeness has not been achieved.

The present writer

takes this opportunity to apologize to all authors, whose work has been overlooked. A second, separate bibliography lists books and papers quoted in the text, but not directly related to Bessel Polynomials. References to the bibliographies are enclosed in square brackets. ing to the second bibliography are distinguished by heavy print. W.H. Abdi - A basic analog of the Bessel Polynomials; while ~]

Those refer-

So [i] refers to: refers to: M. Abramo-

witz and l.E. Segun - Handbook of Mathematical Functions. Within each chapter, the sections, theorems, lemmata, corollaries, drawings, and formulae are numbered consecutively.

If quoted, or referred to within the same

chapter, only their own number is mentioned.

If, e.g., in Chapter 10 a reference

is made to formula (12), or to Section 2, this means formula (12), or Section 2 of Chapter 10.

The same formula, or section quoted in another chapter, would be refer-

red to as formula (I0.12), or Section (10.2), respectively.

The same holds, mutatis

mutandis, for theorems, drawings, etc. ~%ile writing this book, the author has received invaluable help from many colleagues; to all of them he owes a great debt of gratitude.

Of particular importance

was the great moral support received from Professors H.L. Krall and O. Frink, as well as A.M. Krall.

Professors Krall also read most of the manuscript and made valu-

able suggestions for improvements. As already mentioned, there is no hope for an absolutely complete bibliography; however, many more omissions would have occurred, were it not for the help received, in addition to the mentioned colleagues, also from Professors R.P. Agarwal, W.A. AISalam, H.W. Gould, M.E.H. Ismail, C. Underhill, and A. Wragg. Last, but not least, thanks are due to Ms. Gerry Sizemore-Ballard,

for her skill

VI

and infinite patience in typing the manuscript and to my daughter Vivian for her help with the Indexes. Part of the work on this book was done during the summer 1976, under a Summer Research Grant offered by Temple University and for which the author herewith expresses his gratitude. July 1978

E. Grosswald

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IX

PART I CHAPTER 1

Historic Sketch ................................................

CHAPTER 2

Bessel Pol~nomials and Bessel Functions ........................ Differential equations, their d-forms and their S-forms. Polynomial solutions. Their relations to Bessel functions. Generalized Bessel Polynomials.

CHAPTER 3

Recurrence Relations ........................................... Recurrence relations for yn,gn,~n. Representation of BP by determinants. polynomials.

,~

Recurrence relations for the generalized

Moments and Orthogonality on the Unit Circle .................... ~5 Moment problems and solutions by Stieltjes, Tchebycheff, Hamburger; the Bessel alternative. Weight function of the generalized BP. Moments of the simple BP. Orthogonality on the unit circle.

CHAPTER 4

PART I I CHAPTER S

Relations of the BP to the classical orthonormal pol~nomials and to other functions BP as generalized hypergeometric functions, as limits of Jacobi Polynomials, as Laguerre Polynomials; their representation by ~ i t t a k e r functions and by Lommel Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . , , . . ° . .

~l~ .,

CHAPTER 6

-

Generating Functions ........................................... Generating functions and pseudogenerating functions. Results of Krall and Frink, Burchnall, Ai-Salam, Brafman, Carlitz, and others. The theory of Lie groups and generating functions. Results of Weisner, Chatterjea, Das, McBride, Chen and Feng, and others. Different types of generating functions.

h"

CHAPTER 7

-

Formulas of Rodrigues Type ..................................... Methods of differential operators, of moments and of generating functions. Combinatorial Lemmas.

~

CHAPTER 8

-

The BP and Continued Fractions ................................. The BP as partial quotients. Approximation of the exponential function by ratios of BP.

~9

CHAPTER 9

-

Expansions of functions in series of BP ........................ Formal expansions in series of the polynomials yn[Z;a,b), or

6a

8n(Z;a,b ).

The Boas-Buck theory of generalized Appell Polynomials.

Convergence and summab~lity of expansions in BP. to expansions of powers and of exponentials.

Applications

PART llI CHAPTER i0 -

Properties

of the

zeros

o f BP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Location of zeros. Results of Burchnall, Grosswald, Dickinson, Agarwal, Barnes, van Rossum, Nasif, Parodi, McCarthy, DoPey, Wragg and Underhill, Saff and Varga. Olver's theorem. Laguerre's Theorem. Results of Ismail and Kelker. Sums of powers of the zeros.

75

VIII

CHAPTER

ii -

On the algebraic irreducibility of the BP ...................... Theorems of Dumas, Eisenstein, and Breusch. Newton Polygon. Degrees of possible factors. Cases of irreducibility. Schemes of factorization. Two conjectures.

")9

CHAPTER

12 -

The Galois Groul! of BP ......................................... 416 Theorems of Schur, Dedekind, Jordan, Cauchy, and Burnside. Resolvent and Discriminant. The Galois Group of the irreducible BP is the symmetric group. Details of the case n = 8.

C~PTER

13 -

Asymptotic properties of the BP ................................ Case of n constant, z -~ O. Case of constant z, n + =. Results of Grosswald, Obreshkov, Do£ev.

~24

CHAPTER 14 -

Applications ................................ i .................. The irrati~-ality of e r (r rational) and of ~ . Solution of the wave equation. The infinite divisibility of the Student t-distribution. Bernstein's theorem. Electrical networks with maximally flat delay. The inversion of the Laplace transform. Salzer's theorem.

d 3~

CHAPTER

Miscellanea .................................................... Mention of the work by many authors, not discussed in the preceding chapters.

q 50

Some open problems

related to BP ...............................

~62

related to BP ..............................

J6/4

PART IV

APPENDIX

iS -

-

BIBLIOCRAPI~

of books and papers

BIBLIOGRAPHY

of literature not directly

SUPdECT

related to BP .......................

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 ~ < 75

NAME INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

"I, 79

PARTIAL

N84

LIST OF SYMBOLS

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

INTRODUCTION Let us look at a few problems that, at first view, have little in common. PROBLEM i:

To prove that if r = a/b is rational, then e r is irrational;

also that

is irrational. Following C.L. Siegel

[$$] (who strea~dined an idea due to Hermite),

one first deter-

mines two polynomials An(X ) and Bn(X), both of degree n, such that eX+An(X)/Bn(X) has a zero of order (at least) 2n + 1 at x = 0.

This means, in particular,

that

the power series expansion of Rn(X) = Bn(X)e x + An(X) starts with the term of degree 2n + i, Rn(X) = ClX

2n+l

+ c2x

2n+2

+ ... say.

and the number of available coefficients, quely defined, up to a multiplicative

By counting the number of conditions

it turns out that An(X ) and Bn(X) are uni-

constant.

By proper choice of this constant

one can obtain that An(X ) and Bn(X ) should have integer coefficients. manipulations %(x)

By simple

one shows that An(-X ) = -Bn(X ) and that

= (n!)-ix 2n+l I 01 tn(l_t)netXdt

construction of the polynomials

The last assertions are proved by effective

involved

(see Sections

It follows from the integral representation that that Rn(X) > 0 for x ~ 0.

14.2 and 14.3 for details).

'.Rn(X) i' ~ (nl)-lixl ' '2n+le[Xl and

If now e r = e a/b were rational,

let q > 0 be its denominator.

As already observed,

also e a would be rational;

Bn(a) and An(a ) are integers,

$o that

m = qRn(a ) = q(Bn~a)e a + An(a)) is a positive integer. Using the bound on Rn(a), 0 < m < q.(n~)-la2n+le a and, by Stirling's 0 < m < q(a2n+lea/nn+i/2e-n(2z)i/2)(l+E),

where e ÷ 0 as n + =.

large n, 0 < m < I, which is absurd, because m is an integer.

formula,

For sufficiently Hence, e r cannot be

rational. Next setting x = ~i, Rn(~i ) = -An(-~i)(-i ) + An(~i ) = (-l)n+ 1 2 n + I n!

:I tn(l-t)nsin zt dt ~0

(the last equality depends on some computations

and will be justified in Chapter 14).

The integrand is positive, so that Rn(ni) # 0.

Let k = [~] where

greatest integer function; then A ( x ) with integer coefficients.

[x] stands for the

+ An(-X) is a polynomial in x 2 of degree k and

Hence, if ~

2

is rational, with denominator q > O, then

qkRn(~i) = qk{AnC~i ) + An(-~i)} = m, an integer, possibly negative, but certainly

X

# 0.

Also, by using the integral representation of Rn(X), 0 < Iml =

(ql/2 2)n

qk!Rn(wi) l < ( n ) - l q k 2n+l ' ~ ~ < nn+I/2e-n(2v)I/2 0 < In! < i for sufficiently large n. Hence 2 ,

(l+e)

(a ÷ 0 for n ~ =), or

This is, of course impossible for integral m.

and a f o r t i o r i w are irrational.

A (highly nontrivial) modification of

this proof permits one to show much more, namely that e r is actually transcendental for real, rational r.

In particular, for r = i, this implies the transcendency of

e itself. PROBLEM 2.

To prove that the Student t-distribution of 2n+l degrees of freedom is

infinitely divisible. We do not have to enter here into the probabilistic relevance, or even into the exact meaning of this important problem. pioneering work of Paul L e ~

Suffice it to say that, based on the

[44.] and of Gnedenko and Kolmogorov [~9], Kelker [65]

and then Ismail and Kelker [60] proved that the property holds if, and only if the Kn_i/2 (~xx) function ~(x) = ~x Kn+i/2(~x)

is completely monotonic on [0,~), which means that

(-l)k¢(k)(x) > 0 for 0 < x < = and all integral k ~ 0. e.g.

~],

Now, it is well-knowm (see,

10.2.17) that if the index of K (z) (the so called modified Hankel func-

tion) is of the form n+i/2 (n an integer), then (2z/~)I/2eZKn+i/2(z) Pn(U) a polynomial of exact degree n.

= Pn(I/z), with

Previous relation can now be written as

Pn_l(X -I/2) Pn_l(X I/2) n ¢(x] - xl/2pn(X_i/2 ) , or, with Pn(U) = u Pn(i/u), ¢(x) (xl/2) Pn

We now use

Bernstein's theorem (see [68]); this asserts that ¢(x) is completely monotonic if, and only if it is the Laplace transfarm of a function G(t), non-negative on 0 < t < ~.

In the present case it is possible to study the pol)~omials Pn(X) and

compute G(t). large.

It turns out that G(t) ~ 0 for small t > 0 and also for t sufficiently

This alone is not quite sufficient to settle the problem, but if we also

knew that G(t) is monotonic, then the conclusion immediately follows.

In fact, by

playing around with G(t), one soon suspects that it is not only monotonic, but actually completely monotonic.

In order to prove this, one appeals

once more to

Bernstein's theorem and finds that G(t) is the Laplace transform of ¢(x) =

(~2 x) -1/2{i Pn(U).

n ÷ j=l X ~j(x+ aj) 2 - i}, where ~l,a2 ..... an are the zeros of the polynomial

A detailed study of these zeros permits one to reduce the large bracket to

the form xn/q(x), where q(x) is a polynomial with real coefficients and such that q(x) > 0 at least for x > -

min l<jsn

l~jl 2.

This shows that ¢(x) > 0 for

XI

0 < x < ~; hence, G(t) _> 0 on 0 < t < - and ¢(x) is indeed completely monotonic, as we wanted to show. PROBLEM 3. (1)

To solve the equation with partial derivatives

1 32V AV = - ~ - c ~t 2

where A is the Laplacian,

32 2 3 i 32 3 A = ~ 3 r + r Dr + -2-r (~-~ + cot 0 ~ ) with the boundary conditions (i)

1

+ r2sin2 e

32

302

(i) and (ii):

V = V(r,e,¢,t) is symmetric with respect to a "polar axis" through the origin, so that, in fact, V = V(r,O,t) only;

(ii)

V is monochromatic,

i.e., all "waves" have the same frequency ~;

and with the initial condition (iii)

the values of V are prescribed along the polar axis at t = 0, say V(r,0,0) = f(r), a given function of r.

Here r, 0, and ¢ are the customary spherical coordinates, t stands for the time and c represents dimensionwise a velocity. Conditions

(i) and (ii) are imposed only in order to simplify the problem and

can be omitted, but the added complexity can easily be handled by classical methods and has nothing to do with the problem on hand. Following the lead of Krall and Frink [68] we look, in particular, at solutions of (1) of the form (obtained by separation of variables) u = r-ly(i/ikr) L(cos 0)e ik(ct-r) Here y and L are, so far, undetermined functions and we shall determine them precisely hy the condition that u be a solution of (I), while k is a parameter related to the frequency ~ by k = ~/c.

On account of (ii), k is a well defined constant.

The

(artificially looking) device of introducing complex elements into this physical problem is useful for obtaining propagating, rather then stationary waves.

The

real components v, w of u = v + iw will be real solutions of (I) and represent waves traveling with the velocity c.

For c = 0 and with x = i/ikr, one obtains, of course,

directly a real, stationary solution of (i). into account that

We now substitute u into (i), by taking

8u/3# = O, and obtain, with x = I/ikr and z = cos 8, that

L(z)(x2y"(x) + (2+2x)y'(x)) + y(x)((l-z2)L"(z)

- 2z L'(z))= O,

or, equivalently, x2~'"(x)+(2+2x))' '(x) = _ (l-zZ}n"(z)-2zn '(z)

y(x)

L(z)

XII

These two functions, each of which depends on a different independent variable, be identically equal only if their common value reduces to a constant, say C. follows

can It

L(z) satisfies an equation of the form

that

(l-z2) L"(z)-2z L'(z) + CL(z) = O. We immediately recognize here the classical equation of Legendre.

If, but only if

C = n(n+l) with n an integer (it is clearly sufficient to consider only n { O, because -n(-n+l) = n(n-l)) does this differential

equation admit a polynomial solu-

tion, namely the Legendre polynomial of exact degree n; we shall denote it by Ln(Z). Incidentally,

if we would not require symmetry with respect to the polar axis, then

we would obtain here the associate Legendre nolynomials

P(q)(z) instead of the simn

pler Legendre polynomials,

and this is the main reason for the present, more restric

tire formulation of the problem. So far, everything has been fairly routine; now, however,

it turns out rather

surprizingly that, with C = n(n+l) the equation (2)

x 2 y"(x) + (2+2x) y'(x) - n(n+l) y(x) = O,

satisfied by y(x), also admits a polynomial solution for n an integer, namely a polynomial of exact degree n, uniquely determined up to an arbitrary multiplicative constant.

We shall denote it by Yn(X) and may normalize it, e.g., by setting

Yn (0) = I. We

have, herewith,

obtained a sequence of solutions to (i), of the form

u n = Un(r,9,t ) = r-iLnfCOS, 3]Vn(I/irk)eik(ct-r),, . With each solution u n and for each constant an, also anU n is a solution of (i), and so is the sum V =

~ anUn, if it converges. n=O

In particular,

along the polar axis,

with z = cos 9 = I, Ln(l ) is equal to i, and we obtain at t = O, with previous substitution x = i/ikr, V

=

V(r,O,O)

=

~ a r -I e -ikryn(i/ikr ) n=O n

=

ik

~ n=O

xe-i/Xy n ( x ) .

a

n

In order to satisfy also the initial condition, we define F(x) by f(r) = f(i/ik~ ikx F(x), so that condition

(iii) becomes

I

ane-i/Xyn(X ) = F(x).

n=O From (2) it follows that, by taking as closed path of integration the unit circle,

=

XIII

Yn (Z)Ym(Z)e-2/Zdz = ~ 2(-l)n+l mn 2n+l ' where the Kronecker delta ~

mn

= 1 for m = n, ~

rml

= 0 otherwise.

It follows that

an = (-i)n+l (n+i/2)#F(z)yn(Z)e-I/zdz. With these values for an , V(r,O,t) =

~ anr-iLn(COS @)Yn(I/ikr)e ik(ct-r) is a forn=0

mal solution of (I), in general complex valued, that satisfies formally all boundary and initial conditions of the problem. series converges.

Precise conditions

It is an actual solution, if the infinite (that depend on the nature - especially the

singularities - of F(z)) are known (see [13]) for this convergence and will be discussed in Chapter 9.

Here we add only that in the more general situation, when we

discard the restrictions

(i) and (ii), the corresponding solution is of the form

.... ik (ct-r) Yn(i/ikr)/r, V(r,e,¢,t) = ~ n~ 0 an,kP~(cos e) sznlm¢+¢0Je k m=0 with the outer sum extended over all values of k = e/c, corresponding to all frequencies ~ that occur. What do these problems have in common?

All three depend on the study of cer-

tain sequences of polynomials, An(X ) (= -Bn(-X)) in Problem I, Pn(X) in Problem 2, Yn(X) in Problem 3.

In fact, the three problems have more in common than just that,

because actually, all three sequences of polynomials are essentially the same sequence.

There are still many other problems, in which this particular sequence of

polynomials kno~m to-day as Bessel Polynomials plays a fundamental role.

It also

turns out that these polynomials exhibit certain symmetries that are esthetically appealing and have therefore been studied for their own sake. a fairly extensive literature devoted to this specific subject.

To-day there exists Nevertheless,

recently, when the present author needed some information concerning these polynomials, it turned out that it required an inordinate amount of time to search through a large number of papers and several books, in order to locate many a particular fact needed.

It is the purpose of the present monograph, to give a coherent account

of this interesting sequence of polynomials.

It may be overly optimistic to hope

that everything known about them will be found here, but at least the more important theorems will be stated and proved.

Originally an attempt was made to obtain all

important properties in a unified way, but this attempt hm~ not always been successful; in fact it could hardly have been expected to be.

After all, it is not sur-

prizing that the structure of the Galois group of Pn(X) requires for its determination other methods than, say, the study of the domain of convergence of an expansion in a series of these same polynomials. The author has made himself a modest contribution to the subject matter, but the aim of this work is primarily expository:

to systematize and to make easily

XIV

accessible the work of all mathematicians active in this field.

But mainly, unless

this book succeeds in relieving the future student of this subject o£ the need for an exasperating, time consuming search for known items, deeply hidden in the litera ture, it will have failed in its purpose.

PART I CHAPTER 1 HISTORIC SKETCH I t may not be easy t o d e t e r m i n e t h e f i r s t

occurrence in the mathematical litera-

t u r e o f t h e s e q u e n c e o f p o l y n o m i a l s t h a t we a r e i n t e r e s t e d a p p e a r e d s p o r a d i c a l l y f o r a r a t h e r long t i m e . t r a n s c e n d e n e y o f e. j e c t o f our s t u d y .

in.

They seem t o have

In 1873 Hermite [2~] proved t h e

The p o l y n o m i a l s used by Hermite are c l o s e l y r e l a t e d t o t h e obSee [83] f o r t h e c o n n e c t i o n .

The p o l y n o m i a l s d e n o t e d by Olds

w i t h Tn and Zn a r e r e l a t e d t o t h e Bessel P o l y n o m i a l s yn(Z) by yn(Z) = Tn(Z) + Zn(Z}; s e e [83] and C h a p t e r 8 f o r more d e t a i l s . These p o l y n o m i a l s a p p e a r a l s o i n 1929, in a p a p e r by Bochner [14] and in one by Romanowsky [92].

S h o r t l y a f t e r w a r d s (1931), but q u i t e i n d e p e n d e n t l y , t h e y o c c u r i n

a long p a p e r [18] by J . L . of certain differential d i d H.L. K r a l l The f i r s t

B u r c h n a l l and T.W. Chaundy, who o b t a i n them as s o l u t i o n s

equations.

W. Hahn [58] runs a c r o s s them i n 1935 and so

[67] i n 1941. s y s t e m a t i c s t u d y o f t h e s e p o l y n o m i a l s i s due t o H.L. K r a l l and O.

F r i n k , who i n 1949 c o n s i d e r them i n a fundamental p a p e r [68] p u b l i s h e d i n t h e T r a n s a c t i o n s o f t h e AMS. They gave t h e s e p o l y n o m i a l s t h e name o f BESSEL POLYNOMIALS, u n d e r which t h e y have been known e v e r s i n c e .

This same d e s i g n a t i o n has o f t e n been

used by v a r i o u s a u t h o r s , even when t h e y a c t u a l l y s t u d i e d r e l a t e d p o l y n o m i a l s , o r different

normalizations, etc.

So, e . g . ,

we f i n d among t h e s e t s o f p o l y n o m i a l s

l a b e l l e d as B e s s e l P o l y n o m i a l s , b e s i d e s K r a l l and F r i n k ' s Yn(X), a l s o x n y n ( 1 / x ) , (-1)

n-1 n x Yn(2/x),

( x / 2 ) n y n ( 2 / x ) , and o t h e r s .

In t h e p r e s e n t monograph, we s h a l l adopt i n g e n e r a l t h e o r i g i n a l n o r m a l i z a t i o n o f K r a l l and P r i n k and we s h a l l a b b r e v i a t e t h e d e s i g n a t i o n B e s s e l P o l y n o m i a l s , by BP ( r e g a r d l e s s o f t h e i r use i n t h e s i n g u l a r , o r p l u r a l ) . K r a l l and F r i n k had been l e d t o t h e c o n s i d e r a t i o n o f t h e BP by a s t u d y o f t h e wave e q u a t i o n ( e s s e n t i a l l y recurrence relations,

our Problem 3).

They i n d i c a t e t h e d i f f e r e n t i a l

equation,

a pseudo g e n e r a t i n g f u n c t i o n and an o r t h o g o n a l i t y p r o p e r t y ;

t h e y a l s o g e n e r a l i z e t h e s e t Yn(X), by i n t r o d u c i n g two p a r a m e t e r s ( o n l y one r e a l l y

significant). Shortly afterwards, and stimulated by Krall and Frink's work, two other papers appeared almost simultaneously.

J.L. Burchnall [17] pointed out that the BP Yn(X)

is related to the polynomials 8n(X ) studied by Burchnall and Chaundy in [18] by 8n(X ) = xn Yn (i/x) of the results

By using the machinery developed there, Burchnall obtains several

o f K r a l l and F r i n k and, i n a d d i t i o n some o f t h e b e a u t i f u l

p r o p e r t i e s o f t h e z e r o s , as w e l l as a g e n e r a t i n g f u n c t i o n .

symmetry

The present author [53] studied asymptotic properties of the BP, also properties of their zeros, the irreducibility of the BP over the rationals and the Galois group of the BP. At about the same time, when Krall and Frink, Burchnall and the present author started the

systematic study of BP, W.E. Thomson studied certain networks used in

multistage amplifiers, and that led to called maximally - flat delay.

a

particularly desirable characteristic,

The investigation of the complex impedance and of

the transfer functions of these networks led to the consideration of certain polynomials, proportional to the transfer function and defined by initial values and a recurrence relation.

These polynomials turn out to be exactly the BP in the normali-

zation of Burchnall and Chaundy.

It is quite unfortunate that Thomson's work [107],

[108] remained essentially unknown to the mathematicians who worked on BP.

Indeed,

Thomson obtained many important properties of the zeros of BP, but he was not particularly interested in the theory.

"Those interested in the theory will find an out-

line in the Appendix", he writes.

It is indeed just an outline, set in particularly

small print.

In addition to many interesting details, some of which were rediscover-

ed only recently [see Chapter 10),the paper [108] contains also a tabulation of all the zeros (real and complex) of the BP up to the ninth degree (inclusive). other similar tabulations see [99], [I00], [70],

[I02],

For

[69], and [116].

A few years afterwards, a real flood of papers appeared, with improvements of the theorems concerned with the location of the zeros, with generating functions and with the relations of the BP to other special functions, especially to Bessel functions and the hypergeometric series; this is not too surprizing, since Yn(X) = 2F0(-n, l+n; -; -x/2).

It is impossible to mention at this place all contributions

made during these years, and quoted in the following pages but a few names and papers come to mind:

Carlitz

Rainville [89], Agarwal

[19]; AI-Salam and Carlitz [7], [8], Ai-Salam [3],

[2]; Toscano [109];

McCarthy

[74], [75]; Nasif [81], Dick-

inson [47], Brafman [15], Ragab [88], Wimp [112] and others. Some of this material was included in abbreviated form in the books by R.P. Boas and R.C. Buck [13] and by E.D. Rainville [90]. In 1962 Do~ev [48] obtained what until recently was the best upper bound for the absolute value of the zeros of BP. 01vet [84],

See Chapter 10 for more recent results by

[~6] and by Saff and Varga [98].

While the interest in BP never completely disappeared, there were a few years, during which the efforts of mathematicians were apparently directed into other channels.

But recently, there seems to have arisen a renewed interest in BP.

In

1969 the author [54] could settle (computer assisted - although in the proof itself no computer work is invoked) a remaining problem concerning the Galois groups of the BP of degrees 9, ii, and 12.

Parodi

[85] r e p r e s e n t s

t h e BP as d e t e r m i n a n t s and o b t a i n s new bounds f o r t h e

r e g i o n i n which one may f i n d z e r o s o f BP. known o n e s , one g e t s r a t h e r Barnes

[12] s t u d i e s

continued fractions

I f one combines t h e s e w i t h some p r e v i o u s l y

strong results.

a g a i n t h e z e r o s o f BP, as w e l l as t h e i r

c o n n e c t i o n with

and w i t h t h e e x p o n e n t i a l f u n c t i o n .

Wragg and U n d e r h i l l

[113] r e l a t e

mants t o t h e e x p o n e n t i a l f u n c t i o n ;

t h e BP t o t h e d e n o m i n a t o r s o f t h e Pad~ a p p r o x i -

they also represent,

f o l l o w i n g P a r o d i , t h e BP as

d e t e r m i n a n t s and, by u s e o f G e r s h g o r i n ' s and t h e B e n d i x s o n - H i r s c h t h e o r e m s , o b t a i n r a t h e r s i m p l e p r o o f s f o r good bounds on t h e z e r o s o f t h e BP. K e l k e r [65] and I s m a i l and K e l k e r [62] r e d u c e an i m p o r t a n t p r o b l e m o f p r o b a b i lity

(the infinite divisibility of certain distributions - essentially our Problem

2) to the complete monotonicity of the function ¢(x) = Yn_l(X-I/2)/xl/2yn(X-i/2). The proof that this property, in fact, holds, has been obtained by the present wTiter [55], [56]. In the meantime, Thomson's contribution, ignored by mathematicians, became rapidly common knowledge among electrical engineers.

They discovered soon the iden-

tity of Thomson's polynomials with the BP and used systematically the properties of these polynomials and of their zeros, as presented in [68], [17], and [53], in order to perfect amplifiers, as well as filters. full credit to Thomson,

In 1954, L. Storch [106a,b] giving

and quoting [108], [68], [17] and [53] treats the topic with

all the details needed for the understanding of the theory and also for the effective computation of the numerical values of the elements of the network. Soon the use of BP in the synthesis of certain networks was treated in textbooks like those of E. Guillemin [57] and D. Hazony [60]. More recently R.R. Shepard [I01] indicated an almost mechanical method for the design of networks with certain preassigned characteristics, by use (among others)

of BP.

For

related work see

also [64], [115], and [116]. At present, the BP are accepted along with the classical orthogonal polynomials among the "special functions" and are often mentioned in connection with either the Bessel functions,

or some other previously studied functions,

ducible (see Chapter 5).

to which they are re-

They are quoted, besides in [90], also in such well-known

collections as [71], [72], [20], [50] or [ll4](but not in [~]), and there does not appear to exist in the mathematical literature any systematic discussion of their properties.

This fact, the author hopes, will be accepted as a sufficient justifica-

tion for the present monograph.

CHAPTER 2 BESSEL POLYNOMIALS AND BESSEL FUNCTIONS: DIFFERENTIAL EQUATIONS AND THEIR SOLUTIONS. I.

The so called "modified" Bessel functions

Iv(z) and Kv(z ) - sometimes improperly

designated as being of imaginary argument -, K (z) also known as the MacDonald function, satisfy the well-known (I)

(see e.g. [61] or [~]) differential

equation

Lw - Z2W '' + zw' - (z 2 + v2)w = O. For fixed v with Re v > 0

the following asymptotic relations hold (see [I]):

For z -~ O, K (z) _= ~1 r(~)(z/2) -v, K~(z) ~-r(v+l)2~-iz v-I (2a) Iv(z) ~ {F(v+l)}-l(z/2) v, l$(z) a (2F(v))-l(z/2)v-l; for z ÷ ~

and larg z I < ~,

(2b)

Kv(z ) ~ (w/2z)i/2e-Z,

Iv(z) ~ (2wz)-i/2eZ.

These relations suggest that functions like

¢(z), or

8(z), defined by w = z-re, or

by w = z-Ve-Zs, with e = eZ¢, may exhibit a simpler behavior than the Bessel functions themselves. 2.

Let us consider first 8 = e(z).

By logarithmic differentiation v

w, = (- ~ - i

w"=

O" (~ +~--

(

2)W

8'

+ ~-)w

v

+ (- z -I +

~--)w'

z O"

8'.2.Jw v -i + ~--)

.0'.2

z

and, substituting these in Cl) we obtain Lw = Cz8" - (2z+2v-l)%'

+ (2v-l)@)zw@ -I = 0.

As zw(z) ~ 0, it follows that e(z) satisfies the differential (3)

z@" - (2z+2v-l)e'

equation

+ (2v-l)@ = 0.

By the general theory of linear differential

equations

(e.g., [~])

the origin

is a regular singular point and there exist (in general) two independent particular solutions of the form @ = z ~

~ cmzm. m=0

Here ~ is any one of the two (in general

distinct) solutions of the indicial equation.

Either by Frobenius' method, or by

general considerations one finds that the indicial equation is (4)

a(a

- 2u)

= O.

The solutions of (4) are indeed distinct, except for v = O.

In the latter case

the two solutions are, of course, proportional to e-ZIo(Z) and e-ZKo(Z), respectively and are of no particular further interest here. F o r u # O, s e t

8 = O(z,u)

=

Cmzm a n d 8 = O ( z , ~ )

= z 2~

~

m=O entiation

and s u b s t i t u t i o n

into

(3),

2u+l-2m (2u-m)m Cm-1 ~ " ' "

cm -

c'zm'm

By d i f f e r -

m=O

we o b t a i n (2u-1)(2u-3)...(2~-2m÷l) = (2~-l)(2~-2)...(2u-m)ml

Co "

s o that 8(z,v) = coCl+z+ (29-i)(2v-3) z 2 (2v-l)(2v-3)...(2v-2m+l) zm (29-I)(29-2) 2! + "'" + (29-1) (29-2) ... (2v-m) m--~ + "'" )" Similarly, C' 2~+2m-I c' = (2v+l)(2v+3)...(2v+2m-l) m - (2v+m)m m-i "'" = (2~+l)(2v+2)...(2,~+m)m!

c0

and 2 %(z,u) = c6z2~(l+z+ 2~+3 Z (2~+I)(2'~+3)...(2~+2m-I) 29+2 2: + "'" + (2~+l)(2v+2)...(29+m) The f u n c t i o n s linear

O(z,~)

combination

(2a) show t h a t

with constant

i f we s e t

o f [3) w i t h u (0) = u ~ ( 0 )

(s)

a n d z~eZK ( z ) ,

both

coefficients

are solutions of these

Consequently,

o f (3) a n d s o i s

functions.

u v ( z ) = cQz~eZK ( z ) - 2 v - l r ( v ) e ( z , ~ ) , = 0.

m z m--~ + "'') "

u (z) v a n i s h e s

Also,

t h e n u (z) identically

any

relations is

a solution

and

8(z,v) = ~vzVeZKv(z) with ~v = c021-~(r(~))-i. In a similar way one verifies that 8(z,~) = ~ zVeZl (z), with ~

= c62~r(~+1).

It is obvious that O(z,~) reduces to a polynomial of exact degree n if 2~ = 2n+l

and t h a t z - 2 ~ 0 [ z , ~ ) the first

case,

reduces

for ~ = n+l/2,

to a polynomial

t h e two i n d e p e n d e n t 2

0 ( z , n + 1) = c 0 ( l ÷ z + 2 n - 2 z 2n-1 2' ÷ "'"

and

of exact

degree n if

solutions

of

2~ = - 2 n - 1 . 65) a r e

m z m-T + " ' " n (2n-2) (2n-4) ...4.2 z + (2n-I) (2n-2)... (n+2) (n+l) ~T. ')

(2n-2) C2n-4)...(2n-2m+2) + (2n-1) C2n-2)...C2n-m+l)

In

2 , 2n+l.. 2n+4 z e(z,n+ i) = C o Z if+z+ 2n+3 2-! +

m

•

•

•

÷

(2n+4) (2n+6) ... (2n+2m) z (2n+3) (2n+4) ... (2n+m+l) m-T + "" ") -

1 In the second case, with ~ = -n - ~ , (2n+4)(2n+6)...(2n+2m) (2n+3)(2n+4)...(2n+m+l)

e(z,-n-i/2)

= e0(l+z+...+

e(z,-n-I/2)

= c~z-2n-l(l+z+...+

m

z___+ ...) m~

and m

(2n-2)(2n-4)...(2n-2m+2) (2n-l) (2n-2)... (2n-m+l)

(2n-2)...2 (2n-l)...(n+l)

z

_ _

+

m'

n z n--~ )"

If we take c o = c6, then (6)

e(z,n+i/2)

= z2n+le(z,-n-i/2)

and e(z,n+i/2)

= z2n+le(z,-n-i/2).

It follows that, in general, it will be sufficient to consider only non-negative values for the half integral second parameter•

Sometimes, however,

e.g., for rea-

sons of symmetry, it may be convcnicnt to be allowed to use also negative values for the integer n and we shall soon find a convenient way to do it. In @(z,n+I/2)

the coefficient of zn equals CO-2nn!/(2n)!.

In following

Burch-

nall [17] and Burchnall and Chaundy [18], we select c 0 = (2n)!/2nn2, so that the leading coefficient becomes one. simply by en[Z).

With this normalization we shall denote e(z,n+i/2)

One now easily verifies

(see [17]) that

n

(7)

@n(Z) =

where all coefficients (8)

a

are integers

n-m

=

n

[ an_mZm = [ amzn'm , m=O m=O (see Theorem 3•1) given by

2m-n(2n-m)! (n-m) .m.''

(n+m)! , am

2m(n-m) .m.' '

So, e.g., we find that 2 e0(z) = i, el(Z) = z+l, e2(z) = z +3z+3, etc. If we have to deal simultaneously with several polynomials,

it is useful to

identify the polynomials to which a coefficient belongs by a superscript. a~ 5) is the coefficient of z 5-3 = z 2 in es(z); by (8) its value is 8!

7:

. . . . 8.2~3! 12

So, e.g.,

(5+3)~ 23(5-3)~3!

_

420.

The coefficients a(n! respectively. m

that occur in (7) and (8) would then be denoted by a (n) and n-m

While the polynomials en(Z ) have been defined so far only for n > O, we observe that, if we replace formally n by -n in (8), we obtain a(-n) = (-n-m+l)(-n-m+2)...(m-n) m 2mm2

= (_l)2m (n+m-l)(n+m-2)...(n-m) 2mm!

= a(n-l). m

Hence, if we extend the definition (7) of en(Z) formally to negative subscripts, we obtain

(9)

n(Z) =

e -

n

[ a(-n)z -n-m = z-2n+l a(n-l)z (n-l)-m m N m=0 m= 0

= z-2n+l nil a~n-l)z(n-l)-m = z-2n+iOn_ l(z); m=0 indeed, on account of the factor n-m in the numerator of a (n) m a (n-l) vanishes.

the coefficient

n

The identity (9) may be taken as definition of @_n(Z). by n+l, we obtain @n(Z ) = z2n+le_(n+l)(Z ).

If we replace in (8) n

Comparison with (6) shows that

@_(n+l)(Z) is not a new function, but is precisely the function e(z,-n-i/2). In some contexts it is more convenient to work with the reverse polynomial n

(i0)

yn(Z) = znenCl/z) =

[ am zm, m=0

a normalization due to Krall and Frink (see [68

]).

If we want to have the first

coefficient reduced to unity, we may factor out a(n)n = ((2n)!)/2nn! = Co and obtain yn(Z) = c0

n~ b(n)zn-m, b(n) = 2! m=0 m m m!

(2n-m)(2n-m-l)...(n-m+l) 2n(2n-l)...(n+l)

By differentiation of @n(Z ) = znyn(i/z) we obtain successively e~(z) =

nzn-lyn(1/z)-zn-2y~(1/z),

and 8n(Z ) =

n(n-1)zn-2yn(1/z)-nzn-3yn(1/z)_(n_2)zn-3Yn(1/z)

+ zn-4yn(1/z).

If we substitute these into (3) with ~ = n+I/2, we obtain after a few simplifications (see [68]) that YnCX) satisfies the differential equation (Ii)

z2y~(z) + 2(z+l)y~Cz)-nCn+l)YnCZ) From (99 and (i0) it follows that

= 0.

Yn [z) = znSn (z-l) = zn'z-2n-l@-(n+l) (z-l) = z-n-l@-(n+l)(Z-1)

= Y-n-i (z)'

so that the polynomials yn(Z) are defined for negative subscripts by the particularly simple relation (see [68] and [47]) (12)

y_n(Z) = Yn_l(Z). We shall speak occasionally of both, 8n(Z) and yn(Z) as Bessel Polynomials

and it will be clear from the context, which one is meant.

(BP)

When confusion could

arise, then yn(Z) will be called the n-th BP, while 8n(Z) will be referred to (following Boas and Buck [13]) as the reverse BP. We summarize (and slightly complete) the results obtained in this section in the following theorem: THEOREM I:

For integral n > O, the differential equation

zw" - 2(z+n)w' + 2nw = 0 has polynomial solutions w = @n(Z ) . specifically On(Z) =

These are polynomials of exact degree n and

n ~ a(n)z n-m, with a (n) m=O m m

(n+m)! 2re(n-m)!m!

If n = O, the general solution of the differential equation

is cle

2z

+ c2

and also in this case ~mong the solutions one finds the polynomial of degree zero

eo(Z ) = i, the same as one obtains formally by 8ettin~j n = 0 in above formulae. The equation with n replaced by the negative integer -n, n > 0 has a rational solution 8_n(Z) that satisfies 8_n(Z ) = z -2n+18 n_l["z"), or, equivalently,

@n(Z) = z2n+ls_(n+l)(Z). The equation

z2w '' + 2(z+l)w' - n(n+l)w = 0 has polynomial solutions w = yn(Z) of exact degree n.

Specifically, with the seQne

n

aCn)m as

above,

yn(Z) =

a(n)z m is a solution, so t ~ t

m=O

m

yn(Z) = zn8 (l/z) and yn(Z) n

and en(Z ) are polynomials reverse to each other. For n = 0 the general solution of the equation i8 cle2/Z + c 2 and o~nong these solutions one has, in particular, the polynomial of degree zero yO(z) = I.

If n is replaced by -n-l, the product n(n+l) remains unchanged, so that y_n_l(Z) = yn(Z). 3.

We turn now to the function ~(z) = e-Z@(z).

tion, e' = eZ~ ' + eZ~, e" = eZ~ '' + 2eZ~ ' + eZ~

From @(z) = eZ~(z), by differentiaand if we substitute these in (3),

it follows that ~ satisfies the particularly simple differential equation (13)

z2~ '' - 2nz~' = z2~ . 2 Denote the linear differential operator z

d2 d dz 2 - 2nz ~

by L; then the equation

L# = z2~ Hence, with ~(z) = e-Ze(z), also

stays invariant under the transformation z -~ -z. ~(-z) = eZe(-z) will be a solution of (13).

The differential operator L, of second order, can be factored with d the help of the first order operator ~ = z ~-~ .

For future use, and also because

they are of independent interest, some of the (well known - see [17]) properties of this operator will be developed here. Clearly

~zn = nz n

and, more generally, (14)

~kzn = nkz n

holds, and also (15)

(~-a)z n = (n-a)z n. More generally, if f(z) is differentiable,

~f = zf' and, if f(z) is n times

differentiable, then ~nf =

n ~ s(m)zmf(m) n m=l

where S (m) are the Stifling numbers of the second kind. n

The general case follows by

induction on n, starting from n = i (which holds by the definition of 5), with the help of the recurrence relation

(see [~] p. 825) S (m) = mS (m) + S (m-l) '

n+l

n

n

"

Next, by using (15) one verifies that the operators ~-a and ~-b commute. sequently, if P[n) is a polynomial with zeros Ul,U2, .... Um, one has (16)

P(~)z k = (k-Ul)(k-u2)...(k-Um)Z k = P(k)-z k.

Con-

10

We observe that for every differentiable function f(z), (~-n) zf(z) = z ~d (zf(z))-nzf(z) = z(zf'-(n-l)f) = z(~-(n-l))f(z) It follows that a factor z may be "moved across" the operator g-n, from right to left, provided that n is replaced by n-l. If we iterate the procedure we obtain the result that (17)

(~-nl)(6-n2)...(6-nr)Zf(z)

In particular,

for f(z)

= e

-bz

, we h a v e

( 6 - n 1) ( 6 - n 2) . . . ( ~ - n r ) ze

(18)

= z(6-nl+l)...(6-nr+l)f(z).

-bz

= z (~-nl+l) (6-n2+1)...

( ~ - n r + l ) e -bz .

It is possible to compute explicitly the right hand side of (18). This has a particularly simple expression, if the constants nl,n 2 .... ,nr are consecutive integers. Indeed,

(6_n)e-bZ

_(bz+n)e-bZ

zn+l d

(z-ne-bZ),

and ~(~_n)e-bZ = (n+l)zn+l so t h a t

(6-n-1)(~-n)e

d

d 2 (z-ne-bZ), dz 2

( z - n e - b Z ) + z n+2

_

_

-bz = z n+2 _ _d2 ( z - n e - b z ) dz 2

An induction on m will now complete the proof of (19)

(~-n-m+l)(6-n-m+2)...(6-n)e

-bz

n+m = z

dm

(z-ne-bZ).

dz m We observe in particular that for a twice differentiable function f(z),

~(6-a)f and i f ,

in particular,

factorization

o f L.

a = 2n+l, then ~(~-2n-1)f From (13) i t now f o l l o w s ,

(20) THEOREM 2.

operator

= z 2 f '' + ( 1 - a ) z f '

6(6-2n-1)# (Burchnall

[17],

= Lf. that

This yields

¢(z)

of

[18]).

Define the differential

then the function ~n(Z) = Q~(~)e -z is a IA

solution of (20). Proof.

the announced

is a solution

= z2¢.

Chaundy and B u r c h n a l l

= (~-1)(6-3)...(~-(2n-1));

,

6(~-2n-l)~n = ~(6-2n-i)(6-1)(~-3)... (6-2n+l)e -z

= (8-1) (8-3)... (8-2n-l)6e -z = -(8-i) (8-s) . . . (8-2n-l)ze -z = -zS(6-2)...(8-2n)e

-z = -z(6-2)(8-4)...(8-2n)Te

-z

= z(6-2)(8-4)...(8-2n)ze -z = z2(8-1)(8-3)...(8-2n+l)e -z = Z2~n . Here the first and last equality are justified by the definition of ~n(Z), the second and fifth one by the eommutativity of the operators 8-a, the third and sixth one by the definition of 8, and the fourth and seventh by (18).

Theorem 2 may also

be proved directly, as follows: By using (16), ,o

(21)

CnCZ) : Qn(8)e -z : % ( 6 )

[

k

(-1)k k:

~

-

k=O

=

(_l)k

~

k:

Qn (8)zk

k=O

(_l)k ~(k)z k

X k,---~--

k=O

and

z 2,n(Z ) =

(22)

~~ (-I) k! k k=0

(-i) k Qn (k-2)z k " Qn (k)zk+2 -- k~ 2 (k-2)' = "

Next,

8¢n ( z )

(-1)k Qn(k)szk

=

~

k=O

=

°~ (-1)k ~k

~ (k-2) ' k=2

=

Jl

Qn (k) zk

(k-l)'.

=

o~ (_l)k =

(k)z k

k=O (k-3)(k-5)...(k-2n+l)z

k

,

and similarly, 62¢n(Z) = k~2= (k-2) (-l)k! k(k-3)...(k-2n+l)z k.

Hence,

(_1)k

(23)

L¢n = (82-(2n+l)6)~n = k!2 (k-2)!

(_l)k

Qn ( k - 2 ) z k = Z2¢n'

(k-2)'

k=2

(k-3) (k-5)... (k- (2n+l)) zk

°

by (22).

For l a t e r use we s t a t e the f o l l o w i n g . THEOREM 3. k = 1,3,5

.....

In the series expansion of ~ n ( Z ) , the coefficients of (2n-l).

k z

vanish for

12

Proof.

Qn(k) = 0 for k = 1,3 ..... (2n-l), so that Theorem 3 follows from (21).

As already observed, but also by ~n(-Z).

(15) and, hence,

(20), are satisfied not only by Cn(Z),

This is evident also from (22) and (23), in both of which one

only has to suppress the factor (-i) k under the summation sign. From ¢n(Z) = Qn(g)e "z and the relation between On(Z ) and Cn(Z) it follows that On(Z) = eZQn(~)e

-Z

,

an identity found already in [17]. For future use it is of interest to record here also the g-forms of the differential equations for 8n(Z ) and yn(Z): (24)

[g(g-2n-l)-2z(g-n)]gn(Z ) = 0 ,

(25)

[26 + z(g-n)(6+n+l)]Yn(Z ) = O. d If one replaces g by z dK ' one immediately verifies that these are, indeed,

the same as (3) and (ii), respectively.

Equation

4.

(Ii) has been generalized in several ways.

Krall and Frink [68] intro-

duce two new parameters and write the equation as (26)

z2y"+(az+b)y'-n(n+a-l)y

For a = b = 2, (26) reduces, of course, to (ii). the polynomial of degree n.

= 0. Following

[68], we shall denote

solution of (26) (if such exists) by yn(Z;a,b), the generalized BP The reverse BP of degree n, znyn(z-l;a,b)

will be denoted by en(Z;a,b).

If we set Y(z) = y(2z/b) and Z = 2z/b, then Y'(z) = (2/b)y'(2z/b), (2/b)2y"(2z/b)

Y"(z) =

and we verify that

z2Y"+(az+b)Y'-n(n+a-l)Y

2z = (2z/b)2y"(2z/b)+(a--~

+ 2)y'(2z/b)-n(n+a-l)y(2z/b)

= Z2y"(Z)+(aZ+2)y ' (Z)-n(n+a-l)y(Z). This shows that if y(Z) is a solution of (26) with b = 2, then y(2z/b) is a solution of the general equation

(26).

In other words, b is only a scale factor for the

independent variable and not an essential parameter.

If we want to keep b = 2, but

let yn(Z) or 8n(Z ) depend also on the parameter a, we write yn(Z;a) and en(Z,a), respectively. By Frobenius' method or by direct verification through substitution, shown that

it may be

13

n

(27)

yn(Z;a) :

7. d (n)zk, where d~ n) = n' (n+k+a-2)(k) k=O k! (n-k) !2 k

In (27) and hereafter we use the notation u (n) to mean u(u-l)...(u-n÷l).

Similarly,

u

will stand for u(u+1)...(u+n-l). No confusion should arise between these standn ard notations and sub, or s u p e r s c r i p t s , l i k e in d~ n) .

n From these the coefficients f~n) of yn(Z;a,b) =

~ f~n)zk k:O

are obtained, by

replacing z by 2z/b; hence, f~n) = n2(n+k+a-2) (k) kl(n-k)!b k

(28)

Ck) One verifies that, for a : b = 2, fEn)r : n.'( n + k ) ( n + k ) : k~ (n-k) '2k 2kk! (n-k) ! --

=

(a~n)

,

in agreement with (8). Some authors like Obrechkoff [82] and Docker [48] have adopted the normalizations b = i, or b = -I rather than b = 2. script.

They also set m = a-2 and write m as a super-

In their notation, therefore, yn(Z;a) = (-l)nPn (a-2)(-x/2) and, in particu-

lar, yn(Z) = (-l)nPn(0)(-x/2).

S.

The corresponding generalization of 0n(Z) is obtained most conveniently by

setting yn(Z;a,b) = z n O n ( Z - 1 ; a , b ) . If we substitute this in (26), we obtain by routine computations the differential equation satisfied by 8n = 8n(Z;a,b ), namely (29)

z0"-(2n-2+a+bz)Snn + hnOn = 0. For a = b = 2, (29) reduces to Z0n-2(z+n) 0n + 2n0 n = 0,

which is, of course, (3) with 2~-I = 2n. The obvious generalization of On(Z) is clearly Cn(Z;a,b) = e-bZ/20n(Z;a'b)" is convenient, however, to increase the flexibility of the presentation, by introducing a new parameter; hence, we set (30)

Cn(Z;a,b,c) = e-CZ0n(Z;a,b).

It

14

From this we obtain as particular ~n(Z;a,b)

for c = b/2;

cases @n(Z;a,b)

itself,

for c = 0, and

in addition also the case c = b will turn out to be of

interest. If we differentiate

(30) twice and use

satisfied by w = #n(Z;a,b,c), (31)

(29) we obtain the differential

zw"-(2n+a-2÷(b-2c)z)w'+(c(c-b)z+(b-2c)n+c(2-a))w For a = b = 2, c = O,

b = 2c = 2,

equation

namely

(31) reduces

again to

= O.

(3) with 2~-i = 2n, while

for

(31) becomes

(32)

zw"-(2n+a-2)w'+(2-a-z)w

satisfied by ~n(Z;a,2) Finally,

= 0,

= e-Zen(Z;a,2).

for b = c,

(33)

(31) simplifies

to

zw"-(2n+a-2-bz)w'+b(2-a-n)w

with the solution w = e-bZe As in the particular

n

= 0,

(z;a,b).

case a = b = 2c = 2, it is convenient

with the help of the first order differential

d 6 = z ~

operator

to "factor" .

(31)

From the defini-

tion of 6 follows ~(~+l-a-2n)w

d {zw' +(l-a-2n)w} = z ~-~

= z(zw"+(2-a-2n)w').

By (31) this equals (34)

z2(b-2c)w'+z{c(b-c)z+(2c-b)n+(a-2)c}w. If b ~ 2c,

(34) may be written as z(b-2c)(~-n+az+B)w,

= c(a-2)/(b-2c),

so that w = ~n(Z;a,b,c)

with ~ = c(b-c)/(b-2c),

is the solution of the differential

equation (3S)

6(~+l-a-2n)w For b = 2c,

(34) equals

{z2(b/2)2+(a-2)(b/2)z}w

8(~+l-a-2n)w In particular,

= -z(2c-b)(6-n+az+8)w.

=

f-form of (32).

(20).

satisfies

{z2(b/2)2+(a-2)(b/2)z}w.

if b = 2, this reduces to ~(6+l-a-2n)w

the

and w = #n(Z;a,b)

=

(z2÷(a-2)z)w,

If we set here also a = 2, we recover,

of course,

once more

15

6.

For b = c,

(35) becomes (see (20) i n [17])

(36)

8(~+l-a-2n) w = -bz(~-n+2-a)w,

verified

by O n ( Z ; a , b , b ) = e=bZo ( z ; a , b ) . n

In particular,

for a = 2, (361 reduces to

(37)

6(6-2n-l)w = -bz(~-n)w,

with the solution w = e-bZSn(Z;2,b ). Without restricting ourselves to a = 2, let us assume nevertheless Burchnall

[17]) that

a

(following

is at least a non-negative integer and consider the func-

tions (381

W(z) = CP(~)e -bz,

with P(6) = (~-n-a+l)(~-n-a)...(6-2n-a+2)

and constant C.

By (191 (39)

W = Cz 2n+a-I

dn (z-n-a+le-bZ). dz n

We claim that for any constant C, W is a solution of the differential equation (36). This can be proved directly, as in the second proof of Theorem i, but an easier proof is by use of (181.

On account of the homogeneity of (36), it is sufficient

to verify the claim for C = i.

We obtain successively,

6 (6+l-a-2n)W = 6(6+l-a-2n)[(6-n-a+l)...(6-2n-a+2)e -bz] =

(~-n-a+l)... (6-2n-a+2) (~-2n-a+l)~e -bz =

(6-n-a÷l)...(6-2n-a÷l)(-zb)e -bz = -bz(6-n-a+2)...(6-2n-a+2)e -bz

=

as claimed.

-bz(6-n-a+2) [(6-n-a+l)...(6-2n-a+2)e -bz] = -bz(8-n-a+2)W,

Here the first equality follows from (38), the second from the commu-

tativity of the operators 6-c, the third from the definition of 6, the fourth from (18), and the last two are obvious. One can easily verify that not all solutions W of (36), or, equivalently, of bz (33), are such that e w = p(z), a polynomial. It follows that at most one of any two linearly independent solutions of (33) (or (361) can be of this form.

However, we do know one such solution, namely (38).

Indeed, ebzw(z) = cebZp(~)e -bz is obviously a polynomial and, specifically, a constant multiple of On(Z;a,b I as already mentioned immediately after (33) and (36).

16

It follows,

using

ebZw(z) = cebZz 2n+a-1

(39) t h a t ,

f o r an a p p r o p r i a t e

dn ( z - n - a + l e - b Z ) . dz n

constant

C, @ n ( Z ; a , b )

In o r d e r t o d e t e r m i n e

=

the constant

C, we

observe that the highest power of z furnished by the Leibniz form of the derivative dn (z-n-a+le -bz) comes from the term z -n-a+l dn (e -bz) and il equal to dz n dz n (-b)nz-n-a+le-bZ; hence, the right hand side equals Ce bz z2n+a-l[(-b)nz-n-a+l+ czn(-b)n+ (lower powers of z).

(lower powers of z)]e -bz =

The coefficient of zn in en(z;a,b) is i, so that

C = (-l)nb -n and this finishes the proof of the Rodrigues-type formula (40)

en(Z;a,b ) = (-l)nb-nebzz 2n+a-I

dn (z-n-a+le-bZ). dz n

F o r b = 2, i n p a r t i c u l a r , en(z;a)

= ( _ l ) n 2 - n e 2 Z z 2 n + a - 1 _ _d n ( z - n - a + l e - 2 Z ) n dz

and (see [ 17 ]), if also a = 2, then en(Z ) =

7.

d n (z-n-le-2Z) . dz n

( 1)n2-ne2Zz 2n+l _

It is clear that all values of b # 0 are admissible and lead to BP.

however,

(26) reduces to z2y '' + azy' - n(n+a-l)y = 0.

r y = z , where the r are solutions of the equation r(r-l)+ar-n(n+a-1) solutions are n and l-a-n so that the general solution of (26) n 1-a-n y = ClZ +c2z

= 0.

These

for b = 0 becomes

For n > a+l, this is a polynomial only if c 2 = O, when it

reduces, essentially to zn. The parameter

For b = O,

The solutions are of the form

a

This is a trivial case of no further interest here.

may take arbitrary values.

This is clear from the form of

the coefficients dk(n) (see (27)) and fk(n) , which remain well defined for all complex values of

a.

For some of the theory to be developed, however, the values

a

-1,-2,... lead to special difficulties and may have to be discussed separately. e.g. if

a = l-n, equation (26) becomes y" _ n-I y' z

b Z

2

n z

and has the general solution y = --~ + ClZ+C2+bz log z.

= 0, So,

17

This is a polynomial

n only if b = 0, when it reduces to y = ClZ +c2, also of no

further interest. Whenever such cases arise, not zero, or a negative

integer.

we shall tacitly or explicitly

assume that

a

is

CHAPTER 3 RECURRENCE RELATIONS 1.

It is well known

(see

[I]) that the solutions K (z) of (2.1) satisfy recurrence

relations, such as

(I)

K _I-K + 1 = -(29/z)K ,

(2)

K'v = -K~-I - (~/z) K ,

(3)

K _I+K + 1 = -2K$,

(4)

K'

From

(2.5)

=

-K~+I

+

(~/z)K v.

it follows that Kn+i/2(z) = (~/2)i/2e-Zz-n-i/2@ (z) and c o = (2n)'/2nn! • n "

If we substitute this in (i) for Kv(z), we obtain after routine simplifications

(see [17]) (5)

@n+l(Z) = (2n+l)@n(Z) + Z2gn_l(Z). On account of @n(Z) = eZ~n(Z), the same recurrence relation holds also for the

functions

Cn(Z):

(6)

~n+l(Z) = (2n+l)~n(Z) + Z2~n_l(Z). If we replace in (5) z by z

-i

n+l , multiply the result by z and recall that

zn@n(Z -I) = yn(Z), then we obtain (see [68]) the recurrence relation for the BP yn(Z): (7)

Yn+l(Z) = (2n+l)zYn(Z) + Yn_l(Z).

The recurrence relations (5) and (7) allow us to compute successively the polynomials 8n(Z ) and yn(Z) with very little effort.

We obtain as first few polynomials,

starting from @0(z) = 1 and @l(Z) = l+z, the following:

@2(z)

=

301(z) + Z2@o(Z) = z 2+3z+3;

03(z) = 5@2(z) + Z2Ol(Z) =

5(z2+3z+3) + z2(z+l) = z 3+6z 2 +15z+15, etc. Similarly, starting from y0(z) = I, yl(z) = l+z, we find y2(z) = 3zYl(Z ) + y0(z) = 3z2+3z+l, y3{z) = 5zY2(Z ) + yl(z) = 5z(3z~+3z+l) + z+l = l$z3+lSz2+6z+l, etc. znOn(Z-I ) = yn(Z).

These results illustrate the relations znyn(Z -I) = @n(Z) and

19

From yO(z) = Co(Z) = i, yl(z) = Ol(Z ) = z+l, (S) and (7) also immediately follows

THEOREM i.

The coefficients of all BP are positive, rational integers.

I f we p r o c e e d i n t h e same way, by u s i n g a l s o

~ K v!

=

(see ( 2 . 5 ) s o l v e d

-vz-V-le-Ze(z,v)-z-Ve-Zo(z,.) + z-Ve-Zo'(z,.)

f o r ~vK ( z ) ) ,

we o b t a i n from (2)

t h a t On,

Cn' and Yn s a t i s f y

(8)

8n(Z ) = On(Z ) - ZOn_l(Z),

(9)

@n(Z)

= -ZCn_l(Z),

and (io)

z2y~(z) = [nz-l)Yn(Z)+Yn_l[Z),

respectively.

For (8), see [17], for (i0) see [68].

One may remark the particular-

ly simple form of (9), which may be new. Similarly, one obtains from (3) the following recurrence relations: (ii)

2zOn(z ) = (2z+2n+l) On(Z) - (Z2en_l(Z)+On+l(Z)),

(12)

2z~n(z ) = -Z2~n_l(Z)+(2n+l)~n(Z)-¢n+l(Z),

(13)

2z2yn[z) = (Yn_l(z)-yn(Z))+(Yn+l[z)-yn(Z))-Zyn(Z) = Yn_l (z) - (2+z) yn (z) +yn+ 1 (z) . Finally,

(4) is a linear combination of (2) and (3).

Either directly, as

before, or by combining [8) and (ii), or [9) and (12), or (i0) and (13) we obtain (see [17] for (14)) the corresponding relations (14)

zen(z ) : (z+2n+l)On(Z)-@n+l(Z),

C15)

z~(z)

(16)

z2yn_l(Z) = yn(Z)-(l+nz)Yn_l(Z).

= (2n+l)¢n[Z)-@n+l(Z),

By combining above formulae, or by starting from any other among the numerous recurrence relations known in the theory of Bessel functions, many other recurrence relations for the polynomials 8n(Z ) and yn(Z), or for the functions Cn(Z) can be obtained. For future use we observe that (7) [and similarly (5) and (6)), can be &eneralized to read

2O

(7')

yn(Z) = Pm(Z)Yn_m(Z) + Qm_l(Z)Yn_m_l(Z),

whore Pm(Z) and Qm_l(Z) a r e p o l y n o m i a l s i n z o f d e g r e e s m and m - l , r e s p e c t i v e l y . Indeed, on m:

(7) i s t h e i n s t a n c e m = 1 and t h e g e n e r a l s t a t e m e n t f o l l o w s by i n d u c t i o n Assuming ( 7 ' ) a l r e a d y v e r i f i e d ,

we r e p l a c e Yn_m(Z) by u s e o f (7) and o b t a i n

Yn = Pro(z) ((2n-2m+l)ZYn-m-1 + Yn-m-2 ) + Qm-l(Z)Yn-m-1 = {(2n-2m+l)ZPm(Z ) + Qm_l(Z)}Yn_m_l + Pm(Z)Yn_m_2 = Pm+l(Z)Yn_m_l + Qm(Z)Yn_m_2. This shows t h a t that

( 7 ' ) h o l d s a l s o f o r m + l , h e n c e f o r a l l m f n and, i n c i d e n t a l l y

shows

Pm+l(Z) = (2n-2m+l)ZPm(Z) + Qm_l(Z), Qm(Z) = Pro(z). Formulae ( 5 ) ,

(6) and (7) p e r m i t us t o e x t e n d t h e d e f i n i t i o n s

and yn(Z) t o n e g a t i v e v a l u e s o f n.

For @n(Z) and y n ( Z ) , t h i s

o f On(Z), ~ n ( Z ) ,

has a l r e a d y been done

i n ( 2 . 9 ) and (2.12) and one has t o v e r i f y t h a t t h e two e x t e n s i o n s a r e c o n s i s t e n t , for instance that

(7) l e a d s t o Y-n = Yn-l"

From (7) i t

f o l l o w s t h a t Yn-1 = Y n + l - ( 2 n + l ) z Y n ; h e n c e , Y-1 = Yl-ZY0 = l + z - z =

1 = Y0' i n agreement w i t h ( 2 . 1 2 ) . for all

subscripts

Assuming t h a t

(7) r e m a i n s c o n s i s t e n t

w i t h (2.12)

n o t e x c e e d i n g n, by ( 7 ) ,

Y-n-1 = Y - n + l - ( - 2 n + l ) z Y - n = Yn-2+(2n-1)ZYn-1 = Yn as we wanted t o show. what p r e c e d e s , 2.

The r e c u r r e n c e r e l a t i o n

as a d e t e r m i n a n t . n-l,

The c o n s i s t e n c y o f ( 2 . 9 ) w i t h (5) f o l l o w s i m m e d i a t e l y from

and can a l s o be checked d i r e c t l y

Indeed,

(7) p e r m i t s us t o g i v e a new d e f i n i t i o n f o l l o w i n g [85] l e t us c o n s i d e r

as a homogeneous l i n e a r

Yn namely ~ U ] - 3,2, and I.

equation in the three

Yn-2 ZYn_1 - 2n_-----[ = 0.

Y2 in the form ~--

one), it is Yl = z+l (= zy 0 + y_l ). Yn' Yn-i .... 'Yl

"unknowns" Yn' Y n - l ' Y n - 2 '

by Cramer's rule.

Yl -zY2 - 7 = O.

1 Yo zy I = ~ (= fL).

t o t h e BP Yn(X),

( 7 ) , w i t h n r e p l a c e d by

Next, we r e p l a c e n s u c c e s s i v e l y

Y3 For n = 3 the equation reads ~--

(inhomogeneously)

by t h e same method.

Finally,

by n - l ,

n-2,...,

For n = 2 we write it for n = 1 (the last

We solve this system of n linear equations The determinant of the system is

in

21

1 2n-I

-z

1 2n-I

0

0

-z

1 2n-3

0

0

1 2n-3

0

0

0

0

0

0

0

0

0

0

D = 1 -Z

• .•

0

and YnD = Dn, where D n is obtained by replacing the first column of D by the column vector of second members all zeros,

except

respectively.

of the equations.

of this vector are, as seen,

for the last two, which are 1/3 (from n = 2) and z+l

(from n = i),

It is clear that D = {(2n-l)(2n-3)...3.1} -I, because D is triangular;

hence it is equal to the product -z 1 2n-3 0 D

The entries

of its diagonal 1 2n-i -z 1 2n-S

elements•

0

...

1 2n-3

" ""

-z

" ""

As for 0

0

0

0

0

0

.

•

=

n •

.

.

Y

0

0

0

...

1 ~

-z

z+l

0

0

0

...

0

1

1

we shift the first column to last place, taking

into account the sign change

(_l)n-l• Next, we change the signs of the entries D

=

in the last column,

so that

(-l)nMn , where

n

1 2n-i

-Z

1 2n-3 0

-Z

1 2n-5

0

0

0

0

1 2n-3

0

0

0

0

0

0

.

.

:

-z

M n

= •

°

0

0

0

1 ~

-z

1__ -3

0

0

0

0

1

-z-1

22

Finally,

yn(Z)

= Dn/D = (-i)n.i.3.5

The corresponding useful,

because

results

...

(2n-l)M n.

for the generalized

they are rather

complicated

and lack symmetry

For On{Z ) on the other hand, we may start instead

of (7) and obtain

in exactly

BP yn(Z;a,b)

from

do not appear (see, however

(5) {with n replaced

equally

[74])•

by n-l)

the same way that

DO n = Dn, where I

- (2n-l)

-z

2

0

2

...

0

0

0

...

0

0

0

0

1

- (2n-3)

-z

0

0

0

0

...

1

-5

-z

0

0

0

0

...

0

1

-3

0

0

0

0

...

0

0

1

- (2n-l)

-z

D=

"2

-- i

and 2

0 2

. ..

0

0

0

...

0

0

0 0

i

- (2n-3)

-z

0

1

-(2n-5)

...

0

0

0

0

0

...

-5

-z

0

0

0

...

1

-3

0

0

0

...

0

i

D =(-i) n n

2

0 -z

2

-(l+z)

with On[Z ) = D n.

3.

From the recurrence

their coefficients

relations

We denote

for the BP, we obtain recurrence

as before

{see

•

in yn(Z),

normalized

relations

(2.8)) by a (n) the coefficient m

by Yn(O)

= i. n

By substituting

(2.10),

i.e. yn(Z)

a (n)

= m=O

I

(17)

a (n) = m

I

m

z TM into

(7) we obtain

for m = 0 ,

( 2 n = l ) a ( n ? l ) + a (n-2) m-i (2n- i) am(nil)

m

for 1 < m < n-2 , for m = n-i and m = n.

for of z m

23 By substituting

I_In-l)

(2.10) into (i0) we obtain

~0

a(n) = m

(18)

Finally,

:

(n_m+l)a(n)+a (n-l) m-i m

for 1 _< m <_ n-l,

a (n) n

for m = n.

if we substitute

(2.10) into (16) we obtain

1

(19)

a m(nl =

for m = O,

1

~ I I)

(m+n-l)a

-

~

for m = O ,

+a n-l)

for 1 < m _< n-i ,

(2n-l)a~nil) In each s e t o f f o r m u l a e all

cases,

4.

Recurrence relations

@n(Z'a)"

similar

However, t h e i r

appear in

reasons

to

usefulness

and ( 1 9 ) ,

the central

formula holds

for

(5) and (7) h o l d f o r t h e g e n e r a l i z e d seems t o b e l i m i t e d

it should suffice

to only list

BP and f o r

and t h e p r o o f s

are routine.

a few o f them h e r e

(some o f

[68]3:

(n+a-1)(2n+a-2)Yn+l(z,a)

(20)

(18),

if we remember that a~ n) = 1 and a (n) = 0 f o r m > n , o r m < 0. m

For t h e s e these

(17),

for m = n.

= [(2n+a)(n-l+a/2)z+a-2](2n+a-1)Yn(Z,a

)

+ n (2n+a)Yn_l (z, a ) , (21)

z2(2n+a-2)Yn(Z,a)

(223

z 2(2n+a-2)yn_ l(z,a)

(231

(n+a-1)(2n+a-2)en+l(z,a)

= [n(2n+a-2)z-2n]Yn(Z,a)+2nYn_l(z,a ) , = 2 ( n + a - 2 ) Y n ( Z ,a)

- {(n+a-2)(2n+a-2)z+2n}Yn_ l(z,a)

= [(2n+a)(n-l+a/2)+(a-2)z](2n+a-1)6

,

n(z,a)

+ n ( 2 n + a ) Z2en_l ( z , a ) , (24)

(l+(a-2)/2n)e~(z,a) Recursion

however, (251

= en(z,a ) - Zen_l(z,a ).

(233 holds unchanged

if we replace

@n(Z,a), by Cn(Z,a)

= en(Z,a)e-Z;

(24) becomes (2n+a-2)¢~(z, a) = -(a-2)~n(Z,a)-2nZ~n-l(z'a1"

One sees from (25), that the surprizing rather accidental.

simplicity

of (91 was, in some sense,

24

5.

The reader may have been surprized that in Chapter 2 nothing was said concerning

generalized BP for negative subscripts.

This was no simple omission.

The recur-

rence relation (20) permits us to define y_n_l(z;a,b) recurs ively, if we know Y-n and Yl-n"

The first step, however, the determination of y l(z;a,b) fails because

in (20) the factor n(2n+a) of Yn-i vanishes for n = 0.

This is not the case when

a = 2; in that case n (in fact 2n(n+l)) is a factor common to all terms and is cancelled. accident.

Also the difficulty with y_n(Z;a,b) for arbitrary real

a

is not an

Indeed, let us consider the differential equation (2.26) with n = -I, z2y '' + (az+b)y' + (a-2)y = 0.

We easily find that it has a single solution regular at z = 0 and normalized by y(0) = i, given by

(26)

Y(Z)(=Y-l(z;a'b))

= m=0~ Cm zm, with cm = (-l) m

This solution reduces to a polynomial if and only if excess of 2.

a

(re+a-3) (m) bm

equals an integer not in

For a = 2, y(z) (i.e., y_l(Z;2,b)) reduces to the constant

1 (= y0(z;2,b)), as we know from (2.12).

The other case of interest is a = I, when

y_l(Z;l,b) = l+z/b = yl(Z;l,b) and, by (20) of exact degree n.

with a = i, y_n(Z;l,b) is a polynomial

However, the theory for a = 1 has never been developed much

further, the suggestion of Krall and Frink ([68], bottom of page 109) notwithstanding.

See, however,

[99] and [i00].

For the other cases when y_l(z;a,b) is a polynomial, that is, for a = 0, or a

equal to a negative integer, y_l(z;a,b) is given by (26) and once we know

y_l(z;a,b) all other y_n(Z;a,b) can then be obtained recursively from (20), and are polynomials of degree n+l-a. Unfortunately, some parts of the theory break down, or become complicated precisely in these cases, when a = 0, -I, -2, ... and so we shall not pursue this matter further.

CHAPTER 4 MOMENTS AND ORTHOGONALITY ON THE UNIT CIRCLE I.

The theory of moments has a long history.

started with Stieltjes

Its modern phase may be said to have

[$4] and it was considerably advanced by the work of Tcheby-

cheff [ST] and Hamburger

[26], [27].

It is intimately connected with the idea of

orthogonality. In the classical setting, one starts with a function w(x), real and non-negative over an interval a ~ x ~ b of the real axis, and considers the "moments"

mn = f ab w (x) x n d x .

Set

An = A n(w) =

m0

mI

... mn_ 1

mI

m2

... mn

mn_ 1

mn

... m2n_2

, A0=

1

and let us assume that An ~ 0 for all n = 1,2,.... Next, we recall that the polynomials with real coefficients form a vector space S over the reals.

We define an inner product on S by setting (f,g) = I ab w(x)f(x)g(x)dx.

Clearly,

(f,g)

= (g,f).

If

(f,g)

= O, we s a y t h a t

f a nd g a r e o r t h o g o n a l

to each

other. One may t a k e linearly

for

S the

powers of x,

x

0

= l,

x,

2 x ....

which are

independent.

By u s e o f t h e Schmidt procedure powers,

as a basis

i.e.,

Schmidt method (see - we c a n f i n d

of polynomials

[35],

a finite

Pn(X)

p.

152])

- sometimes called

sequence of linear

of successive

one is orthogonal to all preceding ones.

degrees n

If Pn(X) :

0,1,2 ....

~ CmX m:0

t h e Gram-

combinations

n-m

of those

, such that

each

then it is, of

course, sufficient to insure that (xk,pn(X)) = 0 for k = 0,I ..... n-l.

If we

n substitute here ~ cmxn-m for Pn(X) and use the definition and linearity of the m=0 inner product, we obtain n homogeneous equations in the n+l coefficients Cm(m = 0,1 ..... n) of Pn(X).

These have a unique solution, up to an arbitrary

multiplicative factor, if and only if their determinant is different from zero. One immediately verifies that this determinant is precisely An and, by assumption, An ~ 0.

We may take advantage of the mentioned arbitrary constant factor of the

Cm'S , in order to obtain also (pn,Pn) = 1.

A set of polynomials that satisfies

26

these conditions,

i.e. (pn,Pm)

= 6nm, is called orthonormal.

One may show (see [if6],

2.2.6) that

Pn(X) = (An_ 1. An)

-1/2

m0

ml "'" mn

l

ml

?2

.mn+l

mn - 1

nln

m2n- 1

1

x

xn

i

For more d e t a i l s see [56]. This situation is usually generalized as follows [63]):

One replaces the interval

(see, e.g.

[56], [69], or

[a,b] of the real axis by an arc (infinite, or

finite, open or closed) of a curve C in the complex plane and defines the inner product by (f,g) = f

w(z)f(z)g(z)Idzl, where w(z) is a (generally non-holomorphic) C function of the complex variable z, real and non-negative on C. One observes that

instead of the commutativity equation (f,g) = (g,f). setting. interest•

(f,g) = (g,f), the inner product now satisfies the

Much of the classical theory goes through in this broader

For the case in point, however,

a different kind of orthogonality is of

It occurs already in [68] and has been studied in detail by John W. Jayne

[63] who calls it the Bessel alternative. possible•

R.D. Morton and A.M. Krall

in fact a distribution.

Still more general points of view are

[80] start from a weight "function" that is,

In the case of the classical orthonormal polynomials

(Jacobi, Laguerre, Hermite), use of the Fourier transform permits to recover the usual weight functions from the distributional successful in the case of the BP.

ones•

This approach is not entirely

A recent paper of A.M. Krall

[66] however,

com-

pletes this work. In spite of the great interest of this problem, further.

Indeed, precisely because of its breath,

it will not be discussed here

it encompasses much that is not

directly related to BP and the interested reader is directed to the original papers. We shall restrict ourselves to problems of moments directly pertinent to BP and, indeed, many of the results of this chapter can be found in [68] and in [17], although not always in the same generality. For arbitrary real or complex

rfa)

n 0 r(a+n-1)

b n (- ~) and l e t pn(Z) =

=

a # 0, -i, -2, ... and z # 0, set p = p(z;a,b) =

n ~

cmzm be an a r b i t r a r y polynomial of degree n.

m=O

The k - t h moment of pn(Z) on t h e u n i t c i r c l e , Mk(Pn;P) = 2 -1~

with weight f u n c t i o n p is d e f i n e d by

I ] z[=l z k Pn (z) (r [g

0

r ( aF+(a) r-1)

(-b/z)r)dz"

Here we may s u b s t i t u t e the e x p l i c i t form of pn(Z) and i n t e r c h a n g e summation and

27 integration

b e c a u s e t h e sum c o n v e r g e s e s s e n t i a l l y

only a finite

number o f t e r m s ,

like

~ n=O

and t h e p a t h o f i n t e g r a t i o n

Ibln n!

, pn(Z) c o n t a i n s

is finite.

Most terms

v a n i s h and we o b t a i n Mk(Pn,p)

=

~ Cm(r(a)/r(a+r-1))(-1)rb k+m-r=-i

r =

n+k÷l ~ (-1)rbrcr_k_lr(a)/r(a+r-1). r=k+l n

We a r e i n t e r e s t e d , where ( s e e ( 2 . 2 8 ) ) ~Titten

in t h e c a s e when Pn (x) = y n ( Z ; a , b )

in p a r t i c u l a r ,

f(n)=m ( ~ ) b - m ( n + m + a - 2 ) ( n + m + a - 3 ) ' ' ' ( n + a - 1 ) "

successively,

~ f(n)zm m m=0

The sum may t h e n be

as f o l l o w s :

n+k+l [ (_l)rbr r=k+l (_b)k+l (k+n+l)~

=

n ~

(-l)V

v=r-k-l=O

(_b)k+l (k+n+l) !

n! ( n + r - k + a - 3 ) . . . ( n + a - 1 ) r ( a ) (r-k-1):(n-r+k+D~r-k-lr(a+r-1) n! (n+v+a-2) . . . (n+a-1) • ( n - v + l ) . . . (n+k+l) v! a ( a + l ) . . . ( a + v + k - 1 )

n nl ~ ( ' l ) V v! v=0

(-b) k+l

(n+k+l) ! (n+v+a-2) . . . (n+a-1) (n-v) ' (k+v+a-1) (a+l)a . . . .

~ (-1) v($) v=0

=

=

(n+v+a-2):..(n+a-1) (k+v+a-1) (a+l)a

so that (1)

Mk(Yn,p ) =

(_b)k+l (k+n+a-1) (a+l)a "'"

n 1" v ' n " ... ~ (-) [v ) ( n + v + a - 2 ) (n÷a)(n+a-1}(k+v+a)(k+v+a+l)...(k+n+a-1). v=0

We claim that the sum is independent k(k-l)...(k-n+l). THEOREM i.

of a and is equal to the constant

If we substitute this value for the sum we obtain

Let Yn = Yn (z;a'b) and p = p(z;a,b);

Mk(Yn;P)

then

r(a) r (k+l) = (-b)k+l r(k+a+n)r(k+l-n)

Our claim concerning the sum may be well-known as no reference appears to be readily available,

(see e.g.

[68], p. 113); however,

a proof of the claim follows.

is based on two lemmas, the first of which is indeed well-known. LEMMA i.

(see (3.4) in [ZO]).

t k-n (k). ~. (n)(t_v) = v=0

It

28

k

n

k-n

Z (k) xt = ( x + l ) k = ( x + l ) n ( x . l ) k - n = Z (n) xv Z ck-n)xWw t=O v=O w=O

Proof.

k

Z xt t=0

k n n k-n Z (~) (k-n)w : Z xt Z (v) (t_v) v+w=t t=0 v=0 O~v~n O~wgk-n

and the lemma follows from the comparison of the coefficients of x t in the first and last member.

The function

LEMMA 2.

f(x)

(23

n

=

(_l)v (v) r(k+~+x+l)

Z

r (n+x)

v=O

(k+v+x) ( k - l + v + x ) . . . (n+v÷x)

i 8 i n d e p e n d e n t o f x, and has t h e v a l u e k ( k - 1 ) . . . one has f ( x ) Proof.

For k > n (2) is equivalent to 7

f(x)

C - l ) v ( ~n (k+n+x) ( k + n + x - 1 ) . . . (k+v+x) . . . (n+v+x) . . . (n+x) "V" (k+v+x) • (n+v+x) ""

v=O f(x)

for k < n,

= O.

n

(3)

in pc~ticulav,

(k-n+l);

is a rational

In fact, degree

function

as follows at most n.

and c o u l d h a v e p o l e s

from ( 3 ) , We s h a l l

all

these

show t h a t

poles

lim

a t mos t f o r x = -m, n < m 5 k+n.

cancel

f(x)

and f ( x )

exists

is a polynomial

for all

these

of

k+l values

of

X -~ - m

m in the It

then

cally,

g i v e n r a n g e and t h a t follows

that

all

these

the polynomial

limits

f(x)

equal k!/(k-n)!,

k'/(k-n)!

o f m. identi-

as claimed.

To find the limit, set m =

n+t, 0 < t < n, t E Z, x = -n-s.

r(k-s+l) lim s-~t

independently

of degree n < k+l equals

f(-n-s)

=

lim S-~t

= r(k-t+l)

(t-s) (t-l-s).

lira

t 7 ~

S+ t

v=O

because for t

< v _< n ,

not cancelled

by a c o r r e s p o n d i n g

lira f(-n-s) s ÷ t

= r(k-t+l)

n ""

(-1)

-s < t - s < v - s

t Z

v=O

(-s) r(-s)

factor

1 v n (-) (v)

V n (v)

v=O [

lim

f(x)

= (k-t)'t~ (k-n)!

and t h e v a n i s h i n g

t. w (t-v)!(k+v-n-t),

=

k! (k-n),

'

factor

of the denominator.

kl t'(k-t)'

(t-s)...C-s)

(v) (k*v--~n-sS_::~-s)

t (t-l)... (t-v+l) (k+v-n-t)'

By Lemma 1, the l a s t sum equals (tk) , so t h a t

X "+ - m

1 v n

(-)

=

Then, by (2),

It

of the numerator follows

(k-t),t, (k-n) l

indeed ' as claimed.

is

that

t 1 r n k-n Z (-) (v)(t-v)"

v=O

29

In the sun in (I) set a-I = x.

Proof of Theorem i.

The sum becomes

n

(-l)V(~)(n+v+x-l)...(n+x).(k+v+x-l)...(k+n÷x) v=0

=

n 1 v n (k+n+x)(k+n+x=l)...(k+v+x)...(n+v+x)...(n+x) [ (") (V) (k+v+x) (n+v+x)

v=0

by Lemma 2.

_

I f we s u b s t i t u t e

this

k2

(k-n)!

"'"

value in (1), the theorem follows.

Concerning the range of validity

o f Theorem i , one o b s e r v e s t h a t t h e r i g h t hand

side has a finite value for all a # -m, m an integer of the interval 0 ~ m < k+n-l. If

a

approaches one of the excluded values, then Mk(Yn(Z;a,b),0) ÷ ~.

2.

From Theorem i follow several corollaries.

COROLLARY i.

(see

Mk(Yn(Z;a,b);p)

(57) in [68]; also Lemma 16.1 in [80]).

For 0 f k < n,

= 0.

Formally, the corollary follows immediately from Theorem I. verified directly.

It can also be

Indeed, now we obtain instead of (I):

~ (yn,p)

(_b)k+l =

n [

v n (n+v+a-2) . . . (n+a-l) (-i) (v) (k+v+a-l) (a+l)a v=0 "'"

and the sun, again with a-I = x, equals n

(x÷l)... (x÷n-l)

v=0

( - )1

This sun vanishes for k < n.

v n

(v) ( n + v + x - l ) ( n + v + x - 2 ) . . . (k+v+x+l).

Indeed,

let F(y) =

n . 1"v'n" n+v+x-i [- ) [vjy ; then v=0

~n-k=l F n 8yn-k--------~= v=0[ (-l)V(~)(n+v+x-l)'''(k+v+x+l)yk+V+X"

The sun to be evaluated is the last one for y = i. F(y) = yn+x-i

n~

On the other hand,

n v = yn+x-l(l-y)n. (-I)n (v)y

v=0 Hence, by Leibniz' rule, the (n-k-l)-th derivative is of the form asy

n+x-l-s.. .n-t If-y) and vanishes for y = i, because n-t > n-(n-k-l) =

s+t=n-k-i

k+l > 1. By

taking for

a

a natural integer and, in particular a = 2, we obtain from

Theorem I the following corollaries:

COROLLARY 2.

(see [ 6 8 ] ) .

For i n t e g r a l

a > O,

30

Mk(Yn(Z;a,b);p) = (_b)k+l

COROLLARY 3.

(see [68]).

(a-l) !k! (k+n+a-l)' (k-n) ! "

k! a = 2, Mk[Y n (z;2,b);p) = (-b)k+l (k+n+l) ~ [k-n) :

For

(_l)n(b/z) n e-b/z n! = so, that Corollary 3 may be

REMARK. For a = 2, p[z;2,b) = n=0 rephrased as COROLLARY 3' 1

2~i

[see [68]).

For

k > n

zkyn(Z;2,b)e-b/Zdz = (_b)k+l

$1z]=l

k' (k+n+l) .'(k-n) '

; the integral vanishes for

k
Nk(Yn(Z),p)

(see [68]). (_2)k+ik! z l = l zkyn{Z)e-2/Zdz = (k+n+l) I (k-n) '

= Mk(Yn,e-2/z ) = ~

fork

>n

and vanishes for k < n .

COROLLARY S.

Proof.

(see

[17]).

i 2~i

I[zl=l Yk(Z;a'b)Yn (z;a'b)0(z;a'b)dz = 0.

if k ~ n, then

Let k < n; then yk{z;a,b)__ =

k [ flkjzm,: ~ with 0 < m < n; m m=0

hence, the result

follows from Corollary I. COROLLARY 6.

(see [68], [17]). 1 2~i

Proof.

~ bn! IIzI=l y (z;a,b)p(z;a,b)dz = (-i)n+l 2n+a-i

By Corollary S, ~ i

1 2~i

f[

zI=l

r (a)

I" ( n + a - l )

"

I]zi= I y~(z;a,b)o(z;a,b)dz

f~n)znyn(z;a,b)p(z;a,b)dz = f~n)Mn(Yn(Z;a,b) ,p)

and the result follows from Theorem I and (2.28). By setting a = b = 2, one obtains from Corollary 6 also COROLLARY 7.

3.

[see [68])

1 2~1

2 IIzI=l Yn (z)e-2/zdz = (-l)n+l

2 2n+l "

- -

If we replace yn(Z;a,b) by zn@n(z-l;a,b) in Theorem i and its cdrollaries, we

obtain the following results.

3~

THEOREM 2.

r(a) r(a+r-1)

Pl ( z ; a ' b ) = z-2 ~ r;0

Let

(-bz) r a n d s e t

= Mk(On(Z;a'b);Pl ( z ; a ' b ) ) = 2 -1~ f l z [ = l

%(en;01)

z -k (z-nen ( z ; a , b ) ) Ol(z;a,b)dz;

then

~lk(On;O1) = (_b)k+l COROLLARY 8.

For k < n, ~(en,Ol)

COROLLARY 9.

For integral a > 0,

For a = 2, pl(Z;2,b)

COROLLARY 10.

= z -2

[ r=0

r(a)r(k+l) r(k+a+n) r(k+l-n)

= 0. {~(@n;Pl)

(-bz) r r!

= z

-2 -bz e

In particular,

for a = b = 2, one obtains

COROLLARY ii.

For a = b = 2, Mk(@n,Z-2e -2z)

COROLLARY 12.

If

i 2~i

COROLLARY 13.

bk+l(a-l)!kl = (-i) k+l (k+n+a-l)!(k-n):

If

ii z

k~ (k+n+l)!(k-n)!

(_2) k+l

k! (k+n+l) ~ (k-n) ' "

then

I=i

k = n,

(z-kOk(Z;a,b))(z-nOn(Z;a,b))Pl(z'a,b)dz '

In particular,

for a = b = 2, the following Corollary holds:

COROLLARY

1 2--~/izl=l

4.

= 0

"

then

n! 1 i Izl=l (z-nOn(Z;a,b)) 201(z;a,b)d z = (-1)n+lb 2n+a-1 2~i

14.

(z -n@n(Z)) 2z-2e-2Zdz = (-l)n+l

In the case of positive integral

a

, Corollaries

r(a) r(n+a-1)

2 2n+]

12 and 13 may be written in

a somewhat different way. Let us observe that in this case 01(z;a,b ) = z -2 r~0 a(a+l-~.]~-(a+r-2) (-bz)r

= (a-1)!z -2

=

= b2 (a_l) : (_bz)-a

=

"

and we obtain

F o r a = 2, Mk(On'Z-2e-bZ) = (-b)k+l

k ~ n,

"

~[ r=0

~ ,~=a-2

(-bz)

b2(a_l):(_bz)-a(e -bz_ ai3 r=0

-(-bz) - ) .r r!

(_bz)r (a+r-2)' "

32

Hence, / [ z [ = l

(z-kOk(z;a'b))(z-nen(z;a'b))Pl (z;a'b)dz =

(-l)ab2-a(a-l) ! llzl= 1 z-a-k-nek(z;a,b) en(Z;a,b)(e-bZ-

= (_b)2-a(a_l)~

ai3 (-bz) r - - ) r~ r=O

dz

I I z l = 1 z-a-k-nOk(z;a,b)On(Z;a,b)e-bZdz.

Indeed, for 0 < r ¢ a-3, z - a - k - n + r has an exponent t = -a-k-n+r ~ - k - n - 3 ; hence, a l l neglected i n t e g r a l s are of the form I l z l = l Z - t O n ( Z ; a , b ) O k ( z ; a , b ) d z vanish, because the highest power of z that One k

and

k+n

can contribute is z

It follows that for a e Z + the integrals in Corollaries 12 and 13 may be written as

(_b)2-a(a_l)I ~1

n!(a-1)' I [ z l = 1 z -a-k-no k t"z"' a ' b ) O n ( z ; a ' b ) e - b z d z = 6nk(-1)n+lb (2n+a-1)(n+a-2)!

As a consequence the following Corollary holds:

COROLLARY 1S. 1 Yl z i= 1 z 2~i

for k # n

(see [17]).

For k = n,

-a-k-n..

Vk [z ;a,b)en(Z ;a,b)e-bZdz =

(_l)n+a+lba-1

n! (2n+a-l).(n+a-2)!'

the i n t e g r a l vanishes.

I f we r e p l a c e here z by z -1 and write yn(Z;a,b)

for z n e n ( Z - 1 ; a , b ) , we obtain

COROLLARY 16. 2~ii ilz[=l za-2yn(Z;a'b)Yk(Z;a'b)e-b/Zdz = 6kn(-l)n+a+Iba-i (2n+a-l)n"(n+a-2)' " For a = b = 2, Corollaries 15 and 16 reduce to Corollaries 14 and 7, respectively° Finally, if we recall that,by (2.291,¢n(Z;a,b) = e-bZ/2en(Z;a,b), we obtain from Corollaries 12, 14 and 15 that, for a c E + 1 ii -a -k -n = ( l)n+a+iba-i n! 2~i zi=l Z (z ¢k(z;a,b))(z %(z;a,b))dz (2n+a-l]-(n+a-2) if k = n, = 0 if k # n. In particular, ~

i

2 /izl= I z-2(z-nCn(Z))2dz = (-I)n+l 2n+l "

33 Note added August 30, 1978.

In the present chapter we discussed only the orthogona-

lity on the unit circle and the corresponding moments.

In fact, it was known from

general considerations, going back to the work of Stieltjes, that a function ¢(x), of bounded variation, ought to exist, such that f~ _~

Yn(X)Ym(X)d$(x) = 0 should hold

for all integers m ~ n, with the integral taken along the real axis.

Unfortunately,

no such function ~(x) was known. At the moment of shipping the present typescript to the publisher, the author receives a (handwritten) manuscript by A.M. Krall, with the following important results.

Define the function ~ (x) by ~(8)-~(~) :

lira

-1

C -~ 0

I B exp( 22~X2)sin( 22~C2)dx; X

+£

X

+¢

then the following holds:

1.

t~(x)

2.

5.

The support of d~(x) is the origin. The moments ~n xnd~(x) (-2)n+i/(n+l)

4.

f-Z Yn(X)Ym (x)d~(x) = (-l)n+126mn/(2n+l), where 6ran stands for the Kronecker

5.

For (a,b) # (2,2), these results generalize as follows:

is of bounded variation. for n

0,1,2 ....

delta.

Let z = x+ie and set 0(8)-~(~) =

lira

1

then ~(x) is of bounded P

l~a Im{(-b/z)iFl [i; a;

-b/z] }dx;

variation, ~n = (-b)n+i/a(a+l)"'" (a+n-l), and

Yn(X;a,b)Ym(X;a,b)d~(x) = (-l)n+In.'b ~mn/(2n+a-l)a(a÷l)...(a+n).

PART II CHAPTER 5 RELATIONS OF THE BP TO CLASSICAL ORTIIONOR~L POLYNOMIALS AND TO OTHER FUNCTIONS. i.

In Chapter 2, the BP have been defined as the polynomial factors of the Bessel

functions K (z) of half-integral order.

In fact, it follows from (2.5) with

= n + 1/2 and c o = (2n)!/2nn! that 8n(Z) = 8(z,n+I/2)

= c021/2-n(F(n+i/2))-Izn+I/2eZKn+i/2(z),

so that @n (z) = / 2z~ zneZKn+l/2(z).

(i)

Many of the properties of BP can be obtained in the easiest way from (i), by using known properties of the much studied functions Kn+I/2(z). the only possible point of view. of hypergeometric functions, of Whittaker functions

This, however, is not

The BP may also be considered as particular cases

(see, e.g., [90], [2], [3]) of Laguerre polynomials

[2], or as limiting cases of Jacobi polynomials

[3],

[2]. They

also may be defined as partial quotients of certain continued fractions (see [83]), or, as seen in Chapter 3, as determinants of some simple matrices (see [85]). While the identity of all these definitions is easy to establish, some of them are more convenient than others for the proof of specific properties of the BP. In the following sections of the present chapter we shall present the connection of the BP with hypergeometric functions, with Jacobi and Laguerre polynomials,with Lommel Polynomials and with the ~ittaker

functions.

The connections with continued

fractions will be discussed in Chapter 8. These connections with well studied functions have been used by many authors, in order to prove specific properties of the BP in the most convenient way. 2.

Let

(al)m(a2)m...(ap) m p F q ( a l , a 2 . . . . . ap; b l , b 2 . . . . . bq;Z) = 1+ m~1 (bl)m(b2)m" (bq) m ®

where ( a j ) m = a j ( a j + l ) ( a j + 2 ) . . . ( a j ÷ m - 1 ) ,

and b l b 2 . . . b q ~ 0.

The c o n d i t i o n s of convergence of t h i s sum a r e w e l l known ( s e e , e . g . [ 6 ] ) , but we do not need them h e r e , because we a r e i n t e r e s t e d case o f t e r m i n a t i n g s e r i e s . c u l a r , by t e r m i n a t i n g ) I t may be v e r i f i e d tion

zn m--i- '

[90], o r

only in t h e polynomial

The f u n c t i o n s

F r e p r e s e n t e d by convergent (in p a r t i Pq s e r i e s are t h e g e n e r a l i z e d h y p e r g e o m e t r i c f u n c t i o n s . (see [90]) t h a t

F Pq

is a s o l u t i o n o f t h e d i f f e r e n t i a l

equa-

35

(2)

[6 ~ (6+bj-1)-z ~Ti:l(6+ai)]w = O,

d where 6 = z ~

a s i n C h a p t e r 2.

In general, however,

if p > q+l,

one o f t h e a i ' s

is

m i a l and (2) i s m e a n i n g f u l

the corresponding a negative

without

series

integer,

diverges

for all

then the series

any restriction

z # O; i f ,

reduces

to a polyno-

on t h e n o n - n e g a t i v e

integers

p

and q. In p a r t i c u l a r ,

for q = 0 (i.e.,

and a 2 = l+n, the general term of

if

F Pq

(bl)n(b2)n...(bq)

n = 1 for all

n),a 1 = -n,

becomes m

(-n) (l-n)... (m-l-n). (n+l) (n+2)... (n+m) so that the general term of 2Fo(-n,l+n;-;-z/2)

Z

m

(n+m) ' -- (-l)m (n-m)~ (n+m) !

reads

z m~

'

m Z

.

2m(n-m) .m.' ' Comparison with (2.8) shows that (3)

yn(Z) = 2Fo(-n,l+n;-;-z/2). We also verify that in this case (2) reduces to [6-z(6-n)(6+n+l)]w

replacing z by -z/2, to [6+(z/2)(6-n)(6+n*l)]w way one verifies

= O, which is (2.25).

= O, or upon

In the same

(see [89], [90] and [50]) that

(4)

yn(Z;a,b) = 2Fo(-n,n+a-l;-;-z/b). Just as the well-known theory of Bessel functions can be used to obtain proper-

ties of BP from (i) so the equally well-known theory of hypergeometric

function can

be used for the same purpose, on account of (3). This has indeed been done successfully [47], [88], [89], [95]).

(s)

In this context,

lim

2Fl(al,a2;X;Xz)

(see, e.g.

[3], [2], [112],

[90], [15],

it is convenient to recall also that = 2Fo(al,a2;-;z).

This follows immediately from the fact that, for fixed n, (al)n(a2) n lim X + ~ 3.

(X)n

xnx n (al)n(a2)n xn -nl n~

We recall that the Jacobi Polynomial Pn(a'B)(z) may be defined

(see [90], p. 254)

by p ~ a , 6 ) (z) =

(i+~) n n-----~

If we set here a+l = X, z = l+2X(y/b), (l-z)/2 = -X(y/b) and

2Fl(-n,l+a+B+n;l+e;

(l-z)/2).

X+8 = a-l, then a = X-l, 8 = a-l-X,

36 = 2Fl(-n,a+n-1;X;-X(y/b))

2Fl(-n,l+a+8+n;l+a;(1-z)/2)

n! (X-l,a-X-1)(l+2X(y/b)) r(a+l)n! p(X-l,a-X-1)(l+2X(y/b)) (l+a) n Pn = r(n+a+l) n r(x)r(n) = n

r (n+X----------~

p(X-l,a-X-1) ( l + 2 X ( y / b ) ) . n

I f we take the l i m i t for X + ~, then the l e f t hand s i d e becomes (see (5)) 2Fo(-n,a+n-1;-;-(y/b)),

or, by (4), y n ( Y ; a , b ).

We have o b t a i n e d t h e r e s u l t

(see

[2]) (with z instead of y): (6)

yn(Z;a,b) =

nB(n,X)P n( X - l , a - X - l ) (l+2X(z/b)) ,

lim X-~

¢0

where as usually B(a,b) - r(a)r(b) r(a+b) By using the classical differential equation, Rodrigues formula, recurrence formulae, generating function, etc. of the Jacobi polynomials, one easily obtains the corresponding results for yn(Z;a,b),by passage to the limit in (6). See [2] for details.

4.

Laguerre Polynomials may be d e f i n e d (see [90], p. 201) by n Ln(a) (x) =

Z m=O

(- 1)m(l+a)n xm m~ (n-m)

n

I t follows t h a t h(-2n-a+l)(2/Z)n =

"! (l÷a)m

°

(-2n-a+2)n 2m

X ( - i ) m m!(n_m),,(_2n+2_a)m z m=O

and n.'(-z/2)nL (-2n-a+l) (2/z) = n

n

m=0 n

m=0

(-l)m+n2 m-n

( - 2 n - a + 2 ) ( - 2 n - a + 3 ) . . . ( - n - a + l ) n , , z n-m m! (n-m) .'[-2n+2-a) (-2n+5-a) . . . (-2n+m-a+l)

(2n+a-2)(2n+a-3)...(n+a-l)n! m,,(n-m)!(2n+a-2)...(2n+a-m-l)

(z/2)n-m

n

X m=0

(2n+a-m-2)...(n+a-l)n! m,,(n-m)!

k=0

(n+k+a-2)... (n+a-l)n! zk k ! (n-k) ~2k

(z/2)n-m

Comparison with (2.27) now shows (see [3]) t h a t

-m

37

yn(Z'a) = n:(-z/2)nL (-2n-a÷l)(2/z) '

n

"

Finally, if we replace z/2 by z/b, we obtain yn(Z;a,b) = n!(-z/b)nL (-2n-a+l)(b/z) n and en(Z;a'b) = znyn (z-l;a'b) = (-l)nn'b-nL(-2n-a+l)n (bz). In p a r t i c u l a r

(see [19], [3]) @n(Z ) = (_l)nn~2-nm(-2n-l)(2z). n

5.

The Whittaker functions Wk,m(Z ) are solutions of the differential equation

w"+{-i/4+kz-l+(i/4-m2)z-2}w

= 0 and may be represented for z # 0 (an empty

product

being set equal to one) by (see [67]): e-Z/2z k

~

Wk,m(Z) =

(m2-(k-i/2)2)(m2-(k-3/2)2)'''(m2-(k+i/2"r)2) + R; 2 r r z

r=O

here R = R(z;m,k,q) + 0 for fixed q and [z] + ~. The series obtained by letting q ÷ ~ is asymptotic. with integral n, then the sum is terminating and R = 0.

However, if m+k = n+i/2 Indeed m2-(k+i/2-r) 2 =

(m+k+i/2-r)(m-k-I/2+r) vanishes for r = n+l; hence, Wk -k+i/2+n(Z)

= e-Z/2zk

~

n(n+l-2k)(n-1)(n+2-2k)...(n-r+l)(n+r-2k) r r=0 r!z

One recognizes in the sum precisely 2F0(-n,n+l-2k;-;(-z-l)), yn(z-l'2(1-k),l), writing

It follows

z -1 f o r z and

~inally, observe,

.

a

yn(Z-1;2(l_k)

for 2(1-k),

i f we r e p l a c e

however,

that

misprint

yn(z;a,1)

on t h e r i g h t

or, by (4),

,1) = e z / 2 z - k ~~ k , _ k + l / 2 + n l Z. ) ,. =

el/2Zzl-a/2W1_a/2,(a_l)/2+n(1/z).

z by z / b we o b t a i n

(see

[2] and [ 5 ] ;

i n (13) o f [ 2 ] ) :

yn(Z;a,b)

=

or

e b / 2 z (z/b)l-a/2Wl_a/2,(a_l)/2+n(b/z).

For a = b = 2, one obtains the ordinary BP: yn(Z) = el/ZW0,n+i/2(2/z ).

38

6.

The Lommel Polynomials Rn,v(z ) may be defined (see [61], §9.6-9.73) by 1 Rn,v(z ) = ~ ~z cosec n~ {Jn+v(z)J_~+l(Z)+(-l)nj_v_n(Z)J

_l(Z)} .

If one replaces here the functions Jm(Z), for m = k + ½,half an odd integer, by Jk÷i/2(z ) = (2~z)-i/2{i-n-leiz yn(I/z) . + in+l e -iz yn(-i/z)} obtains

(see Dickinson [a7]):

For . = s + ~z , Rn,"

1 {i-nyn+s(iZ)Ys_ l ( - i z ) + z n y n + s ' ( - i Z ) Y s _ l ( i z ) } . function Jk+l/2(z)

and t h e BP yn(~ i / z )

from the d i f f e r e n t i a l

=

The r e l a t i o n

follows

between t h e Bessel

(see Chapters 2 and 3), e . g . ,

Kn+l/2(z-1. ) = (nz/2) 1/2e-1/Zyn(Z ) and t h e r e l a t i o n s in [68], d i r e c t l y

(z-l)

(see [68], (ii)) one

by

between K (z) and J ( z ) , or, as

equations.

7. We formalize some of the results obtained so far in this chapter in the following theorem. THEOREM i.

With previous notations for hypergeometric functions, Jacobi Polynomials,

Beta Function, Laguerre Polynomials and ~nittaker functions, the following relations (conong others) connect the BP with these special functions: (i)

Yn(Z;a,b) = 2Fo(-n,n+a-l;

[ii)

yn(Z) = 2Fo(-n,n+l;.;_z/2) ;

(iii)

yn(Z;a,b) =

lim X

(iv)

yn(Z) =

-; -z/b) ;

nB(n,x)p(X-l'a-x-l)(1+2x(y/b))

;

-~- ~

lira nB(n,x)P (x-l'l-x)(l+xy) x

->

n

oo

(v)

yn(Z;a,b) = n!(-z/b)nh (-2n-a+l)(b/z) n

(vi)

yn(Z) = n~(-z/2)nL (-2n-l)(2/z). n

(vii)

Yn(Z;a,b) = eb/2Z(z/b)l-a/2w

(viii)

yn(Z) = e I/z W

(ix)

8n(Z;a,b)

l-a/2, ~1 (a-l)+n

(b/z);

1 (2/z); O,n + ~-

= (-1)nn'b-nL (-2n-a+l)(bz)" n

(x)

Sn(Z ) = ( - 1 ) n n ! 2 - n a ( - 2 n - 1 ) ( 2 z ) n

REMARK.

In t h e s t a t e m e n t of t h e theorem a l l r e l a t i o n s

referring

to t h e case

a ~ b = 2 have been o m i t t e d , because t h e s e are immediately o b t a i n a b l e by s e t t i n g b = 2 in t h e formulae f o r y n ( z ; a , b ) and 0 n ( Z ; a , b ). The formulae f o r yn(Z) and On(Z ) a r e i n d i c a t e d , because o f t h e i r r a t h e r f r e q u e n t use.

39

8.

In the preceding sections we have discussed the relations of BP to other special

functions essentially from the point of view of identifying BP with these other functions,

for particular values of their parameters.

A different kind of connection is pointed out by the following remark that, apparently,

was never made before.

Let Pn(X) be the n - t h Legendre Pol>momial. case of Jacobi Polynomials, (7)

This i s , of course, a p a r t i c u l a r

but can also be defined by the generating function

(I-2zx+z2) -I/2 =

n ~ Pn(X)Z . n=0

If P(r)(x) stands for the r-th derivative of PnfX~,.. then the following theorem holds n n

THEOREM 2.

yn(Z) =

[ p~r)(1)zr; r=0 n

@n(Z) =

I P~r)(1)zn-r

r=0

The two statements are, of course, equivalent, immediate consequences LE~ZIA i.

on account of (2.10).

Both are

of

a(n) = p(r)(1). r

n

On account of (2.8), Lemma 1 is equivalent to LEMI~ 2.

p(r)(1) = (n+r)! n 2rr][n_r)!

This lemma appears to have been known to I. Schur, but this writer is unable to locate a reference in Schur's work. known, by N. du Plessis,

However, many essentially distinct proofs are

[48], E. Rainville

writer [ZS], among others.

[S0], I.N. Herstein

[51], and this

Here is the particular simple one due to du Plessis:

We differentiate

(7) r times with respect to x and obtain

(8)

zr.l.3 .... (2r-l) (l-2zx+z2) -r-I/2= [ p~r)(x)z n. n=0

By setting x = i and expanding the binomial, we obtain on the left of (8) zr.l.3...(2r-1)(1-z) -2r-I = z r.1.3...(2r-1)

= 1.3...(2r-i)

1.3... (2r-l)

[ (-2~-l)(-z)m m=O

[ (-l)m(-2~ -I) zm+r = m=0

~ m=0

(2r÷l) (2r+2) m!

"""

(2r+m)

z

m÷r

=

40 co

1.3... [2r-l)

[

(n+r) (n+r-l)...<jrn_r~: (2r+l)

zn.

n=r co

For x = i, the right hand side of (8) becomes

~ P(r)(1)zn and Lemma 2, hence n n=O

Lemma i and Theorem 2 follow b y c o m p a r i s o n o f t h e c o e f f i c i e n t s

o f zn i n

(8).

CHAPTER 6 GENERATING FUNCTIONS 1.

A v e r y l a r g e number o f g e n e r a t i n g

series,

others

represent

o b t a i n e d by more o r l e s s tions,

w h i l e some o t h e r s

i s made t o l i s t

here at

functions

f o r BP a r e known.

functions

within

classical

methods, others

still

certain

domains o f c o n v e r g e n c e ; are obtained

a r e o b t a i n e d by t h e t h e o r y

least

t h e more i m p o r t a n t ,

Some a r e f o r m a l

of Lie groups.

o r more u s e f u l

some a r e

by a d - h o c c o n s i d e r a An a t t e m p t

o f t h e known g e n e r a

ting function, but even for many of the listed ones it will be necessary to refer the reader to the original 2.

Let us consider the function f(t,z) = exp{[l-(l-2zt)i/2]/z}.

differentiation Ff '

literature for complete proofs.

with respect to t we obtain

(t,z) = ( l _ 2 z t ) - l / 2 ,

differentiation

By logarithmic

so that f'(t,z) = ( l _ 2 z t ) - l / 2 f ( t

,z), and by a further

f"(t,z) = {(l-2zt)-l+z(l-2zt)-3/2}f(t,z) = {a~l)(l_2zt)-l+a~l)z(l_2zt)-3/2}f(t,z),

with a~ I) = a~ I) = i, the two coefficients

of yl(z).

These formulae are the instances k = i and k = 2 of the PROPOSITION. il)

Fom k > i, f(k)(t,z) = f(z,t)

k-i 7. m=0

a(k-l) zmil-2zt) -(k+m)/2 m

where f(k) it,z) = __~kf and where the a (j) are the coefficients of y~ iz), as given by J ~t k m

i2.8). Proof.

We a l r e a d y

know t h a t

t h e p r o o f by i n d u c t i o n We differentiate fik+l)(t,z

(1) h o l d s f o r k = 1 and k = 2, and p r o c e e d t o c o m p l e t e

on k. (I) and obtain:

k-i ) =- ~ m= 0

a(k-1)zm(1-2zt)-(k+m)/2-(1-2zt)-l/2f(t,z) m

+ fit,z)

= fit,z)

k-i ~. a ( k ' l ) z m + l ( k + m ) ( 1 - 2 z t ) m=O m k ~ m=O

-(k+m+2)/2

(a ( k - l ) + (k+m- 1) a ( k - l ) z m i l - 2 z t ) - (k+m+l)/2 m m-I

By using (3.19), this can be written as

42

k

a(k) zm(l-2zt) - (k+m+ I)/2

f(k+l) (t, z) = f(t,z) m=O

m

and this is precisely (I), with k+l instead of k.

The Proposition is proved.

As immediate consequences we note the following corollaries. COROLLARY i.

f(k+l)(0,z) = yk(z).

COROLLARY 2.

(see [68]7.

f(t,z) =

Z tkyk_l (z)/k!, k=0

where (see (2.12)), for k = 0, y_l(Z) = y0(z) = i. It follows that the function f(t,z) is a generating function for the polynomials Yk_l(Z).

By differentiation with respect to t we obtain (see [17]):

(2)

~f ~t

- (l-2tz)-i/2f(t'z)

=

~ tkyk (z)/k' k=0

The ~nction ~f/3t is, therefore, a generating function for the yk(z). 3.

If we replace in (27, yk(z) by zkek(z-l), we obtain co

(l-2tz)-I/2exp{[l-(l-2tz)i/2]/z} or, replacing z by z

= kZ0= tkzk@k(z-l)/k!

,

-i.

co

(l_2tz-l)-i/2exp{z(l_(l_2tz-l)I/2 } =

~ (tz-l)k@k(Z)/k! k=0

.

We now denote tz -I by v and this leads to (3)

(l-2v)-i/2exp{z(l-(l-2v)I/2)}

= ~ vk@k(Z)/k!

.

k=O We may a l s o , in

following

(3) we o b t a i n

(see

Burchnall

[17],

set

l - ( l - 2 v ) I / 2 = 2u.

I f we s u b s t i t u t e

this

[17]) co

(4)

(l-2u)-le2ZU =

[ {2u(l-u)}kOk(Z)/k! . k=O Burchnall calls this a pseudo-generating function for @k(Z ). The corresponding pseudo-generating function for the generalized

BP ek(z;a,b)

is (see [17] and also [2]). co

(S)

(l-2u)-l(l-u)2-ae bzu =

~ {bu(l-u)}kOk(z;a,b)/k! k=O

.

By replacing here @k(z;a,b) by zkyk(z-l;a,b) and then z by z -I, we obtain co

(l-2u)-l(l-u)2-ae bu/z =

~ {buz-l(l-u)}kyk(z;a,b)/k!, k=0

or, with uz -I = v,

43 co

(l-2vz)-iCl-vz) 2-aebv =

Z {bv(l-vz)}kyk{z;a,b)/k! k=O

.

Finally, still following Burchnall [17], let 2v(l-vz) = t, so that l-2tz = {l-2vz) 2 and we obtain the following generating function for the polynomials yk(z;a,b) : (6)

(i-2tz) -1/2{

[i+ (l-2zt) 1/2]/2 }2-aexp{ (b/2z) (i- (l-2t z) i/2)} oo

= ~ (b/2) ktkyk (z; a,b)/k : . k=O A simple proof of (6) is also given in [43] and [2] and, for b = 2 in [16], see also [50]. 4.

Burchna11's generating function has been generalized by Rainville (see [89] and

[90]), as follows: Let ~n(X) =

q+2Fp(-n,c÷n,l-81-n . . . . . 1-gq-n; 1-al-n ,1-~2-n . . . . . 1-C~p-n;(-1)P÷q÷lx);

then On(X ) admits a generating function of the form (l-4xt) -1/2{1 [l÷(l-4xt) 1/2] }l-CpFq (C~l,C~2 (7)

. . . . .

C~p;81 , 82 . . . . . gq; [2t (1+ (1-4xt) 1/2) -1] )

[

~n(z)tn

(~l)n...(~p) n

n=O

n!

(61)n...(Bq)n

"

Burchnall's generating function (6) is essentially (7) with p = q = 0, c = a-l, x = zb -I, and t replaced by bt/2. A different type of generating functions for BP is obtained by Rainville [89] as follows: (8)

Set ~nCX) = p+2Fq(-n,c+n,~l,...,~p; 81 .... ,Sq;-X); then (l-t) -c p+2Fq(C/2,(c+l)/2,~l ..... ~p;B 1 ..... 8q; 4xt(l-t) -2)

= ~ On(X)(C)ntn/n~ • n=0 For p = q = 0, On(X) = 2F0(-n,c+n;-;-x) = Yn(bX,c+l,b), so that (8) is the generating function for the yn[Z;a,b), with z = bx, a = c+l, and the same b. 5.

W.A. AI-Salam [3] uses two generating functions for Jacobi Polynomials due to

Feldheim [16] and the relation between BP and Jacobi Polynomials (see Chapter S),

44

in order to obtain (in a slightly different notation and somewhat lesser generality) the generating function (9)

et(l-tz/b) I-~ =

[ tnyn(Z;~-n,b)/n~ n=0

,

and the formal (divergent) generating function (i0)

(l-t)-12F0(a-2,1;-;tz/b(l-t)) ~

On account of (5.4) for integral (i0')

a

Z tnyn(Z;a-n-l,b). n=0

(i0) may be written also as

(l-t)-ly2_a(Z/(l-t-l);a,b) -~ [ tnyn(Z;a-n-l,b ). n=O By using the connection with Laguerre Polynomials (see Chapter S) and known

generating functions of these polynomials, AI-Salam [3] also obtains

(l+tz/b)e-2exp{t/(l+tz/b)} =Z tn Yn (z;~-2n,b)/n!

(ii)

"

n=0 The characteristic feature of these generating functions is that they lead t o sums o f BP y n ( Z ; a , b ) , where the e n t r y a = a ( n ) , and i s not a constant. oo

Next

the relation L(a)(x+t) = et •

n

Z (-l)ktkL(~+k)(x)/k: leads (see (6.10) in n k=O

[~]) to (12)

etCl-tz/b)nyn(Z;a, c) =

Z tkyn(Z;a-k,b)/k' k=O

,

where Z = cz/(b-tz). One may observe that the parameter c in the first member is idle and can be replaced by 2• provided observe,

that

that

one then sets Z = 2z/(b-tz).

It also may be worthwhile to

in the second member the subscript of all BP is n, while the "a"-para-

meter varies, depending on k and the sum is over k. Ai-Salam gives also several• rather complicated bilinear generating functions• as well as the formal (no convergence) generating function L~

(13)

(n+a-2 n ) t n yn(Z;a,b] ~ (l_t)l-a2F0(a-12 , ~a ,. -,. - bzt - )

n=O

(l-t) 2

In case (l-a)/2 = m is an integer, this is equivalent (by (5.4)) to .n-2m-l. n [ n ) t yn(z;l-2m,b) ~

(13')

(l-t)2mym(-4zt(l-t)-2 ; ~3

n=O For a = 2, on the other hand, (13) reduces (see [3]) to (13")

I bzt ~ yn(Z;2,b)t n -~ (l-t)-12Fo(~ ;1;-; 2) , n=O (I-t)

essentially obtained already by Krall and Frink [68].

-2m,b).

"

45 From these formulae corresponding ones for en(Z;a,b) can be obtained by simple manipulations. A somewhat different generating function, involving Hermite polynomials is found in [3], (6.13). changing n and k.

It can be obtained by setting a = l-k in (13) and inter-

With some simplifications (also some misprints in [3] need correc

tion), this reads (_l)k(~)tkyk(Z;l_k,b) = e - ~ i n / 2 ( t z / b ) n / 2 H n ( ~

(~)i/2).

k=0 6.

Brafman [15] considers the Cauchy product of the series representations for

1 1 (t-(t2-4xt)i/2), obtains formally 2Fo(~,c-~;-; ~ (t+(t2-4xt) I/2) and 2Fo(~,c-~;-; a sum over 3F2's, transforms the latter by use of a formula of Whipple [65] and obtains in this way the formal result 2Fo(e,c-e;-; ~i (t_(t2_4xt]i/2))2F 0 (~,c-~;-; ~1

(t+(t2-4xt)i/2)).-"

( (~)n (c-~)n/n!)tn2F0 (-n,c+n; - ;x) . n=0 By observing

that 2F0(-n,c+n;-;x)

= yn(-bx;c+l,b) we obtain the formal generating

functions for BP: (14)

2F0(a,c-a;-; ~i

(t_(t2_4xt)i/2) )2F0 (~,c-~;-;

~1 (t+(t2-4xt)i/2)) -~

Z cntnyn(-bx,c+l,b), n=0 where en = Cn(=,c ) = (~)n(C-~)n/n:.

If ~ = -m, a negative integer, then also the

first member of (14) can be written as a product of BP, namely Ym(Zl,C+l,2)Ym(Z2,c+l,2), where z I =

-t+(t2-4xt)I/2,

We may observe

that in

z2 =

-t-(t2-4xt)I/2.

this case the second member reduces to a finite sum,

because now (S)n = (-m)(l-m)...(n-m-l) and vanishes for n > m, and the formal equality becomes an actual one (see [3] in somewhat different notations) and reads: (15)

ym(Zl, a, 2)Ym(Z2, a, 2)

=

m [ [.m-n+l. n )(a+m-n)ntnyn(-bx'a'b)' n=O

with previous values for z I and z 2. 7.

Carlitz [19] starts from Corollary 2, sets %(z) = znyn_l(z-l) = Zen_l(Z) and

observes

that Corollary

2 can be written as

46

(16)

exp{z(l-(l-2t)i/2) } =

Z fn (z)tn/n! n=O

,

or, if we set l-2t = (l-2u) 2, (17)

e 2uz =

Z 2nfn(Z)(U-u2)n/n! n=O

.

If we replace in (17), z by z -I, so that fn(Z -I) = znyn_l(Z ) and then set 2uz -I = x, (17) becomes

(see [68])

(18)

eZ =

Z (z-xz2/2)nyn_l(x)/n! n=O

•

In this surprizing formula, the right hand side is in fact independent of x and, for x = O, it reduces trivially to the left hand side. By using Brafman's formal equality

(14) in the case c = i, Dickinson

[47] obtains

a generating function for the modified Lommel polynomials. Here we have reached a point, where Jt is debatable, whether we are still dealing with generating functions in their original sense.

Formulae like (17), or

(18), may be considered more properly as belonging to the theory of representation of functions by series of BP, a topic that will be discussed in Chapter 9. 8.

This chapter would be incomplete without the mention of a most important general

method that permits us to obtain a large number of generating functions, most of the previous ones.

It is based on the theory of Lie groups.

among them

It is, unfor-

tunately, not possible to develop here the complete theory and the interested reader is advised to consult, e.g., the excellent presentation of the method in [73]. method has been introduced by L. Weisner

[62], [6~],

The

[64] and probably the clearest

presentation of the basic idea is still the one of [62].

On the other hand, the

only complete presentation knoml to the present author of all details needed for the non-specialist

is to be found in E.B. McBride

can be found in the work of S.K. Chatterjea Following Weisner

[73].

A somewhat different approach

[35], and M.K. Das [43].

[62], we consider a linear, ordinary differential

that depends also linearly on a (not necessarily

equation,

integral) parameter ~, say

d L(x, ~-~ ,~)v = 0,

(19)

with a ¢ A, say, where A is some denumerable set. If y is another independent variable, then for every B, with v = v (x), also u(x,y) = yBv (x) is a solution of (19).

Let now, in general, u = u(x,y) be a func-

tion of the two independent variables x,y that satisfies ~ u = au; here we have set Y ~y = y ~

, in analogy with the notation of Chapter 2.

L ( x , ~-~ d , e) i n ~, a n d b e c a u s e

By.ye = ~ y ~ , i t

follows

By the linearity of that

a function

u(x,y),

that

47

satisfies

6 u Y

= eu,

and

also

(20) is

Lu

of the

fairly

form u(x,y)

obvious:

n(x, ~-

=

= y~va(x),

i f v (x)

o f t h e s y s t e m ~yU = a u ,

is a solution

Lu = 0.

L e t u s now a s s u m e t h a t find

a solution

(ii)

expand g(x,y)

ttence,

furthermore,

tion

in a series

of the

the series

operator

so that

{ga(x)} to

(213,

solutions

of

(19).

then

This,

furnish

distinct

least

distinct

a L i e g r o u p and t h a t

a the degree, operators

and v e r i f y

for

orthogonal

that

raise,

been used by various

we c a n

well so that

c a n be j u s t i f i e d ,

of (193.

function

it

a termwise

follows

applica-

that

g(x,y)

way we h a v e o b t a i n e d ,

for a family

in the

of the

reduces

In t h i s

= 0,

fact

{ga(x)}

that

form u = g(x,y)

(21) t o a t r i v i a l

¢ A of

most usual

methods for

= g a ( x ) y a and sums o f identity.

If,

however,

functions

independent)

solutions

solutions

of

group of transformations is possible

g(x,y)

of

(20)

(20) c a n be f o u n d i s t h a t (it

to guess a set

always admits the groups

of differential

on h a n d o f t h e i r

commutation re!ations

these

commute w i t h t h e o p e r a t o r

generators

an a p p r o p r i a t e

function

~(x).

or of Bessel

n and a t t e m p t or lower n.

authors,

form, not of the type ~ ga(x)y a, a for the class {ga(x)} E A of solutions

functions.

polynomials,

or order

(19),

ga(x)y a.

(and l~nearly

Sometimes it

with ~(x)L,

classical

of

(20) c a n be f o u n d i n c l o s e d

when (20) a d m i t s a n o n - t r i v i a l

generate

of solutions

and

case in which non-trivial

order

is a solution

(20).

o f what p r e c e d e s

approach consists

generating

x + ¢, y + i n ) .

= yava(x)

is

form

of solutions

genuine generating

A particular

of the first

,6y)

of course,

of

Moreover,

of

sufficiently

lead to solutions

u = g(x,y)

(21) y i e l d s

(193.

of this

o f (20)

such solutions. solutions

{v ( x ) } i s a f a m i l y

~ sEA

converges

a generating

The d i f f i c u l t y the solution

then u(x,y)

The c o n v e r s e

d d ) = 0, w h e n c e L ( x , ~-~ , a ) ( g a ( x ) y a) = y ~ L ( x , ~-~ , a ) g a ( x )

is a family

according

=

(193.

if

(20);

of the

L = L(x, ~ ~

L ( x , ~-~, 6 y ) ( g a ( x ) y

of

of

of

(19),

of solutions

g(x,y)

If,

0,

=

of

independently

u = g(x,y)

(213

~y)u

with va(x ) a solution

then the se~ {yav (x)} is a family

(i)

,

In the particular

functions,

to use as generators

A variety

in order

of special

to obtain

their

of the

results

they L, o r a t

case of the

one may t a k e

methods,

operators

that

as parameter

Lie group the shortcuts,

(see,

e.g.

etc. [35],

have [43]).

48

In general, however, a rather careful study of the Lie algebra, its commutation relations, conjugacy classes of operators, etc., is needed (see, e.g. [62], [63], [g4],

[73]). As an e x a m p l e ( s e e the n-th

[73] and [ 4 3 ] ) ,

BP, and s e t F n ( X , t ) = e ( n + l / 2 ) t U n ( X ) .

confusion with Yn(X)). for F . n

l e t Un(X ) = x l / 2 e - 1 / X y n ( X ) ,

We o b s e r v e t h a t

(We w r i t e t r a t h e r

R = et(x ~

+ x

than y, to avoid

2 8 ~x) is a right

shift

operator

Indeed, R-Fn = e t { x ( n + l / 2 ) e ( n + l / 2 ) t U n + X 2 e ( n + l / 2 ) t U n

= e

w h e r e Yn(X) i s

}

(n+3/2)t ...... 1/2 - 1 / x 2.1 -1/2 -1/x -3/2 -1/x. 5/2 - 1 / x , . tLn+l/zjx.x e Yn+x t ~ x e +x e jyn+X e yn ~

= e(n+3/2)te-1/Xxl/2{((n+l)x+l)Yn+X2y~} by (3.16).

= e(n+l+l/2)txl/2e-1/Xyn+

1

= Fn+ 1,

It follows that n

(22) e~-RFm(x,t ) =

~

n

~

~ ~-F. RnF~ = ~ ~ n=0 - m n=0

F

m+n

= xl/2e -I/x

n

~ ~y. ~ e(m+n+i/2)tYn+m(X) n=0

On the other hand, e~BFm(X,t ) = Fm(e~Bx, eWBt). ~ Here _Rx = etCx ~- + x 2 ~x)X = etx 2,

8 + "x 2 8__} R2x = et{x ~8x (etx 2) = 3e2tx 3,

and, by an easy induction on n, Rnx = 1 . 3 . 5 . . . ( 2 n _ l ) e n t x Consequently,

emBx = n=0 [ 1 . 3 . . n! .(2n-1)

Y

Similarly, e~gt = t +

m

mn e n t x n + l = x ( 1 - 2 x m e t ) - 1 / 2 = Xl, s a y .

p'-'nt = t +

n=l 1 = t + [

n+l

7~

2 . 4 . . n! .(2n-2)

~n x n e n t

n=l ~

(2~xet)n t - 1 n = ~ log (1-2~xe t)

= tl,

n=l Consequently,

Fm(e~Bx , emgt) = F m ( X l , t l )

exp{(m+i/2)(t-i/2

=

log(l-2mxet))}{x(l-2mxet)-i/2}I/2. exp{-(1-2x~et)i/2x-l}ym(x(l-2x~et)-i/2 )

=

exp{(m+i/2)t}(l_2~xet)-(m+I/2)/2xl/2(l_2x~et)-1/4 • exp{-(l_2x~et)i/2x-l}ym(x(l_2xmet)-I/2)

say.

49

and (22) becomes, after some obvious simplifications, (l-2x~et)-(m+l)/2exp{[l_(l_2x~et)i/2]x-l}ym(x(l_2xmet)-i/2)

(23)

n

L n=O In particular,

nt

n:

Yn÷m(x).

if we set t = O, we obtain n

(l_2x~)-(m+l)/2exp{[l_(l_2x~)i/2]x-l}ym(x(l_2x~)-i/2) Finally,

=

[ m Yn+m(X) HT.' n=O

if we take m = 0 and write t for m, this becomes

(l-2xt)-I/2exp{[l-(l-2xt)I/2]x -I} =

tn [ ~ n=O

which is, of course, precisely the generating function

Yn (x),

(2) of Burchnall.

Instead of the right shift operator, we could have used, of course, the left shift operator L = e-t(-x ~-~ + x 2 ~-~..) 8x ' but we obtain exactly the same result. On the other hand, we may verify that

JR, L] = RL - LR = 0, so that these operators

commute and e K = e uB+vL = eUBe vL, with u,v any two independent variables.

By pro-

ceeding with K as we did before with R, we now obtain (see [43]) (l_2xu)-(m+l)/2(l_xv)m/2exp{ [l_(l_xu)i/2(l_xv)1/2] x-l}ym(x(l_2xu)-I/2(l_2xv)-l/2)

=

unv k [ ~ n,k=o " "

Ym+n-k (x)"

In the second member, for k > m+n, Ym+n_k(X) has to be interpreted Yk_m_n_l(X).

(see (2.12)) as

The formula simplifies somewhat for m = O.

The generating function

(2) had been obtained also by S.K. Chatterjea

very similar approach but after he first reduces equation

[35] by a

(2.11) to its Sturm-Liou-

rifle form d d-~ {p(x)u'} + l¢ (x)u = O. In a somewhat different notation and by following very closely the exposition in [73], Ming-Po Chen and Chia-Chin Feng [39] obtain the following generating functions: (l-xt)nebtyn(Z;a'b) with z = ~

x

=

~ k=l

(bt) k k! Yn (x;a-k'b)'

, which is essentially the same as (12); and

50 .n+k+a-2. k . x l k )t Yn(X;a+k,b), with z = l-t "

yn(Z;a,b) = (l_t) n+a-I k=O

For a = b = 2, in particular, the latter becomes:

Yn(X/(l_t))

=

(l_t)n-1 ~ ~.n+k. k )

tkyn(X;2+k;2)"

k=O Finally, with z - xt(t-~x)-l(l-t) -I, (t_~x)neba/t yn(Z;a,b)

(l_t)n+a-I =

9.

~ m=O

~ m+k-m+a-2. ~ k ) k=O

(~b) m tn+k-myn(X,a_m+k,b) m.--~--

"

In addition to the generating functions presented here, one finds either different

ones, or different proofs for those discussed here, in the following papers: [3], [109],

[46], [36], [76], [77], [78], [79], [16], [30], [31], [44].

Chapter 15.

See also

CHAPTER 7 FORMULAS OF RODRIGUES' TYPE i.

d In Chapter 2, by use of the differential operator 8 = z ~

, that is by following

essentially Burchnall's [17] method, we obtained the Rodrigues' formula (2.40), i.e.:

On(Z;a,b ) =(-1)'nb-nebzz 2n+a-1

(I)

dn (z-n-a+le-bZ). dz n

We mentioned also the important particular

cases of (I), corresponding to b = 2,

and to a = b = 2, which are @n(Z;a) =(-l)n2-ne2Zz 2n+a-I _ d_ n (z-n-a+le-2Z) dz n

(i') and

On(Z ) =(-l)n2-ne2Zz 2n+l

(I")

dn (z-n-le -2z) , dz n

respectively. It would appear reasonable to obtain corresponding formulae for yn(z;a,b) and its particular cases from (i), (i'), (I"), simply by writing 1/z for z and then multiplying by zn.

This is, in fact, possible, but the computation of

dn { d ~ (u-n-a+le-bU)}u=i/z S).

to which one is led is not entirely trivial (see Section

For that reason and also in order to present a variety of approaches we shall

start out differently and follow essentially [68]. 2.

We shall need some known facts, which we state as Propositions and, in order to

make the presentation selfcontained, we indicate short proofs of them. Let C be the unit circle [z I = I and let w(z) be a weight function.

With

respect to w{z) we define the k-th moment of the power zj by mjk = ~ 1

/C zk.zj .w(z)dz.

With these moments we form the determinants mOO

mOl

"""

mo'n-I

I

! An(W) = [[mJkl]O-I =

PROPOSITION i.

mlO

mll

mn-l,O

mn-l,l

"'"

..-

ml'n-i

I

I

mn_l,n_l]

If w(z) is such that An(W) ~ 0 for all n, then the conditions

i 2~---{ IC W(Z) zkun(Z) dz = 0 for k = 0,I ..... n-i define a polynomial Un(Z ) uniquely,

up to an arbitrary multiplicative constant.

52

n

Proof.

n

[ aj

L e t Un(Z ) =

l

[

a.z j with a

j=O

]

n

= 1; t h e n the i n t e g r a l

equals

n

~-~

f C zk+Jw(z)dz =

j =o

ajmjk and the conditions become

Z j =o

n-I

Z

j=0

ajmjk = -anmnk = -mnk (k = 0,I .... n-l).

The determinant of the coefficients

aj (j = 0,I ..... n-l) is An(W) and, by assumption, An(W ) # O. a.'s are uniquely determined. 3

If the coefficient a

n

It follows that the

is allowed to be arbitrary,

then all the aj's have their previous values multiplied by an and Proposition 1 is proved. , PROPOSITION 2.

Proof.

If w(z) = e -2/z

1 mjk = 2-~

" 2J+k+l then mjk = (-i) 3+k+l (j+k+l)!

" 1 fC zk+Je-2/Zdz = ~

(-2) r

1

r~

2~i

fC

zk+j

1 (r=O ~ ~'

~) ( r)dz

IC zk+j-rdz"

r=0 The inversion of the summation and integration is easily justified.

All inte-

grals vanish, except one, where k+j-r = -I, that is, r = k+j+l, and this equals 2~i; Proposition 2 is proved. PROPOSITION 3. Proof.

If w(z) = e -2/z, then An(W) # 0 for all n.

It is easy, although somewhat computational to determine the actual value

An(W ) = (_l)(3n-l)n/22 n

2 n-i [ ] ~=i

2n-i (9~)/ ~ - - (~!), 9=n

from which An(W ) # 0 follows trivially. COROLLARY l.The conditions

For a proof of this formula, see [25].

f C e-2/Zzkun(z)dz = 0 for k = 0,i ..... n-I define a poly-

nomial Un(Z ) of degree n uniquely, up to a multiplicative constant.

Proof is immediate on account of Propositions 1 and 3. PROPOSITION 4.

Un(Z ) = e 2/z - -dn dz n

(z2ne -2/z) is a polynomial of exact degree n, with

constant term 2n.

Proof.

d-~d (z2ne -2/z) = 2n z2n-le -2/z + 2z2n-2e -2/z = e-2/Z(2n z 2n'l + 2z2n-2) ",

53

this is the instance m = 1 of the general statement

where P2n_m(Z) is a polynomial of degree 2n-m. on m.

d m (z2ne -2/z) = e-2/ZP2 n m(Z) dz m ,

One completes the proof by induction

Next, by considering Leibniz's Rule for differentiation, we observe that the

lowest power of z is obtained from the term z 2n (2mz2(n-m)+...)e -2/z.

dm e -2/z = z2n{(2/z2)m+...}e -2/z = dz TM

For m = n, in particular, the lowest term is the constant 2n.

LE~MA l.The first n moments of Un(Z), with respect to the weight function w(z) = e -z/2, vanish. Proof.

dn - -dn (z2ne-2/z) = - dz n dz n

=

[

~~ (-i) v u=O

- -2~ z 2n = ~~ (-i) ~! ~ u!z u u=O

2~ - -dn dz n

z -~+2n

(-1) ~ ~2 ~ (-~+2n)(-~+2n-1)...(-~+n+l) z-~+n.

One observes, in particular, that the powers -u+n corresponding to ~ = 2n, = 2n-l,...,w = n+l do not occur, so that (see remark to that effect in [17]) dn

(z2ne -2/z) =

dz n

n 2u n-u ~ (-i) u ~., (2n-~)(2n-l-u)...(n+l-9)z ~=0

+ ~ 9=2n+i

2~

9--~ (~-2n)(~-2n+l)'''(~-n-l)z-~+n"

This is the sum of a polynomial of exact degree n and of an infinite series of decreasing powers, that starts with z -n-I k

It also follows that fk,n(Z) =

dn

z

(z2ne -2/z) is the sum of a polynomial of exact degree k+n and of a series of dz n

decreasing powers starting with zk-n-l.

Consequently, for k = 0,1,...,n-l, the

residues of fk,n(Z) vanish, so that I c e -2/z zkun(z)dz = IC e-2/Zzk{e2/Z

d n (z2ne-2/Z)}dz = dz n

I C zk - -d n (z2ne-2/Z)dz = I C fk,n(Z)dZ = 0, as claimed. dz n

On the other hand, we know from Corollary 4.4 that f C e-2/Zzkyn(z)dz = 0 for k = 0,1,...,n-l.

From Corollary 1 it now follows that Un(Z) can differ from yn(Z)

only by an arbitrary multiplicative factor.

54 THEOREM i.

(see

[68]).

The BP yn(Z) admits the Rodrig~aee' formula

2-ne2/z

yn(Z) =

(2") Proof.

dn dz n

(z2ne-2/z).

On account of what precedes, it only remains to determine the factor of

proportionality.

We know from Proposition 4, that the constant term of Un(Z ) is 2n.

Also (see Chapter 2) the constant term of yn(Z) is 1 and Theorem 1 is proved. A somewhat different proof of the Lemma may be given, following [68]. depends on the remark that all functions involved are uniform.

integrating around C, say, from +i to +i and use integration by parts. terms vanish, because they are singlevalued.

It

We proceed by All integrated

Specifically,

k-I dn-I IC zk _ _dn (z2ne-2/Z)dz = z k dn-I (z2ne-2/z.,+l dz n dzn-i )J+l -k f C z dzn- 1 (z2ne-2/Z)dz = ... =

(-l)rk(k-l)...(k-r~l) f C zk-r dzn_ dn-rr (z2ne-2/z) dzIn particular, for 0 < k = r < n, this becomes dn-r (-l)rr! fc dzn-r

dn'r-I +i (z2ne-2/Z)dz = (-l)rr' dzn-r-----~(z2ne-2/z)]+l = 0.

The rest of the proof of Theorem i may be kept unchanged. 3.

THEOREM 2.

The general BP YnCZ;a,b) P~s the Rodrigue8' formula b-nz2-aeb/Z

(2)

dn dz n

yn(Z;a,b) =

For a = b = 2, (2) reduces to (2").

(z2n+a-2e-b/z).

It also is clear that not much generality is

gained by allowing arbitrary values of b # 2.

One could well consider instead of

(2) simply (2')

yn(Z;a)

=

2"nz2-ae 2/z

dn (z2n+a-2e-2/z), dz n

from which (2) ilr~ediately follows by a change of variable.

However, the proof of

(2) is not more difficult than that of (2'), so that we proceed to prove Theorem 2 as stated. In the proof we shall need LE~

2.For k ~ n,

k Z (-l)S(~)(2n+a-2-s)(n)=n(k) (2n+a-k-2)(n-k)', for k > n, s=0

sum vanishes. Proof.

For integers n,c,k,

the

55

dn

dn

k

dx n

s=O

{xC(x-1) k} =

dx n

For x = 1, in p a r t i c u l a r ,

k

k

X (-1)S(s)xC+k-s =

t h e second member becomes

o t h e r hand, by L e i b n i z ' s r u l e ,

(-1) s (ks) (c+k-s) ( n ) x c * k - s - n .

s=O

the f i r s t

k s k (n) [ (-1) (s) (c+k-s) s=O

On t h e

n dn-r dr member equals r=O[ (;) ( d x - - ~r xC) ( J (x-1)k)"

The terms with r > k v a n i s h , because ( x - l ) k i s of lower d e g r e e than r ; f o r r < k the l a s t f a c t o r c o n t a i n s a p o s i t i v e power o f x-1 and v a n i s h e s at x = 1; c o n s e q u e n t l y , only the term w i t h r = k can be d i f f e r e n t k _< n

from z e r o .

It exists,

and then i t s v a l u e i s ( k ) c ( n - k ) k ' = n ( k ) c ( n - k ) .

integers n,c,k,

k [ (-1)S(s)1" ( c + k - s ) ( n ) s=O

k < n,

however, only i f

We have proved t h a t f o r an),

= nkkJc k n - k "J ~ " " and t h a t t h e sum

vanishes for k > n.

For c = 2n+a-k-2 this is precisely the Lemma.

Proof of Theorem 2.

We expand the right hand side: b-nz 2-a m~0~ bm = m!z TM

b-nz 2-a

[ ~b m m=O

Z -m

dn dz n

~[ (-I)s s=0

~bS z2n+a-2-s

~[ (-1)S ~-., b s ( 2 n + a - 2 - s ) ( n ) z n+a-2-s s=O

Set m÷s = k; then the double sum can be written as bkzn-k k=O

k

( - 1 ) s (~) (2n+a-2-s) (n).

s=O

the inner sum equals n (k)(2n+a-k-2)(n-k) for k f n, zero otherwise;

By Lemma 2,

hence, the sum becomes n

bk-nzn-k n(k) k! (2n+a-k-2) (n-k) =

k=O

n b-k k.n. (k) [ z [k) (n+k+a-2) k=O

and this is precisely yn{Z;a,b), as follows from (2.28). 4.

From the Rodrigues' formula (i)

for @n(Z;a,b) immediately follows the correspond

ing formula for Cn' namely: ~n(Z;a,b,c) = (_l)nb-ne(b-C)Zz 2n+a-I _ d_n (z-n-a+le-bZ), dz n whence, in particular #n(Z;a,b,b) = (_l)nb-nz 2n+a-I d n dz n

(z-n-a+le-bZ);

56

@n(Z;a,b) = C n ( Z , a , b , b / 2 ) •

=

Cn(Z;a) = Cn(Z;a,2) = (-l)n2-neZz 2n+a-I Cn(Z) = On(Z;2) = (-l)n2-neZz 2n+l

S.

dn ( z - n - a + l e - b Z ) ; dz n

( _ l ) n b - n e z b / 2 z 2n+a-1

dn dz n

dn dz n

(z-n-a+le-2Z) • and

(z-n-le-2Z ) .

In Section 1 it was suggested that a natural way to obtain Theorem 1 and 2 would

be to use (i), already known from Chapter 2, and the relation yn(Z) = zn@n(z-l). In the present section we shall sketch such a proof and similar ones.

We shall show

that they require certain combinatorial identities and they may appear less trivial

t h a n one may have e x p e c t e d . We a l r e a d y know (see ( 6 . 2 ) )

that

(1-2zt)-l/2exp{[1-(1-2zt)

i/2]/z) =

Z Yn(Z) t n / n I n=O

dn

so t h a t yn(Z) = - - ( ( l - 2 z t ) - l / 2 e x p { [ 1 - ( 1 - 2 z t ) l / 2 ] / z } ) t = O dt n

,

.

As z is not involved in the differentiation, (3)

dn = n!c n, e-2/Zyn(Z ) = _dt_ n ((l-2zt)-i/2exp{_[l+(l-2zt)i/2]/z)t= 0

where cn is the coefficient of tn in the expansion of the large bracket. v = [l-2zt) I/2.

Then

1 e-(l+v)/z (-i) k

- k=O k!z k

Let

i

Z (_l)k (l+v) k

= ~ k=0

k!z k

(r k)(l-2tz)(r-l)/2 =

Z k=O

r=O

i

( )vr =

[ zk k=0 k:

= ~ (-i) Zkk k!

(k) r=O

r=0

7. (_l)m m=O

r-I

= ~ (_l)mtm.2mz m k~O (-l)k m=0 = k'z k

~ (r k) r=0

It follows that the coefficient c of tn is n C = [-l)n2nz n ~ (-l)k ~ (kr) n k=0 k!z k r=0 Yn(Z) = (-l)ne2/Zn:2nzn (-l)kkZ0= ~ k!z n

(7)

(?)

=

~ Cm tm m=0 "

, or, by (3),

r=0~ (kr) (r0

By (2") this implies the validity of the following identity:

i

2mzmt m

57

(4)

(-l)n2nzn k~0= k!z k

dn However, - -

(z2ne-2/z)

dz n

2k

[ (-I)k ~ k=0

~ ~

=

(?)

r=0

=

2k ( - 1 ) k k!

dn

k=O

2n-k

dz n

n

(2n-k)(n)zn-k =

z

n.' dz n (z2ne-2/z)"

2k

~ (-i) k ~ k=0

(2n-k)

(n) zn-k

+

~

2k

~ ~ k=2n+l

(k-Zn)(n)Z

-k+n

,

because the products (2n-k)(2n-k-l)...(n-k+l) with n < k ~ 2n all vanish. By equating the coefficients of equal powers of z in the two members of (4), it follows that (4) holds if, and only if

(a)

f o r k ~ n, (-1)n22nn~

(b)

for n < k < 2n,

k~

(~)

= 2k ( 2 n - k ) ( 2 n - k - 1 ) . . . ( n - k + l ) ;

~ (~) r=0

>_

= 0;

~ ( ) r=0

= 2k(k-2n)(k-2n+l)...(k-n-1).

These three conditions are jointly equivalent to

(S)

~

(k)

_k-2n .k-n-l. =

Z

(

n

)

"

r=O This formula appears to be new.

The closest result available in the literature

seems to be Carlitz' formula (see [ZO])

k

(5')

r=0

k

(r) (

i

r/2 n

)

k .k-n-l.2k-2n (n ~ O) = ~ [ n-I ) = 2k

(n = 0)

In fact (S) and (5') can be obtained from each other, but not trivially

[22].

Both are particular cases of sums of the form k (+ l)r(~)((k+~ )/2) (a = i or 2, s = rational integer) r=0 studied by H.W. Gould [21]. We have proved the purely combinatorial identity (5), valid for all integers k, by using Rodrigues' formula (2"). Conversely, however, if one knows that (5) holds (and (S) can be proved also directly, although not trivially - see [22]), then one can trace all steps backwards and one obtains the Rodrigues consequence of the generating function (6.2) and (5).

formula (2") as a

This is perhaps a rather

58 artificial procedure and for this reason we abstain from proving (2), by starting similarly from the generating function (6.6). Finally, let us see how one can obtain (2") from (i"). @n(Z) = znyn(i/z) = zn.2-ne2Z{dnud--n(u2ne-2/U)}u=I/z.

By (2"),

If we also replace On(Z)

with the help of (I") and suppress common factors, we obtain the identity dn dn (z-n-le-2Z) { d ~ (u2ne-2/U)}u=I/z = (-l)nzn+l dz n "

(6)

Clearly, by going backwards, (2") is a consequence of (I") and (6), if the latter can be established directly.

This we now proceed to do.

The left hand

member can be computed by (4) to be equal to

{

~( kk!u (-1)k r k) r=O ~~ (-1)n22nn:un k=O

}u=l/z (-1)n22nn' k=O ~ (-1~ zk-n r=O ~ (k)C~2 ~n //

Assuming once more that (5) holds (this was already needed in order to establish (4) directly), (6) becomes dn dz n (z-n-le -2z) = 22nn!

(7)

co (-1)k k-2n-12k-2n(k-n-l) ~ ~ z k=O

The left hand side equals dn {z -n-I dz n

=

~ (kl~)k 2k dn k=0 dz n

zk-n-i

[(-~I k k=0

=

2kz k}

2k ~ (-l)k ~. k=0

(k-n-l)(n)z k-2n-I

,2k The coefficient of zk-2n-l on the right side of (7) equals (-l)k nk, so that (6) is indeed an identity.

(k-n-l) (n) n:

As already pointed out, (6) and (i") imply (2").

This method, however, requires the use of the combinatorial identity (5) and is hardly easier than the direct proof. It is clear how one can prove the more general result (2), by starting from (i) instead of (i") and making use of (5), but we abstain from reproducing it here.

CHAPTER 8 THE BP ~ND CONTINUED FRACTIONS I.

Let [a0,a I .... ,an .... ] stands for the continued fraction 1 a 0 + al + ... +

1

m

a n+...

of partial quotients an and denote the n-th convergent (pn,qn) = I.

[a0,a I .... ,an] by pn/qn , where

Consider, in particular, the expansion into a continued fraction of the

function e 2/z + 1 -~7~/z - 1 e

(i)

= [z, 3z, Sz ..... (2n+l)z .... ],

known from the work of Lambert [4Z], (see also [~$]).

See [83] for a recent proof.

If we denote the numerators and denominators of its successive convergents by pn(Z) and qn(Z), respectively, we find, e.g., P0 (z) z Pl (z) 1 3z2+i q0(z-----y= z = y ; ql(z-----y= z + 3--{ = 3z

, etc.

It is well known (see, e.g. [47]) that pn(Z) and qn(Z) can both be obtained, for n ~ 2, from the same recurrence relation (2)

Xn = anXn_ 1 + Xn_ 2,

where one has to start with the initial values p0(z) = z and pl(z) = 3z2+I for pn(Z), and with q0(z) = i, ql(z) = 3z for qn(Z). follows from [I).

In either case an = (2n+l)z, as

One obtains, in particular, for n = 2:

p2(z) = pl(z)a2+P0(z ) = (3z2+l)(bz)+z = iSz3+6z, q2(z) = ql(z)a2+q0(z ) = (3z)(bz)+l = iSz2+l. One may observe that (2) actually holds also for n = i, provided that one defines p_l(Z) = I, q_l[Z) = 0. [i)

One is led to make the following remarks:

All powers of z that occur in qn(Z) are of the same parity, which is that

of n, and all powers that occur in pn(Z) are of the parity opposite to that of n. This is verified directly, on above examples, for n = 0,1,2, and holds in general by induction on n, on account of (2).

60

(ii)

p0(z) + q0Cz) = z+l = YlCZ); PlCZ) + qlCz) = 3z2+l+3z = Y2Cz), and also

p2(z)+q2(z ) = Y3Cz). n = 0,1,2.

The identity pn(Z) + qnCz) = Yn+l(Z) holds, therefore, for

It appears reasonable to make a change of subscripts and set Pn+l(Z) =

pn(Z), Qn+l(Z) = qnCz), and to conjecture THEOREM I.

PnCZ) + Qn(Z) = yn(Z).

This statement holds, as seen, for n = 1,2,3, and, with previous convention, p_l(Z) = P0(z) = I, q_iCz) = Q0(z) = 0, n = 0.

it is verified to hold also for

The proof of Theorem I for all n is completed by induction on n as follows.

Identity (2), written out explicitly for PnCZ) and Qn(Z) reads: Pn+ICZ) = (2n+l)z PnCZ) + Pn_l(Z)

(3) %+iCz)

= (2n+l)z QnCZ) + Qn_ICZ).

If we assume that Theorem 1 holds for all subscripts up to n, then (3) yields Pn÷l(Z)+Qn+l(Z ) = (2n+l)Z(Pn(Z)+Qn(Z)) + (Pn_l(Z)+Qn_l(Z)) = (2n+l)z yn(Z) + Yn_l(Z), by the induction assumption.

But, by (3.7), the right hand side equals Yn+l(Z), so

that Theorem 1 holds also for n+l, as claimed. In this decomposition of yn(Z), Pn(Z) is the polynomial containing all the terms of yn(Z) with powers of z of the same parity as n, while Qn(Z) contains the terms with powers of z of parity opposite to that of n.

In other words,

1 1 Pn(Z) = ~- {yn(Z)+(-l)nyn(-Z)}, Qn(Z) = ~- {yn(Z)-(-l)nyn(-Z)}.

With this we have

proved THEOREM 2.

The

n-th

e 2/z + 1

convergent of the continued fraction [z,3z ..... (2n+l)z .... ]

Yn(Z)+(-l)nyn (-z) i8

of

e 2/z - I

2.

In what follows we shall need the following

LEMMA i.

yn(Z)-(-l)nyn(-Z)

Pn+l(Z)%(z)-Pn(Z)Qn+l(Z)

= (-i)n+l.

The lemma would follow immediately, if we knew that Pn(Z) and Qn(Z) are coprime. This is indeed the case, both, in the algebraic sense (Pn(Z) and Qn(Z) do not possess any non-trivial polynomial divisor) and in the arithmetic sense (for integer m, the

61

integers Pn(m) and qn(m) are coprime), as stated, e.g. in [II], p. 83, but it is more convenient to prove the Lemma directly, without the need to invoke much classi cal theory of continued fractions;

the coprimality will then follow as an easy

corollary. We verify that Pl(Z)Qo(Z)-Po(Z)Ql(Z ) = poq_l-P_lq 0 = 0-i 2 = -I,

p2(Z)ql(Z)-pl(Z)Q2(z)

= (3z2+1) -1-z(3z) = I,

P3(z)Q2(z)-P2(z)Q3(z)

= (iSz3+6z)(3z)-(3z2+l)(ISz2+l)

so that the Lemma holds for n = 0,1,2. verified for all subscripts up to n.

= -i,

Let us assume that it has already been One then observes that, by using (3) and the

induction hypothesis one obtains: Pn+l(Z)%(z)-Pn(Z)%+l(Z)

: [(2n+l)ZPn(Z ) + Pn_l(Z)]%(z)

- Pn(Z)[(2n+l)z%(z)+Qn_l(Z)] =_

= Pn_l(Z)%(z)-Pn(Z)%_1(z)

[Pn(Z)Qn_l(Z)-Pn_l(Z)Qn(z)]

= (_i)n+l

and the Lemma is proved. COROLLARY i. Proof.

For all integers n and k, (Pn(k), Qn(k)) = I.

If d = (Pn(k)), Qn(k)), then d divides the left hand side of the identity

of Lemma I; hence d[(-l) n+l, so that d = i, as claimed. COROLLARY 2. Pn (z)

Pn+l (z)

(_I) n

O,n(Z)Qn÷l(Z) #

0.

%cz) holds f o r a l l z E £,

such that

Proof follows trivially from the Lemma. We recall

(see [47]) that any real number falls between any two consecutive of

its convergents. COROLLARY 3.

More precisely, we have

I f n is even and z = x is real and positive,then

Pn+l (z)

e2/X+l

%÷iCz) if n is odd, the inequalities are reversed.

Pn (x)

62 Proof follows from the preceding remark and Corollary 2, by observing that e2/X+l is the n-th convergent o f e2/X_l

Pn(X)/Qn(X) COROLLARY 4.

and that Qn(X)Qn+l(X)

> 0 for x > 0

Regardless of the parity of n ar~ for all real x such that

Qn(X)Qn+l(X) ~ O,

0 < Proof.

I

Pn(X) On(X )

e2/X+l I < e2/X_l

The condition Qn(X)Qn+l(X)

i

Qn(X)Qn+l(X) •

~ 0 implies x ~ 0.

The first inequality then

follows from the fact that Pn(X)/Qn(X) is a convergent with following partial quotient different from zero (in fact, by (i) a non-vanishing multiple of x); while the second inequality follows from Corollaries 2 and 3. e 2/x +i 2 We now observe that - -i = - e 2/x -i e 2/x -i Pn(X) %(x)

yn (x) + (-l)nyn(-X) -I =

and

(-l)nyn (-x) -i =

Yn (x) - (- I )ny n (-x)

, so that by Corollary 4, Qn (x)

n

2

(-1) yn (-x)

I J- z ; - 1

%cx)

I<

12 Q n ( X ) - ( - 1 ) n y n ( - X ) ( e 2 / X - 1 ) [

l Qn(X)Qn+l(X)

, or, equivalently,

< (e2/X-1)/qn+l(X).

I f we r e p l a c e here x by 1/x and t h e n m u l t i p l y by x n we o b t a i n (4)

]2xnQn(1/x)-(-l)nxnyn(-1/x)(e2X-1)l

In t h e f i r s t

member, xnQn(1/x) = ~ ( x ) ,

@n(X), w h i l e ( - 1 ) n x n y n ( - 1 / x ) x

+l(X),

< (e2X-1)x2n/xnQn+ l ( I / x ) .

say, c o l l e c t s

= (-x)nyn(1/(-x))

t h e terms with odd powers o f

= @n(-X) and xnQn+l(1/x) =

a p o l y n o m i a l with o n l y even powers of x.

The f i r s t

member o f (4) may

now be written as 12 ~(x)-@n(-X )(e2x-l)l = 12Qn(x)+@n(-X)-@n(-X)e2Xl. 2~(x)

= @n (X) -@n (-X ) so that 2Q(x)+@n(-X ) = @n(X).

xn%+l (l/x)

x

2n

" [ m=l

=

Next,by a previous remark,

[n~2] . x2 n (e2X-l)/xnQn+l(i/x) j=0 c.x ) 2], so that

(2x) TM [n~2] cjx2 j m' / . j=0

Here c o

_

(2n+2) ' " 2n+l ( n + l ) '

But

=

as f o l l o w s from Chapter 2.

63

This value is here without further relevance, than the smallest zero of the denominator, can be written as a series x 2n+l

Consequently,

except for c O ~ 0.

If Ixl is less

it follows that the right hand side of (4)

[ km xm and has a zero of order 2n+l. m=0

~ also the entire function Rn(X) = @n (x) -@n (-x) e 2x, which, except

perhaps for sign, is the left hand side of (4), has a zero of order at least 2n+l at x = 0.

Let us replace here x by x/2 and set Rn(X/2) = Rn(X).

The result obtained

so far may be written as (6)

On(X/2)-@n(-X/2)eX

Finally,

let

us c h a n g e n o t a t i o n s

= R (x) = x 2n+l ~ c!x J. n j=O J

and s e t An(x) = @n(X/2), Bn(X) = - @ n ( - X / 2 ) ,

Then (6) r e a d s : Rn(x) = A n(x) + Bn(x) e This is

x

2n+l

~ c!x ~. j=0 j

precisely the expression encountered in Problem 1 of the Introduction and

shows that indeed, except for normalization, defined, are BP.

the polynomials A n(x) and Bn(X) there

In fact, a count of the number of constants involved shows that,

except for an arbitrary multiplicative -@n(-X/2)

= x

constant, An(X) = @n(X/2) and Bn(X) =

are the only polynomials of a degree not in excess of n, and for which

the linear form Rn(X ) has a zero of order ~ 2n+l at the origin.

CHAPTER 9 EXPANSIONS OF FUNCTIONS IN SERIES OF BP i.

The topic of expansions of arbitrary functions either of a complex, or of a real

variable in series of BP has not been explored fully and many open problems remain to be answered.

To date the most comprehensive treatment is that of R.P. Boas, Jr.

and R.C. Buck [13], but

the literature available on this subject cannot be compared

to that on expansion in, say, Fourier series, or in series of classical orthonormal polynomials,

etc.

In view of the orthogonality of the BP on the unit circle, as

presented in Chapter 4, it is reasonable to expect that the natural problem to investigate

is that of the expansion of functions f(z), defined for a complex

variable z, and continuous,

or at least integrable on a set containing the unit circle,

or some compact neighborhood of the origin. On the other hand, one may ask just for some formal expansions, f[z) ~

~ CnYn[Z;a,b), n=0

or f(z) N

~ n=0

Cn@n(Z;a,b),

say

and then inquire in what sense

the series "represent" f(z), under what conditions they converge and, if they do, when do they converge to f(z).

Even if the series do not converge,

it may still be

possible to find summation methods, under which the series are summable to f(z). Finally, one may raise questions concerning the uniqueness of such expansions. The question of formal expansions into series of the BP yn(Z;a,b) has been asked and answered already by Krall and Frink [68]. relations of Chapter 4, it follows that,if f(z) =

I f(Z)yn(Z;a,b)p(z)dz r

By using the orthogonality

I CmYn(Z;a,b), m=0

then

= f ( [ CmYm(Z;a,b))Yn(Z;a,b)~(z)dz, r m=O

where P is any simply closed curve that encircles the origin, and p(z) = (2~i)-1e -2/z if a = b = 2; otherwise If

r

O(z) = O(z;a,b), as defined in Chapter 4.

belongs to a connected domain on which the convergence of the series

CmYm(Z;a,b) i s s u f f i c i e n t l y

strong to justify

the inversion of the order of

m=O

summation and i n t e g r a t i o n ,

the last

integral

is equal to

~ Cm g Ym(Z;a'b)Yn ( z ; a ' b ) ° ( z ) d z m=0 r by Corollary 4.6.

~ (-1)n÷lb.n! r(a) = Cn ( 2 n + a - 1 ) r ( n + a - 1 )

'

In the particular case a = b = 2, this simplifies to (see [68])

i 2~i

f f(Z)Yn(Z) e-2/Zdz = (-l)n+l 2 v 2n+l

Cn'

65

or

(i)

Cn

If f(z) =

= (_l)n+l

~ bmzm, b m = f(m)/0)/m! m=0

n 2~i + 1/2

f F

f(Z)Yn (z)e-2/zdz"

, then (i) becomes

(see [81])

co

Cn : (_l)n+l(n+ i)

~ m=0

f(m)(0) {l__!__ f F zmyn(Z)e-2/Zdz}. m! 2~i

If we replace here the curly bracket by its value from Corollary 4.4, we obtain

"

f(m)(o)

Cn = (-l)n+l(n+ 9 ) m~0= m~

fCm) (0) (-l)n(2n+l)

[ m=0

m!

(-2)m+l (m+n+l)!(m-n)!

(-2) TM

(m+n+l)!(m-n)'

In the sum all terms with m < n vanish; in the others we set m = n + v, ~ = 0,I,... and obtain Nasif's

[81] formula: cn = (2n+l).2 n

~ ~ v=0

This, of course, is all formal work. the series

~ CnYn[Z;a,b) n=0

converges,

f (n+v) (0) v!

(-2) 9 (2n+v+l) '

In general, there is no guarantee that

or even if it converges, that its sum is f(z).

The determination of necessary and sufficient conditions on f(z), for the convergence, or the summability to f(z) of the series appears to be an open problem. 2.

Similar considerations hold if we consider expansions of functions f(z) into

series

~ n=0

cn en[Z;a,b).

It is clear, however,

from the work of Burchnall

(see

Corollary 4.15) that the en(Z;a,b) themselves do not form a system of polynomials orthogonal on the unit circle, because the factor z-(m+n+a)e -bz is not independent of the polynomials

em and en involved.

A somewhat different approach, however, permits

one to obtain expansions of entire functions in series of the polynomials It turns out that we obtain particularly simple results, sionally, pn(Z)

(=pn(Z;a))

zed Appell Polynomials,

=

2n ~., en(z;a,2 ).

en(Z;a,b).

if we set, at least provi-

In this case, the theory of generali-

due to Boas and Buck [13], [5], permits

one to obtain not

only the coefficients c n of the formal expansions, but also to determine the number

66

of distinct expansions (the expansions, unfortunately,are in general

not unique),

their respective regions of convergence and the regions of sun~nability for these series of generalized Appell Polynomials.

The main results and sketches of the

proofs will be given here, but for detailed proofs it is suggested that the reader consult the original presentations [5] and [13].

3.

The fundamental ideas may be traced back, on the one hand to Whittaker [66], on

the other hand to the classical work of E. Borel, G. P61va, • Phragmen z and Lindelof (see [32], vol.2 Let A(w) =

and [4]). [ anwn n=O

a 0 ~ O; and g(w) =

[ gn wn n=l

holomorphic in some neighborhood of the origin.

gl ~ 0

be given functions

Let ~ be the largest simply connec-

ted region containing the origin, in which A(w) is holomorphic, and let ~w ~ ~ the largest simply connected subregion of ~, in which ~

= g(w) is univalent.

also Aw as the largest disk centered at the origin and such that h w ~

~w~.

be Define Due

to the univalence of g(w) we have a i:i correspondence between the points w e ~w and the images ~ = g(w).

In particular, let ~

of Aw, so that A ~ c _ ~

Iz] = r.

~ n=O

be the image

in the ~-plane.

To each entire function f(z) =

transform F(w) =

be the image of fi and A w

n!fn n+l w

[ f zn we associate its Borel (or Laplace) n=O n

(see [iO], p. i13), holomorphic outside some circle

We recall the inversion formula: i eZW F I 2~---~ IC (w)dw = ~

(2)

z m w TM

1 2~i

IC (mE =0

~

n !f

__~.t )(n !

w~l

0

IC

~

)dw =

oo

=

[

n=O

f zn = f ( z ) ,

n

where C is a circle of radius r + e.

eZW

~ n!fn [ --~ n=O W

dw =

m

[ z 1 IC n-m+l dw I n !fn • 2-~ m=O ~., n=O w

67 ~-plane

w-plane

W

~

( Figure 1

We shall be concerned mainly with entire functions of exponential type, i.e., functions f(z), such that If(z) I ~ Me alzl all complex z.

for finite, positive constants M, ~ and

If T = inf a for which this inequality holds, f(z) is said to be of

exponential type T.

By D(f) we denote the complement of the set of regularity of

the Borel transform F(w) of f(z). We now define generalized Appell Polynomials pn(Z) by the generating function (S)

A(w)eZg(w) =

[ pn(Z)W n. n=O Two remarks are in order. The first is that the theory has been developed by Boas and Buck [13] in a more general framework, in which the exponential function et is Ont n , s u b j e c t o n l y t o t h e r e s t r i c -

r e p l a c e d by any " c o m p a r i s o n f u n c t i o n " ~ ( t ) = n=O t i o n s t h a t none o f t h e c o e f f i c i e n t s

~n v a n i s h and t h a t t h e r a t i o

~n+l/~n_ d e c r e a s e s

68

monotonically to zero.

Clearly, the exponential function qualifies as "comparison

function" with ~n = i/n~ # 0 and ~n+I/~n = i/(n+l) ÷ 0.

The condition on ~n+i/~n

implies, of course, that ~(t) (here et) is an entire function. The second remark is that the case g(w) = w corresponds to the classical Appell polynomials.

Indeed, (3) then becomes

(3')

A(w)eZW =

[ pn(Z)W n n=0

and we obtain by differentiation with respect to z: A(w)e T M =

A(w) wezw =

[ p~(z)w n, so that n=0

~ p~(z)w n-l, or, equivalently, n=l

(4)

A(w)e T M =

[ p~+l(Z)W n n=0

and a comparison with (3') shows that the polynomials pn(Z) satisfy the differential equation Pn+l(Z) = pn(Z) that characterizes Appell polynomials. We now return to the general case.

By use of the univalent map ~ = g(w), we

can reduce the general case to a more manageable one, close to the classical (4). Indeed, let w = W(~) be the map inverse to To any compact set C ~ A ~

~ = g(w) and set A(w) = A(W(~)) = B(~).

corresponds a class of entire functions f(z), such that

D(f) C C; let us denote that class by K[C].

Let F be a simply closed curve in A ,

enclosing C and passing through no zeros of B(~).

Now (3) becomes:

eo

(5)

B(~)e~Z =

~ pn(z)W(~) n, n=0

co

and it follows that on r,

~ pn(Z)W(~)n/B(~) converges uniformly to e ~z. n=O

We now set (6)

Cn = in(f ) =

i

W(~) n

F(~)d~

and obtain, using (2), that

n--0 CnPn(Z) = ~

1

F(~) PnW(~)n)d~ IF ~ (n 0

= ~1

IF F(~)eZ~d~

=

f(z).

The interchange of summation and integration is justified for f c K[C], by the oo

uniform convergence of

[ n=0

pn(z)W(~) n.

In order to have f e K[C] we need to know that D(f) c

C~A~.

In particular, if

r I is the radius of DI, the largest disk, centered at the origin and inside A~, a

69

sufficient condition for D ( f ) ~

D1 ~A

is that T < r I.

largest disk centered at the origin, such that D 2 ~

We consider also D2, the

; let r 2 be the radius of D 2.

The following theorem holds and most of its assertions follow readily from the preceding discussion.

THEOREM i.

Let A(w) =

n [ an wn, a 0 # 0 and g(w) = [ gn w ' gl ~ 0 be holomorphic n=0 n=l

in some neighborhood of the origin, with g(w) univalent on nw; let W(~) be the function inverse to g(w), defined on ~

, the image of ~w' set A(W(~)) = B(~)

and

define the polynomials pn(Z) by (5).

Let £(z) =

~ f zn be an entire function of exponential type with Boreln=O n

Laplace transform F(~), and define the constants cn by (6). sented by the series f(z) N

~ CnPn(Z). n=0

Then f(z)

i8 repre-

Furthermore, if D I, r I and D 2, r 2 are

defined as above and if the type • of f(z) satisfies r < r I, then f E K[C] for some and the series converges to f(z).

C~D[f)

If T < r2, then the series is Mittag-

Leffler svz~nable to f(z). RE~RK

i.

These conditions insure convergence, or summability for all entire func-

tions of exponential type T, bounded by rl, or by r 2, respectively.

However, what

one needs is only the existence of simply closed curves F, that encircle D(f). Hence, for specific functions, for which D(f) is explicitly known, one can often establish convergence, or summability, when the sufficient conditions T < r I, or T < r 2 do not hold. RE~RK

2.

We abstain here from a proof of the Mittag-Leffler summability in the case

r I < r < r2, for which the reader may want to consult [13]. We observe that F(C)

has an essential singularity at the origin (unless f(z) is

a polynomial, in which case T = 0 and F(C) has a pole at the origin), so that the cn do not all vanish.

B(~) may, or may not have zeros.

If it has no zeros, then

a~y F selected as before, will lead to the same values for cn = in (f) and the series obtained coincides with Whittaker's basic series [66].

If, however, B(~) has zeros

in A , outside D(f), then the values of the c n will, in general, depend on the set of zeros of B[~) enclosed by

F, so that the expansion of f(z) need not be unique.

Further different expansions may be obtained, if one selects F outside ~

(so that

70

g(w) = ~ is no longer univalent),

but, fortunately,

these complexities need not

worry us in the present case. 4.

We start from Burchnall's generating function

(6.6).

As usually we shall set

b = 2 (otherwise, we make the change of variables z'/2 = z/b) and, writing also 2z for z, obtain i/2]/2z} {~1 [1+(1_4zt)1/2]}2-a(l-4zt)-l/2exp{[1-(l-4zt)

=

Z

Yn(2Z;a)-- tn n!

n=0 The polynomials Yn(2Z;a)(=Yn(2Z;a,2)) zed) Appell polynomials,

as here defined, do not appear to be (generali

according to the definition

(3).

Let us replace, however,

t by w/z; we obtain

wl [l+(l_4w)i/2}Z-a(l_4w)-i/4exp{[l_(l_4w)i/2]/2z} = Z

i n n=O z

{~

2n

Z n! n=O

1 (2z) n

~ 2n Yn (2z'a)wn = Z ~., ' n=O

Yn (2z;a) n!

@n((2z)-I a)w n

wn

,

or, replacing i/2z by z, 1 [l+(l_4w)i/2}2-a(l_4w)-i/4exp{z[l_(l_4w)i/2]}

This is precisely n 2 and pn(Z) = ~.,

(3), with A(w) = 1

=

2n

n

Z ~., On(Z;a)w • n=O

[1+(l_4w)i/212-a(1_4w)-1/4, g(w)

On(Z;a).

In order to insure that A(w) and g(w) are holomorphic, the singlevaluedness

of (l-4w) 1/2.

it is sufficient to insure

A convenient way to obtain this is to cut the

complex w-plane along the positive real axis from + 1/4 to ~.

that g(w) the d i s k

= l-(l-4w) I/2,

One easily verifies

is also univalent in the cut plane; hence, the cut plane is flw and A w is

lwl <

1/4.

The image ~

of ~w under ¢ = g(w) = l-(l-4w) I/2 is found, by observing that the

image of the boundary [1 < w < ~ in the c-plane.

i(4w-l) I/2 = 1 ; iv (0 < v < =)

It follows that fl~ is the half-plane Re ~ < i.

w = W(~), inverse to g(w) 1 w = $ ~(2-¢).

of flw becomes 1 %

is obtained from (l-4w) I/2 = i-~

The image of Aw, i.e., of the disk lw] <

of the inside of the lemniscate left loop of it.

1 T

The function

l-4w = I-2~+~ 2 is, therefore, that part

1c(2-c) I < 1, that is contained in ~ ,

i.e., the

The lemniscate is symmetric with respect to the x-axis, which it

meets in the points ~2-2~± 1 = 0.

For the plus sign we obtain the double point

71

t = i of the lemniscate,

on the boundary of at; for the minus sign we obtain the

vertices of the lemniscate, ~t' while ~ = I-/2

at t = 12 ¢~.

= -(fr2-1), is in ~ .

for, a radius r I = ¢~ -i.

The first one,t = 1 + /2 , is not in The largest disk D 1 inside A

This requires some verification,

has, there-

in particular, that no

point of the lemniscate is closer to the origin than this vertex at _(¢r~ -I).

E-plane

w-plane

D2

4"

Re t = 1

-i

1 +/2"

= ~w

fit

Figure 2 The disk D 2 has the radius r 2 : I. {~1 [1+1-~]}2-a(i-~)-1

A simple computation shows that B(t) = A(W(~)) =

t/2)2-a/(l-~)is

= (i-

singlevalued and does not vanish in ~ , 2n

It follows that the s e r i e s

where Re g < i.

Cn@n(Z;a) has uniquely defined

n=O coefficients 1 i 2~---i-IF ~

c

n

.

For

f(z)

~n(2-g)n(1- t) (1_~/2)2-a

2nl

2~ii

=

Z fn zn, Ln(f ) becomes successively n=O

F ( ~ ) d ~

=

1 2n

- -

1 2~i

- -

I F ~n(1-g)(1-~/2)n+a-2F(g)d~ m!f --~)d~m =

I F ~n(l-~)(l-~/2)n*a-2( m=O

1 2-ff

~

1

[ m! fm 2-~ m=O

(1-~)(1-~/2) n+a-2 IF

m-n+l

dE.

72

In particular,

n+a-2 (-i) r (n+a-2. r=O 2r - r )( r_ r+l)

for integral a ~ I, the numerator equals n+a-2 [ r=0

and the integrand becomes

(-i) r -n+a-2) 1 2r [ r { m-r-n+l

vanish, except for m = r+n and m = r+n+l.

1 m-r-n

}.

The integrals

The first case contributes

n+a-2 r=0 [ (-l)r2 r (n+~-2) fr+n(r+n)! ' while the second contributes _n+i-2 (_i) r r=0 2r (n+$ 2)fr+n+l(r+n+l)!,

n+a-2 (-i) r .n+a-2. r=O 2r [ r )(r+n)!{fr+n-(r+n+l)fr+n+l}.

i Cn = Ln(f) = 7

(7)

In the important particular c

°i

2n

n

so that

case a = 2, this formula simplifies to

i (1)r r=0

2-7--

!{fr+n-(r+n+l)fr+n+l}

and the coefficients of en(Z) in the expansion of f(z) are n 2n [ (-I)r n-T Cn = r=O 2 r

C8)

(r+n)! r!(n-r)!

(r+n+l) {fr+n-

fr+n+l }

n

=

[ (-l)ra~n)(fr+n-(r+n+l)fr+n+l), r=O

with the a (n) given by (2.8), as coefficients

of yn(Z) =

r

On account of the fact that r I = / 2 - i

n ~ a (n) zr. r=0 r

and r 2 = i, it follows that, as long as 2n

a

is a natural integer, the expansion f(z) =

~ ~., Cn@n(Z;a ) with the cn given by n=0

[7) are convergent and converge to f(z), at least for entire functions f(z) of exponential type T < a - i .

If Y < i, then the series is at least Mittag-Leffler

sum-

mable to f(z). 5.

As an application,

let us find the expansions of powers of z in series of BP.

fCz] = z k, then fk = i, fr = 0 for r ~ k, so that, by (8), k z

=

[ n+r=k 0<_r
(-I) r 2r

k! r:(n-r)' 8n(Z) [ " n+r+l=k 0
k' (-l)r 2r

r:(n-r)!

en(Z)"

In the first sum r < n < k-r, 0 < n-r <_ k-2r and the sum extends over 0 < r <_ k/2;

If

73

in the second

sum, similarly,

0 < n-r < k-2r-i

and the sum extends

over 0 E r < (k-l)/2

Consequently,

k z

k! k 2 =

(_l)r

r=0

(k-l)/2

1

2r

ek-r(Z)-k" '

r!(k-2r)' •

In the second

sum replace

=

sign,

(k+l)/2 and reads,

1

(r-1)!(k-2r+l)!

ek_r(Z ).

@k-r (z)

For 1 < r < k/2 we can combine the two sums.

2rr~(k-2r+l) l

The general

term is k!

Finally,

(-i)

2r-1

(_l)r.2r

r=l

ek-r-l(Z)"

r

r=l

(~+1)/2

1 r! (k-2r-l)' .

2r

r+l by r, which will then run from 1 to

(~+1)/2 including

(-1) r

~ r=0

(-l)r 2r

(k-2r+l)+2r r!(k-2r+l)'

ek-r(Z)

•

(k+l)' " 2rr!(k_2r+l)!

= (-l)r

we verify that for r = 0 and r = (k+l)/2 this general

cisely the omitted

terms,

the one for r = 0 of the first

r = (k+l)/2 of the last sum, respectively,

(9)

z

k

= (k+l):

[

(-1)

r

r=0 found by C a r l i t z

[19].

term represents

pre-

sum, and the one for

so that we obtain the final result:

(k+l)/2

e k - r (z) ,2rr!(k-2r+l)!

The c o r r e s p o n d i n g r e p r e s e n t a t i o n

nomials yn(Z) i s (in a somewhat d i f f e r e n t

@k-r (z)"

notation)

of z k by sums o f t h e p o l y -

in Dickinson [47] and A1-Salam

[3] and reads z

k

= 2kk!

k [ (-1)r r=0

2r+l (k-r)'(k+r+l)!

Yr (z)"

This formula may be proved by u s i n g (1) t o g e t h e r with C o r o l l a r y 4.4. Let us v e r i f y z

(9) f o r i n s t a n c e in t h e p a r t i c u l a r 3

, 1

= 4.{~-., e3(z ) = e3(z)

If we substitute that indeed 6.

application,

+ __/__1

222!0,

el(z)}

- 6e2(z ) + 3 e l ( z ) .

for ej (z) the corresponding

[z3+6z2+iSz+lS)

As another

1

2-1!2! e2(z)

case k = 3:

polynomials

- 6(z2+3z+3)+3(z+l)

(see Chapter

= z 3.

let us find the expansion

of the exponential

k

f ( z ) = e az =

~ fkzk. k=0

2n Cn = n-T

Clearly,

n~ ( - 1 ) r r=0 2r

fk = ~a

and, by (8),

(r+n)! {a r+n r! (n-r) ! (r+n) !

3) we verify

~r+n+l(r+n+l) (r+n+l) !

}

function

74

n

[

(-I)

r=0

2r

r

ar+n

(l-a)

rI(n-r)!

_ (l-~)a

n

n

n!

n

(2)r

r=0~ (-l)r()

(l-a) (l-a/2)n~n/n~

.

It follows that the expansion of e ~z in a series of BP @n(Z) is: (i0)

e az - (l-a)

[ (a-a2/2)n@n(Z)/n: n=0

.

One easily verifies that (i0) is precisely the result that one obtains, if one diffe rentiates Carlitz' formula(6.17)with respect to u and then sets 2u = a. As for convergence, it is clear that the right hand side of (i0) converges for all complex values of a and z.

Also, we recall that the Mittag-Leffler summability

method is regular, so that, for convergent series, it leads to the same value as the direct summation of the series. Hence, for lal = T < i, the series (i0) converges to the function eaz

~at

happens for I~I ~ i?

One observes that for s = I, the right

hand side of (i0) vanishes, while the function e z does not.

Hence, although the

series converges for all a, for lal > 1 it does not converge to the function e

~Z

One may observe, for instance, that for a = 2, the sum of the series equals -i, which is different, in general, from f(z) = e

2z

If we set z = 0 in (i0) and replace @n(0) by its value (2n) I/2nn2, we obtain after a short computation that, for lal < i/2, (ii)

~ n=O

(2n n ) (a-a2) n = l__!___= l-2a

[ 2mare. m=O

From (ii) one can obtain many combinatorial identities. comparing the coefficients of a k, it follows that k~2 r=0

..r.2k-2r. k-r 2k. (-lJ [ k-r ] ( r ) =

In particular, by

PART III CHAPTER i0 PROPERTIES OF THE ZEROS OF BP i.

In this chapter we shall be concerned mainly with properties of the zeros of the

simple BP, either yn(Z) respectively.

J

or On(Z), which we denote by a~ n) and Bk(n) (k = 1,2 .... n) ' "

W~enever possible (and convenient) we shall extend the results to the d e n o t e d by a ~ n ) ( a ) ,

zeros of yn(Z;a) , or en(Z;a), bitrary

a

From t h e s e ,

(often,

h o w e v e r , we s h a l l

of course,

have to restrict

analogous results

(which we may d e n o t e by o ~ n ) ( a , b ) -

o r 8 k( n ) ( a ) ,

respectively,

ourselves

to values

for the zeros of yn(z;a,b),

and g ~ n ) ( a , b ) ,

respectively)

for ara ~ 2).

or en(Z;a,b)

can be o b t a i n e d

imme-

diately by a change of variable and we shall usually refrain from writing out explicit results for b # 2.

Also, whenever possible the subscripts and/or superscripts

will be suppressed. From n

(1) it

z Yn(1/z) is clear

that

t h e z e r o s o f On(Z) and of Cn(Z) a r e t h e same and a r e t h e r e c i p r o c a l s

of the zeros of yn(Z), valid

so t h a t

t h e z e r o s o f any one o f t h e s e zeros of all

(see

2.

([17],

functions

c a n be t r a n s l a t e d

These considerations Hence,

remain

any i n f o r m a t i o n

immediately

z e r o s o f BP h a v e a t t r a c t e d

Others,

into

about

information

see also

(or 8k of On(Z)) are simple.

are formulated

[47].)

(b).

quite [Ok], or

and i m p r o v e d o n l y l a t e r .

known r e s u l t s

[53],

the attention

s u c h as good u p p e r and l o w e r b o u n d s f o r

were d i s c o v e r e d

Some o f t h e o l d e s t

THEOREM 1.

even for Cn(Z;a,b,c).

of the

[17] and [ 5 3 ] ) .

[Ski, respectively,

n).

o f them.

Some o f t h e p r o p e r t i e s early

Ok8 k = 1 (k = 1,2 . . . . .

e v e n f o r a ~ 2 / b, i n f a c t ,

about the

= 0n(Z) = eZCn(Z)

(a)

in the following

three

A l l z e r o s o k (k = 1 , 2 . . . . .

No two consecutive polynomials y n ( Z ) ,

(or On(Z), en+l(Z), respectively) have a zero in comwnon.

(c)

theorems. n) o f y n ( Z ) Yn+l(Z)

No BP of even degree

has a real zero and those of odd degree have exactly one real, negative zero. THEOREM 2. for ~i)

(see [53]).

(a) All zeros of the BP yn(Z) satisfy [o~n)[ < I, except

=-i; correspondingly,

18~n) l > i, except for 8~i) = -i.

(b)

The real

zeros °(2m-l)m o~. the polynomials Y2m_l(Z) form a monotonically increasing sequence:

76

-i = al(1) <

a~3)

THEOREM

([17];

3.

.

< ..

< ~(2m-l)m

see also

< O.

[62] and

[2]).

The zeros a r

=

a r(n) o f yn(Z) s a t 4 s f i y

the equations n

(2)

n

Z %:-i,

n

(2')

n

-

Z ~rI

=-l,

r=l

= 0.

By ( 3 . 1 0 ) ,

(a)

s o , by i n d u c t i o n it

zero of Yn_l(Z). a = -1.

If

a

for

m

2,3 . . . . .

=

zero of yn(Z),

= 0 and by ( 3 . 1 6 ) it

is also a multiple

zero of yl(z)

= z+l,

i f a i s a common z e r o o f Yn+l(Z)

By i n d u c t i o n

However, y 2 ( - 1 )

on n i t

is also

# 0 and t h e a s s e r t i o n

e.g.

By (1) i t

~],

p. 377 and F i g .

is sufficient

@n(Z) and g ( z )

t h e n Yn(a) =

i t now f o l l o w s

= @n(-Z) a r e b o t h s o l u t i o n s

and y n ( Z ) ,

a zero of y2(z) follows.

(c)

f o r @n(Z).

(easily

Yn_l(a)

= 0

z e r o o f Yn_l(Z) and (b)

then it

By

is also a

and o f y l ( z ) ,

i.e.

This assertion

follows

K (z) f o r ~ = n + 1/2

9 6 ) , b u t can a l s o be p r o v e d d i r e c t l y ,

to prove the statement

!

that

which is absurd.

a l m o s t i m m e d i a t e l y from t h e t h e o r y o f t h e B e s s e l f u n c t i o n s (see,

n.

= 0 for m = 2,3 ..... n.

is a multiple

zero of yn(Z),

on n, a m u l t i p l e

follows that

l-2m

Z ~r

also Yn_l(a)

Hence, if a is a multiple

(3.7)

0

r=l

P r o o f o f Theorem 1.

y~(a)

=

r=l

the zeros 8 r = 8 r(n) of @n(Z) satisfy

Correspondingly,

3.

2m-i

Z %

r=l

We r e c a l l

as follows. that

f(z)

s e e n t o be i n d e p e n d e n t )

=

of (2.13).

It follows that

whence z

d

(f'g - fg')

zg" - 2ng'

= zg,

- fg")

= 2n(f'g

- fg'),

or, by integration,

f,g - fg, = Cz 2n.

While i r r e l e v a n t lim

= zf,

= z(gf"

(3)

Z

zf" - 2nf'

for our present

z-2n(f'g-fg

purpose,we observe that

' ) = 2 ( - i ) n+l and (3) shows t h a t

this

(2.2b)

and ( 2 . 5 )

lead

to

value remains unchanged for

~

all

z, so that

C = 2(-1) n+l.

We now c o m p l e t e t h e p r o o f o f ( c ) , On(Z), r e p l a c e

f and g i n (3) by t h e i r

d z -z d--{ (e Z e n ( Z ) ) - ( e e n ( - Z ) ) - ( e On(Z))-

by f o l l o w i n g respective

d ~-~ (eZen(-Z))

Burchnall

values

[17].

We r e t u r n

to

and o b t a i n

= (-l)n+l.2z 2n.

This

simplifies

77 t o (see

[17])

(4)

-2On(Z)On(-Z)+O~(Z)On(-Z)+On(Z)O'(-z)

The coefficients of On(Z) are all positive.

= 2 ( - 1 ) n + l z 2n.

Hence, any real zero of 0n(Z) is negative

Let -a, -8(a > 0, ~ > 0) be two consecutive real zeros (assuming that such exist ). Then (4) shows that 8~[-a)en(a ) = 2(_l)n+l~ 2n,

and similarly e n C - S ) e n ( 8 ) = 2 ( _ l ) n + l B 2n. As 0n(a ) and 0n(B ) are both positive, O~(-a) and @~(-B) have the same sign, which is impossible for consecutive zeros. real zeros is false.

Hence, the assumption that there exist

two

Finally, the reality of the coefficients of en(Z) shows that

complex zeros occur as complex, conjugate pairs and Theorem 1 is completely proved. 4.

Proof of Theorem 2.

[14] and Kakeya [40].

The proof of Theorem 2 is based on a Theorem of Enestr~m We shall state it in the stronger version due to Hurwitz [34],

in the form of a sequence of three Theorems.

For completeness, we shall sketch their

proofs, but the reader, willing to accept their statement, may skip these. n

THEOREbl A. no zero with Proof.

Let b 0 > b I > ... > b n > O; then the polynomial _ _

Izi <

f(z) =

~ b.z J has 9_,=~ v J

i.

Let Izl = r < i; then, with b_l = bn÷ I =

0,[f(z)(1-z)i=

i n " I n n+l b° - j!l (bj l-bj)zJ-bnzn+l > b 0 - j=l~ (bj-l-bj)rJ - bnrn+l = - j=0~ (bj-l-bj)rJ"

consequently, THEOREM B.

n-i >- - j=O [ (bj-l-bj) r j = f(r).(1-r)

{f(z)(1-z)] (Kakeya).

> 0 "

Let the coefficients bj of f(z) be positive and set

Maxj (bj/bj+ I) = O, minj (bj/bj+l) = o; then all zeros zk (k = 1,2 ..... n) of f(z) satisfy the inequalities e < ]Zk] < p. Proof.

Consider f(zp) =

n ~ bjpJz j = g(z). j=0

Then

b.p j J bj+ipj+l

b. 1 hence, ~ ~ Maxj bj+l = i;

the coefficients of g(z) are increasing, those of h(z) = zng(I/z) are decreasing and Theorem A applies to h(z). is a zero of f(z).

If u is a zero of h(z), i/u is a zero of g(z) and p/u

By Theorem A, lul ~ i, so that any zero zk of f(z) satisfies

78

[Zk[ = [ p / u [

! o,

consideration THEOREM C.

as claimed.

The o t h e r

inequality

follows

in the

s a m e way by

of znf(~/z).

The e~ation f(z) = 0 with positive, monotonically decreasing

(Hurwitz).

coefficients has a zero of absolute value e ~ a l to one, if and only if the coefficients can be partitioned into sets of m consecutive e ~ a Z coefficients, where m is a divisor of n+l. Proof.

For a zero z, from f(z)(1-z)

= -

n+l i ~ (bj_1-bi)z~ = 0 follows

n+l (bj-l-bj )z J = b o"

For Iz[ = 1 also follows

j=l n+l l(bj_l-bj)zJl

=

j--i Consequently,

real, pj

n+l n+l ~ (bj_l-b j) = b o = i I (bj_l-bj)zJl. j=l j=l

all summands have the same argument,

> O.

In fact,

n+l ~

a = O, b e c a u s e

-

j=l

From bnzn+l = Pn > 0 m be the smallest

1-bi)zJ -

n+l [

= e ia

"

pj

= b .

j =1

o

Izl = i, zn+l = i.

Let

integer such that zm = I; then m divides n÷l and m >_ 2 (otherwise

j=0 for m ~ j ,

(bj_l_bj)z j = pje ia ,

now follows zn+l > 0 and Using also

. b.z J =

n ~

z = 1 and 0 =

follows,

(bj

i.e.

J that

n ~

bj

> O, a c o n t r a d i c t i o n ) .

From ( b j _ l - b j )

z-3

= pj

> 0

j=o b j _ l - b j = O.

In particular,

b o = b I = ...

= bm_l;

bm+ I = ... = b2m_l, etc. which proves the necessity of the condition.

bm =

On the other

t km m-I " hand, if the condition on the b.'s holds, then f(z) = ( ~ bkm z )( [ zJ), with J k=O j =0 t = (n+l-m)/m, sufficiency.

so that f(e 2~ir/m) = 0 for r The proof

of Theorem 2(a)

is

= 1,2,...,m-l, now i m m e d i a t e .

and this proves the From ( 2 . 8 )

it

follows

n

that

the

a ° = 1.

coefficients Hence,

o f @n(Z) =

Theorem A is

~ an_mZm a r e m=0

applicable

with

decreasing

from an = (2n)!/2nn

b m = an_ m a n d a l l

zeros

a

1 2

[Bk[ >_ 1.

(m+l)[2n-m) n-m

Going beyond this,

(m = 0 , 1

''"

.,n-l),

we v e r i f y

with

m

that, r m = an-m-1

p = Max r

= n(n+l)/2

8 k o f @n(Z ) b

n-nl

satisfy

_

= bm+l

and ~ = min r

m m

Theorem 2(a) is now a simple corollary of the stronger

: to

= 1. m

m

79 THEOREM 4.

For n > 1, a l l

The z e r o s a k o f yn(Z)

zeros

8 k = 5~ n) of 8n(Z) s a t i s f y

(n > 1) s a t i s f y

2/n(n+l)

1 < IBkl E n ( n + l ) / 2 .

~ Jak[ < 1.

For n = 1, t h e u n i q u e

zero of e1(z) and of YlCZ) is a~1) = B~I) : -1. Proof.

On account of p = n(n+l)/2, o = 1 and of Theorem B, the only statements of

Theorem 4 still to be justified are the omissions of equal signs for n > I. that we use Theorem C.

For

The equal sign requires that all coefficients fall into sets a

of at least two consecutive equal coefficients.

However, the ratio r m

n-m

an-m- 1

equals (m+l)C2n-m)/(2n-2m) > m+l >_ I, with equality only for m = 0, when r 0 = i. Hence, equality can exist only between the first two coefficients.

The sets of

consecutive equal coefficients required hy Theorem C reduce, consequently, to a single set of two coefficients, so that n = i, when yn(Z) = 81(z ) = z+l, and when we have indeed a I = 81 = -I. Theorem 4 and with it Theorem 2(a) is proved. The proof of Theorem 2(h) is based on (3.7). of yn(Z)

(n odd).

Let a (n) be the unique real zero

Then (3.7) with n-I instead of n and z = a (n-2) becomes

YnCa in-2)) = (2n-l)a(n-2)yn_l(a(n-2)).

By Theorem i, ~(n-2) < 0 and Yn_ICa (n-2))

>

0

because n-i is even, Yn_l(Z) does not change its sign and Yn_l(0) = 1 > 0; consequently, }'n(a i n - 2 ) )

a (n-2) S.

< 0, YnCa in))

= O and, by Theorem 1, yn(Z) > 0 f o r z > a (n),

so t h a t

< a (n) < 0, as c l a i m e d .

Proof of Theorem 3.

(see [17]).

By logarithmic differentiation of (i) we obtain,

successively: @n'CZ)

8'CZ)n

%c=---Y + l = -

@nCz)

n

=

1

n

=-

r=l Z-Br

1

Z

1

n

~

- - - -

I

l

r=l fir l-Z/Br

r=l

k=l

zk-I Bk r

CS) n =

-

m

Z zk-i where ~m = Z ~r" k=l U-k' r=l

By Theorem (2.3), ~n(Z) contains no odd powers of z with exponent less than 2n*l; it follows that neither ~ ( z ) ,

nor ~ ( z ) / @ n ( Z ) contains even powers of exponent less

than 2n.

Hence, by comparing coefficients in (S) for k odd, 0 < k-I < 2n, one has

O_k = 0.

For k = I, by (S), 1 = -a_l.

n

Consequently,

~ Bi-2mr = 0 for m = 2,3 ..... n, r= 1

n

while

8 -I r=l

r

= -i.

We recall also that the a ' s

and with this Theorem 3 is proved.

r

are the reciprocals of the _ 8r'S

For a proof of the converse, namely that every

8O

solution of the system (2) (or (2') consists of the zeros B$n) of @n(Z) (or a(n)r of yn(Z), respectively) 6.

taken in some order, see [17].

It is easy to improve some of previous results.

First we observe that the proofs

of Theorem l(a) and (b) go through for yn(z;a), provided only that we use (3.21) instead of (3.10);

(3.22) instead of (3.16).

Next, the statement of Theorem l(b) can be strengthened to read as follows

(see [47]). If n # m, then no zero of yn(Z) can be a zero of ym(Z); in fact, no

THEOREM l(b').

zero of yn(Z;a) can be a zero of ym(Z;a).

Proof.

For the proof of the first statement

(see [47]), it is sufficient to use

(3.7') instead of (3.7) and for the second one to use the obvious generalization of (3.20) instead of (3.7'). We have proved THEOREM

i'.

The statements of Theorem l(a) and l(b') hold for the generalized BP.

REN~RK.

The proof of Theorem l(c) does not translate immediately into one for

yn(Z;a).

The problem of determining the values of

a

for which Theorem l(c) holds

appears to be open. Also Theorem 2(a) can be extended to the case a # 2.

Indeed, by (2.27), with

d b the coefficients d k = d~ n) of @n(Z;a ), rm = dn-m-ln-----~m = = bm + l

(m+l)(2n-m+a-2)2(n_m) . If we

consider for a

moment m as a continuous variable and take the derivative, we verify a-2 that in general r m increases monotonically with m, from o(a) = 1 + ~ for m = 0 to o(a) = n(n+a-l)/2 for m = n-l.

An exception can occur only for n < -a+2, i.e.,

only for a < 2, and even in those cases only for finitely many values of n. these can be studied individually, n ~ -a+2.

As

we shall ignore them here and assume that

By using Theorems A and B as before we obtain

THEOREM 2'.

For n B Max(-a+2,2)

(i.e., if a > O, for all n > 2)

1 + (a-2)/2n ~ IBk(a)[

< n(n+a-l)/2,

2/n(n+a-l)

2n/(2n+a-2).

For n = i, the unique zero of yl(z;a) is

S~ I) = - a / 2 .

~)

= -2/a and the zero of @l(z;a) is

81 7.

In previously mentioned paper [47], Dickinson, by using the connection of BP with (see Chapter 5), showed that no a kon) is purely imaginary.

Lommel polynomials

We

shall use a different method (see [12]) to prove the stronger result (see [iii] and

[9s]) of TIIEOREM 5.

For a l l n, the z e r o s a~n) o f yn(Z) s a t i s f y

Re a~ n) < O.

In the proof we use the following known theorem.

Let P(z) =

TIIEOREM D.

n . (n+~)/2 n-2k+l ~ a.z n-3 and set Q(z) = L a2k_iZ j =0 3 k=l

All the zeros of P(z) have negative real parts, if and only if, in the notation of Chapter 8, Q(z)/P(z) = [0,ClZ÷l, c2z ..... CnZ ] with the coefficients cj (j = 1,2 ..... n) all positive.

For a proof of Theorem D, see, e . g . Proof of Theorem 5.

[111].

In the notations of Chapter 8, 1

yn(Z) = Pn(Z) + Qn(Z), with Qn = 2 {Yn (z) - ( - 1 ) n Yn(-z)}" Yn (z) Pn (z) By Theorem 8.1, Qn(Z) - 1 + Qn(Z----~ e2/Z+l =

e

1 + [z, 3z . . . . .

By Theorem 8.2 this is the

(2n-1)z . . . . ] = [l+z, 3z . . . . .

n-th convergent of

(2n-1)z . . . . ].

2/z 1

Hence,

%(z3 yn(Z )

- [0, l+z, 3z . . . . .

(2n-1) z].

As a l r e a d y observed i n Chapter 8, Qn(Z) c o n t a i n s p r e c i s e l y a l l powers of yn(Z) o f p a r i t y o p p o s i t e to n, so t h a t we may i d e n t i f y Qn(Z) and y n ( Z ) , with Q(z) and P(z) r e s p e c t i v e l y of Theorem D and the r e s u l t f o l l o w s . Wimp [112] proved t h a t Theorem 5 holds a l s o for the zeros akn)(a)" of yn(Z;a) at l e a s t for a > 2. 8.

Another result of Dickinson

[47], namely that the origin is a limit point of

zeros of BP yn(Z), as well as Nasif's McCarthy's

[74] generalizations

[81] bound

[e~n)] ~ {(n_l)/(2n_l)}i/2

and

of some of these results are all consequences of a

result of DoPey [48], that we shall prove directly in the slightly more general form of TIIEOREM 6.

For any complex

satisfy the inequality

a

are positive b the zeros a ~n) (a,b) of yn(Z;a,b)

82

b

la n ) ( a ' b ) [ F o r a = b = 2,

COROLM~RY 1. We o b s e r v e

that

this

result

~ n - 1 + Re a

[~n)[

~ 2/(n+1).

is sharp,

because

for n = 1 it

The u p p e r b o u n d o f T h e o r e m 6 may be c o m b i n e d , w i t h t h e and t h e r e s t r i c t i o n the best

presently

and @n(Z;a), [98],

[46],

of Theorem 5 in order known s t a t e m e n t s

respectively, and [84],

that

the

latter

to l e a d u s c l o s e

concerning

are valid

for

with better

the all

locations n.

See,

results,

holds with the equal

sign.

lower bound of Theorem 2' to what are essentially of the zeros however, besides

but valid,

of yn(Z;a) [85] a l s o

only asymptotically,

f o r n ~ ~. THEOREM 7.

For

a > 2,

all zeros ak(a ) of yn(Z;a) belon~ vo the semi-~nulus d e ~ n e d

by the inequalities: 2 n(n+a-1)

i <- '~k ( a ) [

2 <- n + a - 1

' Re ~k < 0.

The zeros Bk(a) of @n(Z;a) similarly satisfy (n+a-1)/2

~ [Sk(a)[

~ n(n+a-1)/2,

Re Bk < 0.

The inequalities for ]ak(a)[ and [~k(a)[ arc sharp and become equalities for n = i. COROLLARY 2.

The zeros ~k of yn(Z) and the zeros ~k of 0n(Z), belong to vhe s~mi-

annuli n ( n2+ l )

<- [akl

< ~

2

, Re ak < 0

and (n+l)/2

5 Igkl

5 n(n+l)/2,

Re Sk < 0,

respectively. Theorem 7 and its corollary are immediate consequences of previously quoted results, of which only Theorem 6 remains to be proved. Theorem E, due to Laguerre.

The proof relies heavily on

Up to now most theorems not directly connected with BP

and whose proof requires more than just a few lines have only been stated for ease of reference and with an indication where a proof may be found.

It is, however, not

entirely trivial to formulate Laguerre's theorem (first published in 1882; see [4~]) in the form here needed.

In fact, the clearest presentation that the author could

find is in a sequence of problems in P~lya and Szeg~'s "Aufgaben und Lehrsatze aus der ~lalyse" ([49], vol. 2, Problems 102-118). Laguerre's theorem will be sketched here.

For these reasons the proof of

83

9.

I]IEOREM E.

(Laguerre

[41]).

X(x) = x-2(n-l)f'(x)/f~'(x).

Let f(x) be a polynomial of degree n and set

Let x 0 be a simple zero of f(x) arZ consider a circle

C (possible a straighv line) ~hro~h the point represenved by the complex number Xo, such that no zero of f(x) belongs to one of ~he two open, cira~iar regions de-

term.ined by C.

Then X (Xo) belo~sz to ~he other region.

Proof of Theorem E. n = (j =i ~ z j)/n.

Let Zl,Z 2 .... ,z n be complex numbers with the center of mass

We generalize this concept and shall say that ~ = ~

as just defined

is the center of mass of the points z. (j = l,...,n) with respect to the "pole" J z 0 = . In order to define the center of mass ~z0 of the zj 's with respect to an arbitrary pole z 0, different from all the points zj, we proceed as follows: First, we map z 0 into the point at infinity, by a linear fractional transformation, say z' =

z!] = -zJz 0 a

a z-z 0 + b.

+ b (a,b arbitrary complex numbers).

Under this map zj is sent into

n We now find ~' = g~' = (j=l ~ zj)/n as before and then map ~' back.

The inverse of ~' under our mapping will be denoted by ~ = ~_ ~0 generalized center of mass of the zj.'s with respect to z 0.

and is the desired

We now show that ~z0

depends only on z 0 and the zj's and is independent of the specific auxiliary map selected.

Indeed, ~ z~ j3

ever, also ~'~

uz 0~

= a ~ (zj-z0) -i + nb, so that ~ j 1 1 + b holds, so that ~ ~. -zJz O J

_

1 0 z -z j

+ b.

How-

1 , and ~Uo~z-Z

-I ~z0 = z 0 + n/~j (zj-z0)

= ~a ~ j

-l = z0-n/~j (z0-z j)

is indeed independent of a and b. n Next, let F(z) = j=iI ]

(z-zj) be a poly~lomial.

F' 1 , so that Then ~ (z) = ~ -z-zj

the center ~z of the zeros with respect to an arbitrary pole z is given by (6)

~

z

= z-nF(z)/F'(z).

From the Gauss-Lucas theorem we know that the (ordinary) center of mass ~= of the zeros of F(z) is inside the convex closure of those zeros and that this is the smallest convex set containing all z~'s.

Each side of the polygon is a straight line

84

through two zeros. the pole z = ~. (zi)'s.

It may be considered also as a circle through any two zeros and

It follows that ~

is inside the corresponding polygon of the

When we map back, the sides of that polygon become circles through any two

of the z.'s and the pole z. J

Of the two circular domains determined by such a circle,

it may happen that one contains no other zeros of F(z). mentally and keep only the one that contains all these remaining circular regions

Let us "delete"

all other zeros.

it then

The intersection

(the image of the intersections

of

of the corres-

ponding half planes through the (zi)'s, taken two by two) yields a curvilinear polygon, Claim I.

say Cz, that contains If the

set of zeros z. (j = 1,2,...,n) J

and if z is outside D, then C Proof of Claim I.

D'.

belongs to any circular domain D

is in D.

z

all the (zj')'s (because that polygon is the smallest con-

all (z~)'s and a circle is a convex set), while z' = ~ is outside

When we map back,

Claim 2.

.

The smallest convex polygon containing all the (z~)'s is inside

any circle D', containing vex set containing

5z' the image of ~

set inclusions

are preserved

Let x be a simple zero of F(z).

and Claim 1 is proved.

The center of mass of the remaining

zeros

with respect to x is

(7)

x = x-E(n-1)F' (x)/F"Cx).

Proof of Claim 2.

Let FCz) =

(z-x)f(z);

then F'(z) = f(z) + (z-x)f'(z),

F"(z) =

2f'(z) + (z-x) f"(z), so that F'(z) = f(x) and F"(x) = 2f(x). We now apply (6) to f(z) X = x-(n-l)f(x)/f'(x).

(whose degree is n-i with respect to x) and obtain

If we replace f(x) and f'(x) by F'(x) and F"(x)/2,

respective

ly, we obtain (7) and Claim 2 is proved. Let D 1 be a circle through x that contains all other zeros of F(z).

As x is a

simple zero, one can deform D 1 into a circle D that leaves x outside, but still contains all other zeros.

Let C

F(z), with respect to x. Cx~D i0.

because x ~ D.

x

be the curvilinear

polygon of these other zeros of

We know from Claim 1 that X = X(X) belongs to C

x

and

This finishes the proof of Laguerre's Theorem E.

From Theorem E i~nediately

follows the

COROLLARY 3. If x 0 is one of the zeros of largest modulus of a polynomial f(x), then

85

No z e r o belongs to the open circular domain ]z I > [x0l; hence,

Proof.

IX(x0) ] > ]x01 LEb~

is not possible and the conclusion follows.

If the polynomial f(z) satisfies a linear differentia~ equation of second

i.

order of the form, P(z)y"

+ Q(z)y'

+ R ( z ) y = O,

then, ybr each zero z0 of f(z),

(8) Proof.

f ' ( z O)

P ( z O)

E ' ( z O)

q ( z O) "

This follows trivially from f(z0) = y(z 0) = 0.

Let a be a zero of yn(Z;a,b); then

LEMb~ 2.

2(n-l)~ 2 X(~)

Proof.

= c~ +

as

+ b

By (2.26), yn(Z;a,b) satisfies a linear differential equation as described

in Lemma i, with P(z) = z

2

and Q(z) = az+b, so that the result follows from (7) and

(8).

If a is one of the zeros of iar;]est modulus of yn(Z;a,b) then

LEbtMA 3.

2(n-l)

ll ÷ D ~ g Proof.

By C o r o l l a r y

the result

follows

3, i f

locus

is

a

z e r o o f maximum m o d u l u s ,

then

Ix(~)l ~ I~1

and

by Lemma 2.

P r o o f o f T h e o r e m 6. yn(Z;a,b).

a

51.

Let v = ~

T h e n , by Lemma 3,

of the complex variable

-1

, where a is

v,

5 1.

f o r w h i c h we o b t a i n

2(n-l) 1 + a+bv

(9)

one o f t h e z e r o s o f l a r g e s t

ll+2(n-1)/(a+bv)l

- e i~

'

We s t a r t

modulus of

by d e t e r m i n i n g

equality.

the

It is given by

0 < * < 27 '

or, solving for v, by v = -b

-i

(a+n-l+i(n-l)cotg ~/2)

= -b-l{Re a+n-l+i

((n-l)cotg ¢/2 + Im a)}.

As # varies from 0 to 27, v describes the straight line L parallel to the imaginary axis, of abscissa -(n+Re a-l)/b.

Consequently, the locus of v -I for which one obtains

the equality (9), is the inversion of L in the unit circle,

q'his is the circle

through the origin, with center on the real axis and passing through -b/(n+Re a-l). Its equation is

86

b

b

Iiz + 2(n+P,e a - l )

] = 2(n+Re a - l )

The required inequality for a corresponds to points inside the circle.

In particular,

[a[ < b / ( n + R e a - i ) and Theorem 6 ( h e n c e , REbLCRK. ii.

a l s o Theorem 7) i s p r o v e d .

Do~ev considers only the case b = -i and denotes a-2 by m.

In 1976 appeared a paper by Saff and Varga [98] with an important improvement of

DeWey's result.

It became available too late for a complete treatment here; there-

fore, only a brief outline will be given, although it leads to the strongest presently known results concerning the location of the zeros of yn(Z;a) and @n(Z;a). The method is based on the following THEOREM F.

Let {Pn (z)}kn=O be a sequence of. ~olynomials of re~rective~ degrees n,

which satisfy the three-tez~ recurrence relation (10)

Z

Z

p n ( Z ) = (~--- + 1 ) P n _ l ( Z ) - ~ - P n _ 2 ( z ) n

(n = 1,2 . . . . .

k),

n

where the b n's and cn's are .~;ositive real nwnbers anJ uhere p_l(Z) = 0, pO(z) = PO ~ O.

Set a =

min i
bn(l-bn_iCn I) (b 0 = 0).

If ~ > O, then vhe parabolic

region P

= (z = x+iy E Cly 2 < 4~(x+a)

x >

-~}

contains no zeros of any of the polynomials pn(Z), n = 1,2 ..... k. Sketch of Proof.

Let z 0 c P , such that none of the pn(Z) vanishes at z 0 and set

~n(Z) = ZPn_l(Z)/bnPn(Z ) (n = 1,2 ..... k). Re ~n E 1 (n = 1,2,...,k).

First we verify that ~n = ~n(Z0 ) has

This is clear for ~I = z0/(zo+bl)' because in P ,

Re z ~ -bl; and for 1 < n < k that property follows by induction on k, by use of (I0) and the definition of P~.

Next, we observe that Pn(0) ~ 0 for all n = 1,2,...,k.

Indeed, by (i0), Pn(0) = Pn_l(0); hence, by induction, for every n, Pn(O) = P0(0) = PO ~ 0.

It follows that if pn(Z0) = 0, then z 0 ~ 0.

Assume now that pn(Z0) =

Pn_l(Z0) = O; then, by (I0) and z 0 ~ O, also Pn_2(z0) = 0 and, by induction Pn_j(z0) = 0 for all j < n, and this if false for j = n. Pn_l(Z) have no common zeros.

Consequently, pn(Z) and

Let now, contrary to the statement of Theorem F,

pn(Zo) = 0 for some n in 1 <_ n < k and z0 ~ Pa.

By (10),Pl(Z) = (z+bl)P0/bl, so

87

that z = -b I is the (only) zero of pl(z). -b ~ P ; hence, 2 5 n < k.

Again by (i0),

_ (Zobnl+l)Pn_l(Z0) this is meaningful,

By the definition of ~, -b I ~ - ~, so that

= Z0cnlpn_2(z0) , or

z0+b n z 0 Pn_2(z0 ) bn - Cn Pn_l(zO) ;

because pn(Z0) = 0 implies, as seen, that Pn_l(Z0) # 0.

This

cn Z0Pn_2(z 0) shows that - (z0+bn) = Un_1(z0) and, by z 0 ~ P , Re ~n l(Z0 ) < I. bn_ibn bn_iPn_l(Z 0) C

c

Hence, Re n bn_ibn (z0+bn) <_ i, or

Re z 0 j -bn(l-bn_ic~l)

~ -~ .

C

n n , whence --Rebn_ibn z 0 <_ i- --bn_l

This, however,

P , where for all z, Re z > - ~.

shows that z 0 could not have been in

Theorem F is proved. n

Consider now the polynomials

as in (2.27).

@n(Z;a)

=

By comparing the coefficients

~ d(n) zn-m, with d (n) = n:(n+m÷a-2)(m) m=0 m m m:(n_m)!2 m ' of zn-m(m = 0,1,...,n)

on both sides,

one verifies that

(ii)

- ~1 (n-l)z On_2 (z; a+2).

@n(Z;a) = (z+n-l+a/2)@n_l(z;a+l) For each polynomial

say n+a = s.

in (Ii), the sum of the degree and the a-entry is the same,

With a = s-n, (ii) becomes

@n(Z;s-n)

= (z+(s+n°2)/2)0n_l(z;s-(n-l))

- ~1 (n-l)z @n_2 (z;s-(n-2))"

If we set b n = (s+n-2)/2 and divide by bn, we obtain (12)

2(s+n-2)-iOn(Z;s-n) Equation

= (zbnl+l)0n_l(Z;s-(n-l))-(n-l)(s+n-2)-iZ@n_2(z;s-[n-2)).

(12) is almost of the required form (10),except for the factor

2/(s+n-2) of @n(Z;s-n). 2n-I/(s+n_3)(n-2)

To eliminate this discrepancy,

and obtain 2n (s+n-2)(s+n-3)...s

Z

(~--+I)

we multiply (12) by

2n-I (s+n-3)...s

@n (z's-n) = ,)n-2 (s-~n-4)...s @n-2 (z;s-(n-2))'

Z

@n-I (z;s-(n-l))

- c

n

n

where we have set c

= (s+n-2)(s+n-3)/2(n-l).

This is precisely of the form (i0),

n

with pn(Z)

=

2n (s+n-2)(n°l)

0n(Z;s-n ) and bn, c n as defined.

Also,

88

bn(1-bn_lCn 1) = ( s - l ) / 2

i n d e p e n d e n t l y o f n, so t h a t a = ( s - l ) / 2 .

By Theorem F, none

o f the p o l y n o m i a l s pn(Z), hence none o f the polynomials @n(Z;s-n), can have zeros i n s i d e the p a r a b o l a y2 5 2 ( s - 1 ) ( x + ( s - 1 ) / 2 )

= (s-1)(2x+s-1).

This may be w r i t t e n as

Izl 2 = x2+y 2 ~ x2+2(s-l)x÷(s-l) 2 = (x+s-l) 2 ) or Iz[ < Re z ÷ s-l. s-i @ _ 1-cos n+a-i O " r 5 r cos @+s-l, or r < l-cos

In polar coordinates

Consequently, for all zeros g~n)(a)

n+a-i @ ' and therefore the zeros (k = 1,2 ..... n-l) of @n(Z;a) we have ]B n)(a)l > 1-cos e~n)(a) of yn(Z;a) are inside the cardioid I ~ n)(a)] < (1-cos @)/(n+a-l).

The bound

(1-cos @)/(n+a-l) is clearly better than 2/(n+a-l) obtained in Theorem 7.

By re-

calling another result of this chapter, we may state here an improved version of Theorem 7, as

COROLLARY 4. For r e a l a > -n+l and b > O, t h e z e r o s ~ n ) ( a , b )

of yn(Z;a,b) satisfy

the inequalities

b n(n+a-l)

~n) 1-cos @ f [a (a,b)[ S n+a-i

b;

~(n)(a,b) of @n(Z;a,b) satisy~4 and the zeros ~k

n+a-1 (1-cos @)b

~n) n(n+a-1) 5 18 (a'b) l ~ b

Under certain conditions we know that Re ~n)(a,b)

< O.

This happens when a = 2

(see Theorem 5), also for real a ~ 2, sufficiently large (see [74]) and, presumably, for all real

a.

Whenever that is the case, also Re B~n)(a,b) < 0 and @ in above

inequalities may be restricted to I~-01 < ~/2. 12.

During the years 1956-59, Parodi perfected an approach for the determination of

bounds for the characteristic values (eigenvalues) of matrices.

It is based on the

following classical theorem of Gershgorin, a proof of which may be found in [45]. THEOREM G.

Let A be an n × n

matrix with complex coej~'~icients ajk.

All character-

n

istic values of A lie in the union D 1 of the disks Ix-ajjl <

lie in the union D 2 of the disks Ix-ajjl S

n ~

~ lajk I. They also k=l k~j [ajkl; hence they lie in the intersea-

j=l jCk tion Dlf~ D 2.

In connection with the representation of BP by determinants (see Chapter 3), Parodi's method leads to bounds for the zeros of yn(Z).

These are weaker than those

89

given by Theorem 7, but the method is simple, elegant, further.

and may well be improved

For these reasons we state here

The zeros ~k of yn(Z) belong to the intersection of the regions D 1 and

THEOREM 8.

D2, defined by D 1 = union of the circles

Izl 5 2/3 and ll+zl 5 i; D 2 = union of the

circles I~l ~ 6/s ~nd ll+zl ~ 1/3. Proof.

Let M

n

be the determinant

the same as those of M . n matrix and

On the other hand M

Then the zeros of yn(Z) are

= IA-Izl, where I is the n x n

unit

n

1

0

2n-i

1 2n-3

A=

defined in Chapter 3.

0

0

...

0

0

0

1 2n-3

...

0

0

0

0

...

0

0

0

0

0

0

i

0

1 2n-5

0

0

1 2n-7

"""

0

0

o

...

y

0

-y

1

0

0

0

...

0

1

-

1

It follows that the zeros of yn(Z) are precisely the characteristic

values of A.

These are located, by Theorem G and looking at rows, in the union of the circles [z[ < ~ 1

,

Iz[

2 f 2n-2k+l

(k = 2,3, .

centric and their union is D I. D 2 and the Theorem is proved.

It is obvious that it can be improved, Parodi shows in the same paper

CnlB-Iz],

by adding the [85], how pre-

If we multiply the next to last column of M n by l+z

and add to the last and expand the resulting determinant that, with an irrelevant

Most circles are con-

In the same way, proceeding by columns, we obtain

condition of Theorem 5 that Re ~ < 0. vious result can be sharpened.

,n-l) . . and . Iz+l] < i.

(but easily found) non-vanishing

where B is the (n-l) x (n-l) matrix

by the last line, we see constant C n, yn(Z) =

90

1 2n- 1

0

of yn(Z), [z[

@+

f 71 '

A more powerful [113].

THEOREM H.

0

0

...

0

0

0 0

0

0

0

0

0

...

0

1 --7

0

0

1 - g (l+z)

31

0

0

0

...

1 ~-

0

0

0

...

0

Theorem

31 -z (l+z)

G we now find that the zeros of of the regions

1 + z(l+z)[ [X

A sketch

method

They use,

see

0

...

[B-Izl, hence,

Di and D½, where

< g1 ; D2, = u n i o n

shows inmlediately,

The result

that

of

Di = union of

lz[

f g1 + g1 and

D I / ' I Di i s much s m a l l e r

can be improved

further,

by removing

Parodi's

approach

to Gershgorin's

[3] and to Hirsch

[33].

also the following

For a more recent presentation

[45].

conjugate and by hj (j = 1,2 ..... v) is characteristic values. (A+A*)

and

C = ~i

rain ~j ~ Re h k < Max j - j

(A-A*) are .qez~itian matrices and denote their respec-

If

and min vj < I m

Vj

j

~ = ~+in is

its complex

Set A = B + iC,

tive characteristic values by ~j and vj, respectively (j = 1,2 ..... n).

TIIEOREM 9.

the

is due to Wragg and Under-

theorem

Let A be an n × n matrix with complex entries, denote by A

where B = ~1

those

in the right half plane.

to improve

in addition

Theorem H, due to Bendixson and proof,

0

1 2n- 7

of D I f ~ D 2, or of D~/~ D½

13.

"""

0

in DI/'~D 2.

hill

1 2n-3

0

z(l+z) l s g 1 ,l + z [ •

portions

0

1 2n-S

to the intersection

than and contained

0

0

5 g1 + g1 i i + z I , and

[z[

0

0

Gershgorin's

belong

...

1 2n-3

B =

By applying

0

a zero

Then

~k < Max ~. for k = 1,2 ...,n. - j 3 '

-

of yn(Z),

t h e n - ~2 5 ~ <

( 2 n - 3 )2 ( 2 n - 1 )

< 0 and

In[ < 8/15. Proof.

Similar

as previously Pij

matrices

defined

= 0 otherwise.

have the same characteristic

and P = llPij[I, Then p-i = IIP~jll

i+j = n, p~j = 0 otherwise.

values.

Let W = p-IAp,

with A

an n × n matrix with Pij = 1 if i+j ~ n+l, with p~j = 1 if i+j = n+l, p~j = -I if

One verifies

that

91

2

1

y

y

o

1

2

1

o

1 ~

2 3~

0

...

0

0

0

0

...

0

0

0

1 7

"'"

o

o

o

W= oo.

0

0

0

0

0

0

0

"""

-i 2n-3

0

...

0

-2 (2n-3) (2n-l)

1 2n-i

-i 2n-I

-i 2n-I

We now apply Theorem H, by writing W = B+iC, with B the diagonal matrix of the diagonal elements

of W, and o

-y

...

o

o

o

•..

0

0

0

0

i --5

0

0

0

i 2n-5

0

-i 2n-I

0

0

0

0

i 2n-i

0

i Y

C

o

=

"

The characteristic

values of B are the diagonal

2 2 Theorem H, - ~ < ~ < - (2n-3)(2n-l) C lie in the union of the circles

Izl _< ~ -1~

entries themselves,

so that, by

By Theorem G, the characteristic 1 1 1 Izl < ~ • Izl <- ~-~-3 + ~

; they are concentric and the largest corresponds

values of

(k = 3,4, "'" ,n) and to k = 3, whence the

theorem. 14.

In 1952 and 1954 appeared two important papers of F.W.J. Olver

[46]).

In them the author discusses

class of differential

equations

expansions

among others the distribution

Specifically,

In particular,

he determined,

(see [i], 9.6.4).

~.

he puts

([84], p. 353)

of zeros of the Hankel functions H(1)(z)

for integral and for half-integral ately because

of functions that satisfy a certain

and studies their zeros.

special emphasis on Bessel functions.

(see [84] and

and H(2)(z),

From these the zeros of K (z) follow immedi

92

I ~i = i ~ie K(z)

= =

~- i H (1)(ze 2

1 i . - ~- ~ i ~ ~le

)(=~ < arg z < ~/2)

1t(2) (ze

~i -2 ) ( _ ~ < arg z < ~).

For half integral values of ~ = n ÷ 1/2, the zeros of K (z) are (see Chapter 2) of course the

as the

that is the reciprocals

of the

The theory o f

Olver and even his precise conclusions cannot be presented here and the interested reader is advised to consult Olver's original papers version of his results, however,

is easy to state,

[84], [46].

A simplified

lie shows that the zeros of

H(1) n+ i/2 (ze~1 ) lie on a convex arc (see Fig. 15 on p. 352 of [84]), symmetric with respect to the imaginary axis. For n ÷ =, the points of intersection with the real axis approach -n and +n, while that with the imaginary axis approaches inz0, where z 0 = t~02 -i ~ .66274 ....

Here t O Z 1.19967...

, is the (unique) positive root of

the equation coth t = t. The zeros of Kn+ i/2(z), i.e. of @n(Z), are then located (see e.g. and Fig. 9.6) on an arc obtained by rotating the previous arc by n/2.

[I], p. 377 Finally, the

zeros of yn(Z) are on an arc obtained from the preceding one by inversion in the unit circle.

(see Fig. 1 ).

-i Let v = z 0 ~ 1.50888... i then the results of Olver have as a

K

corollary the following statements. THEOREM i0. All the zeros ~ n )

of yn(Z)

are located on an arc symmetric with respect to the real axis, that runs in the left half-place. For n ~ ~, the intersection of that arc with the imaginary axis approach ± i/n, while that with the real axis approaches -v/n (see Fig. i). The absolute largest zero of the polynomials yn(Z) of odd degree is the real one, say ~(n)(m = (n+l)/2) and a(n) = m

i n

Figure 1

-vn -i + 0 (n-2).

m

93

To give an idea of the degree of approximation

n]

-v/n

in Olver's results, we may consi-

n

der the following tabulation:

[

~m

I

-1.50888..

3

.5029...

-i

5

- .3017...

.4305... 2742.

While Olver's results contain an element of finality,

they yield exact results

only in the limit, as n + =. Numerical CONJECTURE.

investigations

have led to the following

For arbitrary a > O, b = 2 and odd n + ~

the real zero e (n) (a,2) of '

yn(Z;a,2)

satisfies the asymptotic

a(nl m ~,2)r,J-2(1.52548 This follows conjecturally the differences 15.

results

(see, e.g.

simple explicit yn(Z).

n + a - (~+1)/~} -1.

from [71], vol. 2, p. 194, by taking into account

of notation and normalization.

Burchnall's

applications

m

equality

(2), (2') turned out to be useful in some rather unexpected [62],

[55]).

In fact, it would be very desirable to have

formulae for the sum of arbitrary powers of the zeros of 8n(Z), or

No such formulae, valid for all powers seem to be known.

A certain amoung of information from Newton's

classical

formulae

about these sums, can, of course, be obtained

(see, e.g.

[SO], p. 208).

of the r-th power of the zeros of a polynomial

Let us denote the sums

by o r and, if we want to put into

evidence that the polynomial is yn(Z), then we write a(n). n r f[x)= ~ ckxn-k with c 0 = 1 Newton's identities are k=0

For a general polynomial

r-i

j =0

c. o .+rc = 0 ] r-] r

for r = 1,2,...,n;

c. o

for r > n.

(13) n

j=O

.1

In t h e c a s e o f y n ( Z ; a ) ,

= 0

r-j

we may t a k e c j = dn_(n)j/d n(n), w i t h t h e dk(n) g i v e n by ( 2 . 2 7 ) .

2n T h i s l e a d s t o c 1 = 2n+a-2 ' c2 = (2) 2n Cn = ( 2 n + a - 2 ) ( n ) -2n 2n+a-2

' °2

=

= 2n

(n+a-2) ! (2n+a-2)! "

- a l C l - 2 C 2 = c21 - 2c2

22 (2n+a-2)(2n+a-3)

By (13) =

.....

2j cj = (~.) (2n+a-2) ( j )

(assuming n large)

4n(n+a-2) (2n+a- 2) 2 (2n+a-3)

.....

we o b t a i n o 1 = - c 1 =

94

o3 =

-8(a-2)(n+a-2)n (2n+a-2)3(2n+a-3) (2n+a-~)

, etc.

significant

simplifications,

simple

In that case cj = (~)

BP.

It is immediately

if we restrict

ourselves

that we obtain

to the case a = 2, of the

2j - - .) and previous (2n) (3

1

apparent

formulae

reduce to o I = -i,

-i

o2 = ~

, o 3 = 0; also,

2

~4 =

' °5 = 0, o 6 = (2n-l) 2 (2n-3)

10n-17

o 7 = 0, a 8 = -

, (2n-l) 3 (2n- 3) (2n-5)

, o 9 = 0, etc.

(2n-l)4(2n-S)2(2n-5)(2n-7) The values Burchnall's

obtained

theorem

for odd subscripts

The sums of negative of the positive confusion

powers

of course,

of the zeros ~ n )

powers of the zeros

in what

are,

follows

~n)

of yn(Z)

of Gn(Z)

In order to avoid

we shall denote by ~r = °(n) the sum of the r-th powers r

(r positive,

zero,

o_r of negative

or negative)

powers

8~n) of 8n(Z),

are now precisely

o2 = -a°-l'2a2

of the zeros a~ n) of the simple

can of course be computed

by use of Newton's

formulae

BP yn(Z).

as sums of positive (13).

The sums

powers

The coefficients

of the

cj in (13)

the a.'s of (2.8). 3

With a 0 = i, a I

2 = al-2a2

' a2 =

(n+l)n 2

(n+2) (4) (n+l) n ~ ,... we obtain O_l = -a I = - - 2 ' 2"2! 1 n (n+l) {n (n+l)-6},

' o-3 =

1 n (n+l) {n2 (n+ i) 2-28n (n+l) +60 } etc. o_S = - --16

°-4 = - ~i n(n+l){n(n+l)-3},

It turns out that we obtain

somewhat

simpler results

1 N(N+I) , o_3 = ~1 N(N-3), Then o_i = -N, o_2 = ~-

say.

by

are the same as the sums

and conversely.

'

zeros

those predicted

(2).

if we set n(n+l)/2

= N,

o_4 = -N(2N-3),

o_5 = _ ~1 N(N2_I4N+IS) , etc • In this way we are able to compute (positive

or negative)

seems to emerge

effectively

of the zeros of the BP yn(Z);

and an explicit

formula

the sums of an}' given power however,

for o (n) and/or r

no obvious pattern

o (n) does not seem to be -r

known. A more convenient (see [62]), valid appearance,

it permits

gives a new proof.

way to approach

at least

for positive

one to go beyond

this problem values

of r.

Burchnall's

is through

a recursion

formula

While of a rather complicated (2), for which

incidentally

it

95 We start by observing that, on account of Theorem 1 all zeros of BP are simple. Hence, On_l (z) n _ _ = ~ On(Z) j=l

A . nj with Anj = lim (z-Bj) @n-l(Z) z-Bj' z + flj @n (z)

@n-l(BJ) @n(~j )

By (3.8) the last fraction equals -B -I , so that J (14)

n~ j=l

-en_l(Z - = )

On(Z) The corresponding by znyn(i/z),

1 gj(Bj-z)

"

formula for yn(Z) is easily obtained from (14), by replacing On(Z)

8j by ~i I and then simplifying the result.

We obtain

2

n

~.

J=]

Z-~.

Yn_l(Z) Yn (z)

3

Returning to (14), we observe that it can be written as

en_l(Z )

n

n

j=l 8~(1-z/Bj)

8i2 r l o ( z~r

j=l

=

zr

q

r=O

n j = l Bj

r=O °r+2Z "

On the other hand, by (3.5) (with n replaced by n-l), On(Z) On_l(z )

en-l(Z) (16)

_ 2n-l+z 2 On-2 (z) @n_l(Z )

I

@n(Z )

i

= 2n_l+Z2On_2(z)/O n_l(z)

or

[

(_l)m

- 2n-~ m= 0

z" (2nXl

en-2(z) m @n_l(Z))

"

We now equate the right hand sides of (15) and of (16) and then replace the ratios @n_2(z)/@n_l(Z)

with the help of (15).

r (n) I ~ ~r+2 r=O

=

~ 2m z ~l)}m I (-l) m { I zk~ m=O (2n-l) 2m+l k=O

2n_l +

I -

2n-i

+

tr

We obtain

[

m=l

(2n-l) m+l

Zr

r=l

k=O

(-i) m

2m÷kl+...+km=r m >- 1 ,

By comparing coefficients,

(I

k .j

zko

(n-l) (n-l)

_(n-l)

(2n-l) m+l °kl+2 °k2+2 ""Ukm+2 •

>- 0

1 we obtain first o~ n) = 2n-I ' then, in general, for r > 0,

96

(17)

(-i) m (n-l) _(n-l) (2n_1) m+l ° k i + 2 " " ° k m + 2 •

o (n) = r+2 m

2m+

[ k.=r j=l j

m ~ i, k.j >- 0 Formula

(17) p e r m i t s

one,

in principle,

a l r e a d y knows t h e sums o ( n - 1 ) .

though (so far,

(17)

can be p u t t o any u s e .

at

least)

if

m

In f a c t ,

m

wonder w h e t h e r i t

t o compute t h e sums o (n) r e c u r s i v e l y , iooks rather

Fortunately,

forbidding

and one may

the answer is positive,

(17) c o u l d n o t be made t o y i e l d

explicit

one

al-

formulae for all

o(n) m

To s t a r t

w i t h , we may u s e

observe that

(17)

in order to obtain

a new p r o o f o f ( 2 ) ,

(2')•

We

f o r n = i and n = 2 we have eo(Z) Ol(Z)

1 l+z

~ r=0

(_l)rz r

o ( 1 ) zr r+2

= r=0

and @i (z) -

l+z

-

i

=

z {i- ~

-3-

1 = y-

1

=

=

82(z)

3+3z+z 2 2 (l-z+z2-z

z2 7+

l+z

l

- -

3

+...)

1

=

3 l+z+z2/3 4 z + ~--

3 l+z2/3(l+z)

2 (l-z+z

z3 2 4 1 5 ~ - - 27 z + ~ - z + ...

3 -z

=

.2 +..

)

...)

-

~ 0 (2) z r r+2 " r=0

We note that o ~2) = 0; this is the instance n = 2 of the claim that o (n) = 0 holds 2~+i for ~ = 1,2,...,n-l.

(For n = 1 this is also true - vacuously.)

o i)= -I and o~ 2)- ~ , o

(183

27

'

o{n) (-I) n 2n+3 = { 1 . 3 . , . ( 2 n _ l ) } 2 ( 2 n _ 1 )

Let us assume that o,k~f ~ = 0 has already been veri2~+i

fied for all k with 2 < k < n-i and all ~, I ~ v < k-l.

Then

r+2 = 2~+i is odd, also o (n) = o (n) = 0 for ~ = 1,2,...,n-i r+2 2v+l if r+2 is odd, then at least one of the k.'s is odd. 3 1 < k.] <- r-2m = 2~-i-2m <_ 2~-3 ~ 2n-5,

product

for ~ < n-l.

Also,

(17) shows that if Indeed,

•

let k = n-l;

all odd k.'s satisfy 3

In particular,

we find in each

_

at least one factor o (n-l) with 3 < k + 2 <

1

=

, so that

(n) (-I) n °2n+1 = { 1 . 3 . . . . . ( 2 n _ i ) } 2

both hold for n = i and n = 2.

(1)

Next o 3

Km

and this vanishes by the induction hypothesis.

k.+2

j

It follows

-

2n3=

2On l) 1

3

that all summands

'

in (17)

97

vanish

and a,n,l~ = 0 for all 9 = 1,2, ,n-i in agreement 29+1 "'"

we wanted

with

(2) and that is what

to prove. so that r = 2n-l,

If r+2 = 2n+l,

then one of the summands

in (17) has m = 1

and k I = r-2 = 2n-3

reduces

(no other k.'s occur) and does not vanish. Indeed, (17) now 3 _(n-l) U 2n-i _(n-l) _(n-i) to o -n" = The numerator is U2n_l = O2(n_l)+l and we have 2n+l (2n_i)2"

that (18) holds at least for the superscripts

verified

the first of (18) has already then s(n) _ i 2n+l = (2n-l) 2

for superscripts

been recognized

(-i) n-I {1"3"''(2n-3)} 2

n , and t h i s

=

concludes its

(-i) n {1"3'''(2n-1)} 2

k 2 = 2n-3.

Hence,

assuming

(n-l) (n-l) 2 O2 °2n-I

(2n_i)2 +

i

that the second

(17) and the induction

_(n-i) = - U2n+l

2n+3

and (18) h o l d s a l s o

non-vanishing terms

in (17),

to m = i, k I = 2n-l; m = 2, k I = 2n-3, k 2 = 0; and m = 2, k I = 0,

and k up to n-l, ~(n)

n-l,

p r o o f by i n d u c t i o n .

I f r+2 = 2n+3, t h e n t h e r e a r e t h r e e p o t e n t i a l l y corresponding

1 and 2; if we assume that

as valid up to the superscript

(-I) n-i

for ~(k) 2k+3

yield

=

(-l)n {l.3...(2n_3)}2(2n_3)(2n_l)2

!:1)_~ n (i__ {i.3...(2n_i)}2 2n-3

i

2"

=

hypothesis

(2n_i)3

2n-3 {1.3...(2n_3)}2

of (18) has been verified

(2n_i)3

2 (2n-i)(Zn-3)q

(-i) n

{l.3...(2n-l)}2(2n-l) and this finishes

the proof by induction

One may, of course, increases

fast and the formulae

We shall explicit

go on.

of (18).

However, become

the number of non-vanishing

rather complicated.

that, although we do not know

finish this section with the remark

formulae

terms in (17)

for a (n) , valid for all integral

values

of r and n, we still can

r

at least prove the following THEOREM

ii.

For n = i, a (I) = (-i) r. r

For n = 2, c (2) = 3-[m/2]h m

m

and a (2) = -m

3[(m+l)/2]hm , where h m depends only on the residue class of m .~dulo 12, as fo~low~: m h

m

Proof.

0

1

2

3

4

5

6

7

8

9

I0

2

-i

1

0

-i

1

-2

1

-I

0

1

For n = i, the result

is obvious.

ii

modulo

12

-i

For n = 2 we obtain by Newton's

first

98

formula

(13), that G 0 = 2, o I = -i, o 2 = 1/3.

verify that also ~3 = 0, o 4 = -1/9,..., theorem.

In general,

Next, by the second formula

Oli = -3 -5 , conform to the statement

om

= -°m-I

1 - ~ °m-2"

We assume that for all values of k _< m-i we have already verified

theorem.

of the

by (13),

(19)

holds.

(13), we

Then we check b y (19) that also o This requires

12 checks,

may reduce these to 6 checks,

in

to the residue class of m (mod 12).

that hm+ 6 = -hm.

out one of these checks,

say for m - 5 (mod 12).

the induction hypothesis

and (13),

We

As an example, wo work

Then m-I - 4 and m-2 - 3 and, by

o m = -°m-i - ~1 °m-2 = _3[(m-l)/2]h 4 _ ~1 . 3- [ (m - 2)/2]h 3 = and this proves that Theorem

ii

takes on the value prescribed by the

according

by observing

that Theorem

ii holds also for m, provided

3-(m-2)/2

=

3-[m/2]

that m - 5 (mod 12), and

in fact, also for m ~ ii (mod 12). The assertion concerning Newton's

formulae

for 82(z)

o (2) -m

could be proved

= z2+3z+3.

in the same way, by using

It is much easier,

however,

to observe that

-m -m ~ m m = 3m-[m/2]h = 3[(m+l)/2]hm , as claimed. a-m = el + ~2 = (~ + ~2)/(~i~2) = 3m°m m No similarly

simple results

seem to hold for o
CHAPTER ii ON THE ALGEBRAIC i.

IRREDUCIBILITY

Much of the present chapter covers material

OF THE BP

found in [53].

there are many errors of detail and the presentation

This is equivalent

rational,

integral coefficients,

duce to a constant polynomial,

It is extremely statement exists.

with

then f = gh is possible only if either g, or h rein fact to a rational

The polynomials

integer,

and the other factor

here considered will, in fact, be monic.

likely that all BP are irreducible,

but no proof of this general

What we can prove are the four theorems of the present chapter.

The symbol p, with or without positive inters.

over the field of rational num-

to the statement that if f,g and h are polynomials

has the same degree as f.

In

in order to make the

but also readable.

The term irreducible will mean here irreducible bers.

in [53],

leaves much to be desired.

the present chapter many procfs have been reworked completely, treatment not only correct,

However,

subscripts,

stands for "an odd prime, while q and m are

Also, for given n, we set k I =

and k = k(n) = rain (kl,k2).

min

(n-Pl), k 2 =

Pl -< n

min

(P2-n-1),

P2 > n

THEOREM i. (a) The BP of degrees n = pm are irreducible. (b)

The BP of degrees n = qp-l, with q < p/2 are either irreducible, or else can

~ v e only irreducible factors of degrees rp or rp-i (r = 1,2 ..... q). then the BP is irreducible. (c)

If p-i > k(n),

In particular, the BP of degrees n = p-I are irreducible.

The BP of degrees n'= qp+l, with q < p/2 are either irreducible, or else can

have only irreducible factors of degrees rp, or rp+l (r = 0,i ..... q). p > k(n) and q is odd, then the BP is irreducible.

if also

In particular, the BP of degrees

n = p+l are irreducible. (d)

The BP of degrees n = qp with q < p/2 can have irreducible factors only of de-

grees that are multiples of p.

If also either p > k(n), or q <_ 17, then the BP is

irreducible. (e)

The BP of degrees n = pk-i can only have irreducible factors with degrees that

are multiples of p-l. (f)

If also p-i > k(n), holds, then the BP is irreducible.

If p is the largest prime factor of n, or of n+l, then the BP of degree n can-

not have any factors of degree less than p-l. THEOREM 2.

For every integer n, the BP of degree n contains an irreducible factor

of degree Ann, with

l im

A n = I.

For every~ n, A n > 16/17 and no BP can have an

n->oa

irreducible factor of degree d, with n/17 < d < (16/l?)n. THEOREM 3.

All BP of degree n ~ 4QQ are irreducible.

There is no doubt that with more work the bound n ~ 400 could be increased. In fact, there are only four values of n, 301 < n < 400 cation was needed.

for which a separate verifi-

Also the fraction 1/17 that occurs in Theorem 2 could be

100

decreased and Theorem i could be improved by the addition of other types of n, for which irreducibility can be guaranteed.

There does not seem, however, to be much

point to such a task, because in any case it would fall short of the proof of the already mentioned

All BP are irreducible.

CONJECTURE i.

2.

The proofs of these theorems rely heavily on Theorem A, due to Dumas [lZ], to be

stated presently.

It is based on the theory of the Newton polygon of a polynomial

f(x), with rational, integral coefficients, with respect to a fixed, rational prime p (see [~Z] and [aS]).

This frequently occurring sentence will be abbreviated to

read "N.p. of f(x), w.r. to p", or simply "N.p."

DEFINITION.

Let f(x) =

the points

Pm = [m'em]"

n ~ m=0

e amP

m n-m x , am,e m rational integers,

p~a m,

and consider

For want of a better name we call these particular lattice

p o i n t s o f the plane the sgots o f f ( x ) .

The N.p. of f(x) w.r. to p i s the unique

open polygonal line w~th vertices only at the spots of f(x), convex dowr~)ards and with no spots below its sides (see Fig. i). THEOREM A.

(Dumas [~2]).

Let f(x) =

n e m n-m ~ amp x , P~am; let h r are v r be the m=O

horizontal and vertical projections, respectively, of the

r-th side of the N.p. of

f(x) w.r. to p and let their greatest common divisor be t r = (hr,Vr).

Then, if M

i8 the number of sides of the N.p. and h r = trSr, all factors of f(x) have degrees of the form

M ~ r=l

RE~RK.

~rSr, with integers ~r' 0 < ~r < t - r"

If v r = O, then t r = h r and sr = i.

COROLLARY i.

~

e 0 = 0, em ~ men/n for m = 1,2 ..... n, then M = i and all factors of

f(x) have degrees of the form un/t, t = (n,en), 1 < ~ < t.

If, furthermore,

(n,e n) = i, then ~ = t = 1 and f(x) is irreducible. Proof of the Corollary

follows immediately from Theorem A.

In what follows it will be convenient to refer (improperlyl) to the horizontal projection h r, as the length of the r-th side of N.p. proof of Theorem A.

We shall not give here a

Indeed, an excellent presentation can be found in [59] (see

also [It], or [60]), and none of the available proofs is really easy. We use square brackets rather than parentheses, in order to avoid confusion with the notation (m,em) used for the greatest common divisor of the integers m,e m.

101

e (¢m) I

,

J

,i

1

2

3

I 4

I

f

I

7

5

I

8

9

i0

ii

12

13

14

15

m

Newton Polygon of the polynomial zls(x) = zlS+S(clzl4+c2zl3+c3zl2+c4zll+cszl0) 7 6..3 S .4. 4 3 2 , 5[cj +52(c6z 9 +cvzS+csZ, +CgZ )+b Cl0Z +b [CllZ +el2 z +CI3Z +C14Z+C15) (j = 1,2 ..... 15), modulo p = 5. Figure i.

102

Sometimes a particularly simple case of Theorem A is sufficient for the purpose on hand.

This is known as Eisenstein's criterion of irreducibility.

THEOREM B.

(Eisenstein; see [59]).

f(x)

then

Proof.

Here e 0 : 0, e n : 1, so t h a t As a l s o e m > _ 1 for

Theorem A a p p l i e s

and f ( x )

To c o n c l u d e t h i s will

there

is

exists

n

+ p

of f(x)

~ amX m= i

n-m

, with P~an;

an N = N ( E ) ,

Bertrand's

further

improvements of this

postulate,

(R. B r e u s c h ) .

line

= 1, t h e C o r o l l a r y

of to

irreducible• we s t a t e

such that,

x < p ~ (l+~)x.

reduces to a single

= (n,1)

also,

for ease of reference,

I t i s w e l l known ( s e e ,

known.

3.

t h e N.p.

1 < m < n and ( n , e n )

section,

be n e e d e d l a t e r .

the interval

THEOREM C.

n

is irreducible.

l e n g t h n.

that

Let f(x) = x

e.g.

for x > N(e),

Unfortunately,

there

N(I/8)

by I .

is at

the explicit

p r o v e d by T c h e b y c h e f f , result

[4~]) t h a t

Schur

least

function

states

[51],R.

that

Breusch

another theorem t o e v e r y E > 0, one p r i m e p i n

N = N(e) i s n o t

N(1) = 1 a n d ,

after

[8] p r o v e d

= 48.

Let us denote by e(m) (= ep(m); we shall suppress the subscript) the exact power

of p that divides m. For any integer n, it is clear (see, e.g. e(n!) : [~]

(i)

+ [~'] P

[24]) that

+ ....

[~-] P

+ ...

where [x] stands for the greatest integer not in excess of x;

there should be no

danger of confusion with Pm = [m'em]' where there are two arguments enclosed in square brackets.

It is sufficient to stop in (i) with the term r = [log Llog n] pj, because

all other terms vanish.

For n = pm, (i) immediately yields

(2)

e((pm) !)

=n-1 p-i "

k ~ ajp J, 0 < a• < p-i be the p-adic representation of n; j=0 -J-

More generally, let n =

then we obtain by direct substitution in (i), e(n!) = + akpk-1

+ a k _ l pk-2

k-2

+ akP ÷

+ ak

+ a2P + a I

k-3

+ ak_iP

° ....

+ akP

+ ...

+ ak_ 1

+ ... + a 2

103

= ak (pk-1

+ p k-2

+...+

= ak

+ ak-1

pk_l_i 2_ 1 p-1 + • "" + al P p-i

k

1

p-i

+ ak _ 1 (pk-2

p+l)

+ p k-3

+ ....

+ 1)÷...+

a2(P+l)

+ a1

k

(j!O ajp3 - j=O[ aj}.

This completes the proof of the following lemma, due to Legendre: n

e(n:) = n-o(n)

LF~IA i.

p-i

where a(n) ( = Gp(n)) = •

~

aj, s(0) = 0.

j =0

From Lemma i immediately follows LEMMA 2.

Set

(3)

Cm

e(Cm) =

then

Proof. 4.

=

(n+m) !

m'• (n-m) !

'

m+a (m) +o (n-m) -o (n+m) p-i

By Lemma i, e(Cm) = ~

{n+m-~(n+m)-m+e(m)-(n-m)+o(n-m)}

and Lemma 2 is proved

It is clear that Yn(X) and On(X) are either both reducible, or both irreducible•

In fact, there is an obvious one-to-one correspondence between their respective irreducible factors.

It is somewhat more convenient to apply previous criteria of

irreducibility to monic pol}momials and, therefore, we shall be concerned here with @niX), rather than Yn(X). serve that On(X/2 ) = 2nOn(X/2) =

2-n

n [ c x n-m m=Q m

We obtain a slight additional simplification, n

(n+m) ! ~ m:(n-m)' m=0

x

n-m

if we ob-

, so that, by using also (3), Zn(X) =

It is sufficient to study the irreducibility of Zn(X) and,

in this chapter only, conclusions referring to BP of degree n will apply to any of the three polynomials YnfX), @n(X) or Zn(X ). Proof of Theorem i. e(Cn) = ~

Part I.

(i).

Let n = pk, then 2n = 2p k and, by Lemma I, or 2

k-i {2n-2-n+l} = n-lp_l= j=0[ pJ.

Clearly,

(e(Cn),n) = i holds, but we ob-

serve that this would no longer be the case for p = 2, n = 2m, where 2n = 2m+l,

104

e(Cn) = n, so that (e(Cn),n)

= n.

This is the reason, why p is assumed to be an

odd prime only. We claim that, for 1 < m S n-l, m m(n-l) e(Cm) > ~ e(Cn) = n(p-l)

(4) Assuming for a moment

(4), it follows that the N.p. of Zn(X) reduces to the straight

line from P0 = [0,0] to Pn = [n, e(Cn) ]. shows that Zn(X), is irreducible.

Let m =

"

On account of (n,e[Cn)) = i, Corollary 1

It remains to prove

(4).

k-i k-i [ ajp J, 0 < a. < p-l, with o(m) = [ a.. j=O ]j=O J

Then n+m = pk +

k-i [ ajp3, j=O

o(n+m) = o(m) + 1 and, by Lemma 2, e(Cm) = p _ ~ (m+o(m)+o(n-m)-o(n+m))

To prove

= p _ ~ (m+o(n-m)-l).

(4), we have to verify that n(m+o(n-m)-l)

> m(n-l),

or that n(o(n-m)-l)

This is obviously true, because o(n-m) and m are both positive integers.

> -m

Part (a)

is proved. (ii)

We now dispose first of a few other easy to prove statements of Theorem A.

If n = p-l, then PlCm for m = 1,2 ..... n and p2~c n ( = p(p+l)...(2p-2)), Theorem B applies and Zn(X ) is irreducible.

so that

If n = p+l, then plc m for 2 < m < n,

2 p ~c n, because c n = (p+2)(p+3)...(2p)(2p+l)(2p÷2).

It follows that the N.p. of

Zn(X ) consists of two sides, the first of length one, from P0 = [0,0] to P1 = [i,0], and the second from P1 to Pn = In,l) of length (n-l) and without spots on it. Theorem A, Zn(X ) is either irreducible, irreducible polynomial sible.

or else splits into a linear factor and an

of degree n-i = p.

The last alternative,

Indeed, the linear factor has a real zero,

gative integer)

however,

is impos-

(which, in fact, must be a ne-

and we know (see Chapter i0) that BP of even degree

has no real zeros.

By

It follows that Zn(X ) is irreducible.

(n = p+l is even)

Let us observe at this

point that we already have proved so far a weaker form of Theorem 3, namely THEOREM 3' Proof.

All BP of degrees n < 20

The statement

are irreducible, except, perhaps, for n = 15.

is obvious for n = i and n = 2.

For 3 S n ~ 20 , n # 15, all

odd integers are prime powers and all even integers are of the form p ± i. the irreducibility [iii)

of ZlS(X) , see Section 5.

In the more general case n = qp-l, q < p/2, one has n = (q-l)p+(p-l),

As for

105

o(n) = p+q-2,

2n = (2q-l)p+(p-2),

and

= (pq-l,q)

r+s.

(n,e(Cn))

= I.

(r+q-l)p+(p-l),

with o(m+n)

= p+q-r-s-2.

Consequently,

Pn = [n'en]

= r+q+p-2,

of two sides,

t I = (hl,Vl)

1 1 slope p_-~> ~ ,

are above that

= 0,1,...,q-l;

9 = 0,1.

case n = qp+l,

that o(n)

[I f m ~ n-l),

= q÷l,

one obtains

n-m = (q-r)p+l(l-s),

I leads to e(Cm)

e(Cm)

Consequently, P0 = [0,0]

and the

and v I = q-l, = 1 and

By Theorem A,

= (~+~)p-~,

=

the N.p.

to PI = [1,0],

= q+r+l,

if s = p-l.

if 2 ~ s ~ p-2, if s = p-l.

= q-r+l-s,

= q.

if s = 0,i;

= q-r+p-s,

r = 0, s = i), e(cl) Pl = [i,0], with slope

of Zn(X ) consists

and e(Cn)

with o(m+n)

with o(n-m)

n = qp+l,

= 2q+2,

if 0 < s ! p-2;

r+2

= r, including

a[2n)

analogous.

= q+r+s+l,

r+l

line through

is entirely

with a(m+n)

if s = 0, or i,

= 0.

if 2 < s < p-l.

As under

i/p contains

while all other spots now of two sides,

of length one, and the second

length h = n-i = qp, v = q, of slope by Theorem

= [(q-l)p,q-l],

that 0 < r ~ q, with r = q only if s = 0,

r

for m = 1 (i.e.,

that the straight

m = rp÷l,

including

that the N.p. of

of the N.p.

q < p/2,

2n = 2qp+2,

with o(n-m)

n-m = (q-r-l)p+(p+l-s),

In particular,

(downward)

[rp,r],

With this the proof of the first and of the last state-

n+m = (q+r+l)p,

This

all other spots It follows

of the form d = pp + ~(p-l)

Next, n+m = (q+r)p+(s+l),

Furthermore,

all spots

1 (b) is complete.

successively,

~[m) = r+s.

line.

side has length h I = (q-l)p,

by the convexity

The treatment of the general

For m = rp+s

n-m = (q-r-l)p+(p-s-l)

one from the origin to P(q-l)p

of Zn(X ] have degrees

ment of Theorem

One finds,

Finally,

= r+l if s # 0, e(c n) = r if s = O.

the second has h 2 = p-l, v 2 = i, t 2 = (h2,v2)

required

as

0 < r < q, one has o(m) =

i/p through the origin runs through

The first

= q-l, while

= (2n-o(2n)-(n-~(n))/(p-l)=q,

if s # 0; n+m =

to the values m = pr (with s = 0), while

other from P(q-l)p to Pn"

all factors

e(Cn)

= r+q+s-l,

if s = 0.

By Lemma 2, e(Cn)

= [(q-l)p+(p-l),q],

Zn(X ) consists

[iv)

with o(m+n)

the line of slope

corresponding

= 2q+p-3,

For 1 < m ~ n-l, m = rp+s,

Also, n+m = (r+q)p+(s-l),

with ~(n-m)

o(2n)

(iii) we observe all spots with

lie above that

one horizontal,

line.

from

from Pl to Pn = [n,q] of

l/p, and t = (h,v) = (qp,q)

= q.

It follows,

A, that any factor of Zn(X ) has a degree d = up+~, with ~ = 0,i ..... q

and ~ = 0,i.

With this the first and the last statement

of Theorem

i (c) are proved.

106 (v)

Let n = qp, with q < p/2; then ~(n) = q, 2n = 2qp, ~(2n) = 2q, e(Cn)

Also, for m = rp+s, with 0 < r ~ q-l, 0 < s ~ p-l, r+s ~ i, one has as before,

= n-q

p-i = q"

~(m) = r+s and,

computes v(n+m) = r+q+s, o(n-m) = q-r if s = 0, ~(n-m) = q-r-s-i if

1 ~ s < p-l.

By Lemma 2, e(Cm) = r+l if s ~ 0, e(Cm) = r if s = 0.

slope i/p through the origin contains all the spots while all other spots

The line of

[rp,r], including Pn = [qP'q]'

[rp+s, r+l], s ~ i, lie above that line.

The N.p. consists of

a single side, from P0 = [0,0] to Pn = In,q], with h = qp, v = q, and t = (qp,q) = q. By Theorem A, all factors of Zn(X ) have degrees of the form d = ~p (~ = 1,2,...,q). This finishes the proof of the first statement of Theorem 1 (d). (vi)

The next t>Te of n to be considered

is n = pk-l.

Now n = (p-l)(pk-l+...+p-l),

k-I

so t h a t g [ n ) =

k(p-1).

Also, 2n = 2pk-2 = pk +

[

(p-1)pJ+(p-2) and a(2n) = ~ ( n ) .

j=l k-I

By Lemma 2, e(Cn) - p-ln _

n-m =

k-i

~ pj. j=0

For m =

~ j=0

k-I ajpj , ~ ( m ) =

k-i . ~ [p-l-aj)p J, with o(n-m) = k[p-l)-g(m). j =0

= o(m)+r(p-l).

l-i-p-1 (m+~(m)+k(p-1)-~(m)-~(m)-r(p-1)} = ~ me (Cn) n

(5)

m

+ [p-l)(k-r)

(6)

k-i ~ ajpj + (ar_l)pr + j=r+l

We shall verify t h a t

Written explicitly,

k-1 ~ a.. j=r J

For given r, the right hand side is maximized by the choice of

strict inequality.

(S)

~ m, or

( p - 1 ) ( k - r ) >_ o ( m ) =

case, and that case only,

k-i . ~ ajpJ-l. j =0

< e (Cm),

with equality if, and only if for all j ~ r, aj = p-l. reads m-a(m)

+

By eemma 2, e(Cm) =

{m-~(m)} + k - r .

p-1

Also,

k Finally, n+m = p

If a~J = 0 for j = 0,i ..... r-l, while a r ~ O, then n+m = pk + r-i k-I [p-l) j=O~ pj and o(n+m) = j=r ~ aj+r(p-l)

~ a.. j=0 J

(6) becomes an equality.

a.

= p-l.

J For any other choice

In t h a t (6) is a

This shows that all the spots that correspond to

k-i m = [p-l) [ pJ (r = 0,i ..... k-2) lie on the straight line y = (e(Cn)/n)x = x/(p-l), j=r of slope i/(p-l) through the origin, while all other spots are above that line.

I07

The N.p. of Zn(X ) consists, therefore of a single side with h = n = pk-l,

v = e(Cn) = pn__~= pk-i p-I ' t = (h,v) = Pk-I p-I ' and s = n/t = p-l.

According to

Theorem A, all factors of Zn(X ) have degrees of the form ~(p-l)

(~ = 1,2 ..... t).

This proves the first statement of Theorem 1 (e). REMARK.

A slightly stronger form of Theorem A can be obtained, by the remark that

any splitting of f(x) into irreducible factors can be effected by splitting f(x) first into two (not necessarily irreducible)

factors, then splitting any factor

that is not already irreducible into two factors, etc.

At each stage the sum of

the degrees of the factors equals the degree of the polynomial being split.

By

keeping track of this equality one can obtain the mentioned stronger version of Theorem A.

Before stating it, it is convenient to define a new term.

~enever

there are spots on a side of a N.p., these divide the side into collinear segments, that we shall call (following Wahab

[60]), the elements of the N.p.

are the lengths (i.e., the horizontal projections)

If bl,b 2 ..... bg

of the different elements, then

we have the following version of Theorem A, which in a somewhat different formulation appears to be due to Wahab. THEOREM A'.

(Wahab [60]).

fs(X) is of the ! ~

Let f(x) = fl(x)f2(x)

ds =

I

... fz(x); then ds, the degree of

~sjbj, with ~sr = 0,i and 6sr6tr = 0 if s ~ t.

3 For a complete proof of Theorem A', see [60]. If we use this version of Theorem A, we have some added information concerning the possible degrees of the factors of Zn(X), n = pk-l.

Indeed we already know

that the spots on the single side of the N.p. of Zn(X) have the abscissae

(m-l)

k-i ~ pJ (r = 0,I ..... k-2). j=r

It follows that the lengths of the elements are

k k-i k-i k-2 b I = p -p , b2 = p -p ..... b k = p-l. k degree ds = J ~

~sj ( p k - j + l - p k - j )

(~s~~ = 0 o r 1).

p r o o f o f Theorem 1, i s m e n t i o n e d h e r e , analysis (vii)

may l e a d e v e n t u a l l y (n+m)! By (3), cm : m!(n-m)!

By Theorem A', any factor of Zn(X) has a

This result,

because it

seems t h a t

to a proof of Conjecture (n-m+l)(n+m) m

c

m-i

and

not needed in the

this

kind of closer

1.

Cm+l =

(n-m)(n+m+l) (m+l)

Cm"

It follows that (without the earlier restriction q < p/2), e(Cm) = e(Cm_l), unless p divides at least one of the factors n-m+l, n-m, or m.

For pln this can

108

h a p p e n o n l y f o r m ~ 0, o r 1 (mod p ) . horizontal

the spots arrange themselves

rows o f e q u a l e ( C m ) ' S w i t h s t e p s u p , o r down, o n l y f o r m ~ 0 , 1

From t h e f a c t

that

t h e N.p. o f any Zn(X ) s t a r t s

vexity of the N.p.

it

w i t h e(Cml ) > 0, a l l no such r e s t r i c t i o n furthermore

T h i s means t h a t

follows that other

exists

follows that

if

a t P0 = [ 0 , 0 ]

in

(mod p ) .

and t h e downward c o n -

[ml, e(Cml) ] i s a s p o t on a s i d e o f t h e N . p . ,

s p o t s w i t h m2 > m1 h a v e e(Cm2) > e ( C m l ) . and we h a v e s p o t s

at all

integers

F o r e(Cml) = 0

m w i t h e(Cm) = 0.

a s p o t w i t h e = e(Cm2) can be a v e r t e x ,

It

o r i n d e e d even a

spot on a side of a N.p. only if m 2 is the largest value of m for which e(Cm) = e(em2 ) [see Fig. 2 ).

/ /

/_

r

ml

m2

m

Fig.

Fig. 2

3

Quite generally, once e(cj) > 0,vertices or even spots on the sides of a N.p. can occur only at abscissae m > j, such that e(Cm+l) > e(Cm). requires m÷l E 0,i (mod p), or m ~ -I, 0 (mod p).

If n ~ 0 (mod p), this

The distance between two consecu-

tive abscissae of this type is either p, or p-l, or p+l, and this is the distance between consecutive spots on a side of the N.p.

This does not necessarily mean that

the length of the complete side of the N.p. has one of these values.

It is quite

possible to have a side continue in the same direction and hit another spot after another interval of length p-l, p, or p+l.

The side of the N.p. cannot continue

with a smaller slope than on a preceding interval, as that would violate the convexith downwards, but it may continue with the same slope. After the side has traversed a number of spots, it either reaches Pn = [n'e(Cn)] and the N.p. terminates, or else the polygonal line bends again upwards, thus starting a new side of the N.p. We may verify that the spots on any side lead to elements whose lengths satisfy the requirement of Theorem A namely to be of the form ~s, ~ ~ t = (h,v).

To

do this we observe that along a given side the spots that occur as interior points,

109

say Q2' Q3 in Fig. 3 are lattice points (they have integral coordinates). points can appear, if (h,v) = i.

No such

Indeed, vj = (v/h)(mj-ml) with h > mj-m I for mj

not the last abscissa so that v. cannot be an integer, unless v/h is a reducible ] fraction, i.e. (h,v) = t > i.

If v = tw, h = ts, (s,w) = i, then vj = (w/s) Cmj-ml)

with sl(mj-ml) , so that mj-m I = ~js f h = ts, ~j f t, vj = w~j, and finally mj-m I = (vj/w)s = ~js, ~j ~ t,

in agreement with Theorem A.

From the point of view of

Theorem A', m 2-m I, m3-m 2 are the lengths of the elements of the N.p.

It follows

that the length of an element (hence, that of the permitted degrees of a factor) cannot be less than p-l. In case n ~ -i (mod p), the factors n-m+l, n+m, and m are congruent mod p to -m, m-l, and m, respectively.

Hence, e(Cm+l) > e(Cm) only for m ~ 0, or I (mod p)

and the N.p. can have spots on a side of the N.p., or vertices, only for m ~ -I, or 0 (mod p). pressed. S.

From here on the proof proceeds as before and its details may be supThe proof of Theorem 1 {f) is complete.

A given integer n may belong to several of the types considered in Theorem I.

So, e.g., for n = 9, we may ~Tite n = pm (p = 3, m = 2), or n = qp-i {q = 2, p = 5). The prime with respect to which one considers the N.p. will of course be different in each representation.

In general, after selection of a definite prime p, Theorem

A indicates factorizations that may he possible for a given n. So, e.g., for ZlsCX), the N.p.w.r.

to p = 13 has two sides; one of the sides,

of length h I = 2, has a spot in the middle, the other one of length h 2 = 13 contains no spots.

The N.p. contains 3 elements and, by Theorem A (or A'), any polynomial

factor of Zls{X ) may only have degrees obtained as sums of the integers i,I, and 13. We state such a result succintly in the form (15) = (1)(i)(13) and call it a scheme of factorization. The N.p. of the same zls{x ) but with respect to p = 3 leads to {IS) = {9)[4){2).

These two results, while different, are not contradictory.

can be reconciled in two ways.

They

Either trivially, as (1+1+13) = {15) = (9+4+2) if

ZlsCX) is irreducible and none of the potentially allowed factorizations actually exists, or else by observing that both schemes allow the non-trivial factorization (i+1) C13] = {2){13) = C2)(9+4). Two such f a c t o r i z a t i o n schemes, with a common, n o n - t r i v i a l f a c t o r i z a t i o n , w i l l be c a l l e d

oompatible.

Otherwise, we c a l l them incompatible.

I t is c l e a r that i f

one can e x h i b i t two incompatible f a c t o r i z a t i o n schemes, one thereby w i l l have proved the C r r e d u c i b i l i t y o f the polynomial considered.

In the case o f zisCx), the N.p.

w.r. t o p = S leads to the scheme (1S) = ( S ) { 4 ) ( 6 ) .

Also t h i s i s compatible with

C15) = { 9 ) { 4 ) ( 2 ) , because (S+4)(6) = (9)(6) = (9)(4+2).

I t i s , however, incompatible

110

with the scheme (15) = (11(I)(13) obtained from the N.p.w.r.

to p = 13 and we con-

clude that Zls(X ) is irreducible. There should be no need now for a more formal proof of the PROPOSITION i.

If the N.p. of Zn(X ) with respect to ~wc different primes lead to

incompatible schemes of factorization, then Zn(X) is irreducible. 6.

In this section we shall prove Theorem 4. This will be needed in the proof of Theorem 2 and only by use of Theorem 2 are

we going to complete the proof of Theorem i. We already know that Zn(X ) is irreducible if n < 20. 3' and the irreducibility of Zl5(X ) proved in Section 5. if n = p.

This follows from Theorem Also, Zn(X ) is irreducible

If n is not a prime, let Pl < n < P2' with Pl and P2 consecutive primes

and set n = Pl÷kl = P2-k2-1.

Then k I = n-Pl < p2-Pl and k 2 = P2-n-i < p2-Pl-i or,

k 2 ! p2~Pl-2.From Theorem C we know that p2-pl ~ Pl/8 Pl we verify that p2-Pl < -~ 6 = (6/23)Pi).

for Pl ~ $3.

For Ii ~ Pl < 53,

(the largest value is attained for Pl = 23, with 29-23 =

This inequality holds by Theorem C for all Pl ~ ii,

which is more

than sufficient because we are concerned only with n ~ 21 (Pl ~ 191. particular,

that Pl÷2kl < Pl+2(P2-Pl ) < 2Pl.

This insures that the N.p. of Zn(X)

w.r. to Pl has the first k I coefficients not divisible by PI" _

Cl

~l(n÷ll (n-l)n(n+l)(n+2) i! ' c2 = 2!

We note, in

''''' Ck I =

Indeed,

(n-kl+l)"'(n+kl) kl!

=

(Pl+l)...(Pl+2kl and Pl < (Pl ÷I)' (Pl ÷2k) < 2PI"

kl!

We claim that all other coefficients are divisible by Pl to exactly the first power. For k I < m < PI' the factor Pl which now occurs in the numerator is not cancelled by any factor of the denominator.

For m = PI'

(kl÷l)[kl÷21""Pl'"(2Pl) c

= Pl

Pl ~

and pl [ 2 cpl' Pl~Cpl

still holds.

None of the fac-

tots by which cpl has to be multiplied in order to obtain the coefficients

Pl < m ~ n, contain multiples of PI' either in the numerator, and the claim is proved.

We may verify it, in particular,

Cm,

or the denominator,

for m = n,

1ii

cn = (n+l)...(2n) = (Pl+kl÷l)...(2Pl)...(2Pl+2kl) 2P1+2k I < 3Pl.

with Pl+kl+l > Pl and,

It follows that the N.p. of Zn(X ) w.r. to Pl consists of 2 sides:

the first one o£ length kl, on the real axis, with spots at all the integers and the second one from Pkl = [kl,O ] to Pn = [n,l], with no spots on it and of length h2 = n-kl = PI"

By Theorem A this leads to the factorization scheme

(n) = (1)(1)...(l~(Pl)

and it follows that z (x) contains an irreducible factor of n

k I times degree at least equal to PI" We study in exactly the same way the N.p. of Zn(X ) w.r. to P2" (n-k2+l)...n(n+l)...(n+k2) =

In particular,

(P2-2k2-1)...(P2-k2-1)(P2-k2)...(P2-1) =

Ck 2

k2~

k2~

and all coefficients cm with 0 5 m ~ k 2 are not divisible by P2' while

Ck2+l =

(P2-2k2) ... (P2-1)P2 (k2+l) !

first power.

and all successive ones contain P2 to exactly the

Indeed, n < P2' so that no factor o£ the denominator can cancel the

factor P2 in the numerator. p2...(2P2-2k2-2)

In particular, cn = (n+l)...(2n) = (P2-k2)...

and the last factor is less than 2P2.

It follows as above that

Zn(X) contains an irreducible factor of degree at least equal to n-k 2.

If we

combine this result with the preceding one and recall the notation k = k(n) = min(kl,k2), we can state THEOREM 4.

For every n, the BP of degree n contains an irreducible ~ c t o r of degree

at least equal to n-k(n) and can contain r~ factor of degree d with k(n) < d < n-k(n) 7.

Proof of Theorem 2.

As already seen in Section 2, for every a > 0 and x ~ N(E),

there is at least one prime p in the interval x < p ~ (l+e)x.

In particular, it

follows with x = Pl' that there is at least one prime P2 such that Pl < P2 ! (l+e)P lLet Pl < n < P2

with PI' P2 consecutive primes; we want to estimate the largest

possible value of k(n).

In k I = n-Pl and k 2 = P2-n-l, set n = rlPl, P2 = nr2' so

that k I = n(l-i/rl) , k 2 = n(r2-1- i/n).

Now, k(n), the smaller one of kl,k 2, takes

its largest possible value, if n happens to fall between Pl and P2 in such a way as to make k I = k 2.

This means, l-rl I = r2-l-n-l.

rlr 2 = p2/Pl < l*e.

Also, nP2 = nrlr2P I, whence

The most unfavorable situation (large k) occurs when the

112

separation between Pl and P2 is largest, of 2 equations

obtained

i.e. for p2/Pl = i+~.

2+e for rl, r2, we find r I = 2+ i/n

From this we obtain k I = n-Pl f n(l-r i I) = ne-I 2+e k2

=

P2_n_l < n(r2_l_n-l) -

=

n¢-i 2+¢

"

fraction k/n can be made arbitrarily indeed take ¢ > 0 arbitrarily

> _

n(l-

r2 =

(2+ i/n)(l+e) 2+ C

and verify that also

Consequently,

k = k(n) < n~-I

,

_

~

<

small by taking ¢ sufficiently

n

e 2-~a) = Ann , say, with A n

For n + ~, one has a + 0, hence

=

e

2+--T

The

"

small.

small, provided we take n sufficiently

By Theorem 4, the degree of an irreducible n-k[n)

'

By solving the system

We may

large, n ~ N(a).

factor of Zn(X ) exceeds

I-

lim

2+¢c A

i I+ g/2 "

= 1 and this proves the first state-

n

ment of Theorem 2. We now recall that, by Theorem C, we may take a = 1/8 for n ~ Pl ~ 53. for n >- 53,

k(n) n

< 2+ 1/8 1/8

i and A n > I+ i1/16 = i--7

Hence,

16 '. this proves the validity - i-Y

of Theorem 2. 8.

Proof of Theorem

I.

Let us assume first that n > 53.

i.

the theorem the N . p . w . r .

Part II.

It is now easy to complete the proof of Theorem We shall show that under the assumptions

to p (of Theorem i) and the N . p . w . r .

to Pl or P2 (largest

prime less than n, or least prime larger than n) lead to incompatible factorization,

so that the irreducibility

follows from Proposition

If we assume that an(X ) is not irreducible

p-I < k(n).

i.

ducible,

factor,

Similarly,

under the conditions

By Theorem 4, d ~ k(n), so that d • p implies p ~ k(n). is impossible,

or else has a linear factor with a real

either d = 1 or d ~ p.

If we know, however,

that

so that Zn(X ) must be either irreducible

(in fact,

integral)

qp+l is even and (see Chapter I0) the last alternative Exactly the same reasoning,

so that Zn(X) is

of Theorem 1 (c), if Zn(X) is not irre-

the minimal degree d of one of its factors satisfies

p • kin), the last inequality

satis-

From Theorem 4 it follows that d < k(n), so that

Under the assumption p-I > k(n), this is not possible,

irreducible.

schemes of

and n = qp-l, q < p/2, as assumed

in Theorem 1 (b), then the lowest possible degree of an irreducible fies, as seen, d ~ rp-i ~ p-l.

of

applied to the conditions

with p • k(n) or p-i • kin), respectively,proves

zero.

If q is odd, then

cannot occur. of Theorem

the irreducibility

1 (d) and (e),

in those cases.

If, instead of p • k(n), or p > k(n) + 1 we know instead that q ~ 17, then, say, in Theorem

I (d) we obtain p = n/q ~ n/17.

that p > k(n).

If Zn(X ) is not irreducible,

~p, ~p ~ p • k(n), and this contradicts

As proved in Section 7, k(n) < n/17, so then all factors have degrees equal to

Theorem 4 and proves the irreducibility

of

113

Zn(X ).

It is clear that, while not stated in the Theorem,

be substituted

for p-i > k(n) also in Theorem

and 2 are completely proved, 9.

i (b) and (c).

With this, Theorems

1

at least for n 5 20 and n > 53.

It still remains to be shown that Zn(X ) is irreducible

proof we may use all previous results, Theorems

the condition q ~ 17 can

except,

for 2 1 S

n s 53.

In the

of course, those statements of

1 and 2, such as those referring to q f 17, that were obtained under the

assumption n > 53. From Theorem 1 (a,b,c), we know that for all n = pm and n = p ± I, the Zn(X) are irreducible

(without the restriction n > 53).

still remain with eight values. inspection.

After we eliminate these n, we

For each of them it is easy to find k = k(n) by

So, e.g., for n = 33, Pl = 31, P2 = 37, k I = 2, k 2 = 3 and k(33) = 2.

For five of these 8 values of n we may use Theorem

l(d) as follows:

21 = 3.7, k = I;

33 = 3.11, k = 2; 34 = 2.17, k = 2; 39 = 3.13, k = i; 51 = 3.17, k = i. case any factor would have to have a degree d that is a multiple the two prime factors of n, and at the same time have d ~ k. n = 3S, d = ~.Ii ~ Ii and d < 2. ible.

For instance,

This is not possible and hence,

z33(x)

if

is irreduc-

The last remaining three values of n are:

35 = 2.17+I = S.7, with factors of degrees with the factorization

17r, or 17r+l only, k = i, and also (35) = (10){25)

45 = 4.11+1 = 32.5, with factors of degrees with the factorization 50 = 3.7-I, with factors of degrees

With this Theorem

llr, or llr+l only, k = i, and also (mod 5);

7r, or 7r-i only, and k = 2. polynomials

Zn(X ) are irreducible.

1 and Theorem 2 are completely proved.

Proof of Theorem 3.

THEOREM 3".

(mod 5);

(45) = (25)(4)(16)

By Theorem A and Theorem 4, all corresponding

10.

In each

of the larger of

We already proved a weak form of Theorem 3, namely

All BP of degree n, with n < 53

are irreducible.

In order to complete the proof of Theorem 3, we proceed as before. suppress

We first

from the list of all integers n, 53 < n < 400, all primes, odd prime powers,

and integers of the form p ± i.

For the remaining

by looking for the primes closest to n. the patterns of Theorem

integers n, one determines

k(n),

Next, one tries to fit each n into one of

i, such as n = qp+~,

6 = -i, 0, +i, q < p/2 or n = pm-l.

This will impose upon the smallest degree d of a factor of Zn(X) the condition d ~ p, d > p-i or d = i. the resulting

On the other hand, d < k(n) and, if d # 1 and k(n) < p-l,

contradiction

is proof of the irreducibility

of the corresponding

If d = 1 and n is even no linear factor may split off, with the same result.

Zn(X )

114

After integers

the

odd p r i m e

n with

powers

53 < n < 200.

one

representation

not

listed,

never

n = qp+~ exceeds

and n = p ± 1 h a v e These

are

listed

or n = pm-l.

been

eliminated,

in the T a b l e

In all

cases

there

at

a n d k(n),

which

p ~ ii

= 5.11

115 = 5 . 2 3

154

56

= 3.19-1

116 = 4 . 2 9

155 = 5.31

S7

= 3.19

117 = 2 . 5 9 - 1

159 = 3.53

63

= 2.31+1

118 = 2.59

160 = 7 . 2 3 - 1

= 32.7

= 5.31-i

64

= 5.13-1

119 = 7 . 1 7

161 = 7.23

65

= 5.13

120 = 112-1

165 = 2 . 8 3 - 1

69

= 3.23

122 = 2.61

170 = 9 . 1 9 - 1

75

= 4.19-1

123 = 3.41

171 = 9.19

76

= 4.19

124 = 4 . 3 1

175 = 6 . 2 9 + 1

77

= 6.13-1

129

176 = 3 . 5 9 - 1

85

= 5.17

133 = 7 . 1 9

177 = 3.59

86

= 2.43

134 = 2.67

183 = 3.61

= 3.43

= 7.52

87

= 3.29

135 = 8 . 1 7 - 1

184 = 5 . 3 7 - 1

91

= 4.23-1

141 = 3.47

185 = 5.37

92

= 4.23

142 = 2.71

186 = 6.31

93

= 3.31

143 = 2 . 7 1 + 1

94

= 2.47

144 = 5 . 2 9 - 1

188 = 4 . 4 7

95

= 5.19

145 = 5 . 2 9

189 = 1 0 . 1 9 - 1

146 = 2 . 7 3

195 = 2 . 9 7 + 1

99

=

105

= 2.53-1

?

147 = 4 . 3 7 - 1

iii

= 3.37

153 = 8 . 1 9 + i

From

Table

ducible

into

by Theorem

that

k<99)

follows.

factor

Also

one must

still

n = 175,

the

integers

and

largest

have

4.

factors

= i and

143,

175,

u p to 200

grees

less

than

6 exists.

grees

less

than

12 exist.

factor

if not

factor 195,

less

all

than

irreducible,

of degree odd

linear

98,

= 11.17

= 15.13

and n = 195,

one

of t h e

however,

largest I0. can

whence

is irre-

prime

On the split the

divisor

other at m o s t

irreducibility

and of type n = q p + l

factors.

is P m = 7 and b y T h e o r e m

F o r n = 143

into

z99(x),

P m = Ii is the

of degrees

that the}, do n o t c o n t a i n prime

fit r e a d i l y

The polynomial

z99(x),

and

187 = 4 . 4 7 - i

= 9.17

Indeed,

an i r r e d u c i b l e

for n = 63, show

= 11.13

for n = 99.

1 (f) a n d T h e o r e m

= k 2 = 101-99-1

a linear

all

i, e x c e p t

so that z 9 9 C x ) c a n n o t

o f 99, hand,

of Theorem

is

1

55

patterns

61

least

6.

Table

1 it a p p e a r s

remain

i, e a c h w i t h

F o r b o t h n = 63 and 1 (f) n o

P m = 13 and n o

factors factors

of deof

de-

115

~ile

not needed for the purpose on hand,

factorization ponding

it is instructive

to see to what

schemes one is led in these S cases, by the N . p . w . r .

largest prime factor Pm"

(143) = (13) ( 1 2 ) ( 1 1 8 ) ,

One obtains

(99) = (!i)(i0)(78),

(17S) = ( 7 ) ( 7 ) ( 4 9 ) ( 4 9 ) ( 4 9 ) ( 6 ) ( 8 )

and (19S) :

to the corres(63) = (7)(7)(49), (139(139(169).

One may remark that, while n : 63 and n = 195 lead to d ~ Pm' for n = 99, 143, and 175, one obtains nothing better than d ~ Pm-l, so that the statement of

Theorem

1 (f) cannot be improved without some added restrictions. The verification

that also all Zn(X ) w~th 200 < n < 400 are irreducible

equally easy - and dull•

For 201 < n ~ 300, all integers fit one of the patterns

Theorem i, except for 209.

Here Pm = 19, k(209) = i, so that z209(x)

by Theorem 1 (f) and Theorem 4. to be considered

separately,

of

is irreducible

In 301 < n < 400 we find four values of n that have

namely n = 323 (Pm = 19, k = 6), 324 (324+i = 52 13

Pm = 13, k = 6), 351 (Pm = 13, k = i) and 391 irreducible

is

(Pm = 23, k = 2).

Three of them are

by Theorem 1 (f) with n = qp and Theorem 4, while for n = 324 one uses

Theorem 1 Cf) with n = qp-i and Theorem 4. In spite of the ease with which we are able to prove the irreducibility single polynomial

Zn(X),

a general method still eludes us.

of any

Such examples as n = 99

and n = 209, etc., show that we cannot take it for granted that we shall be able to fit all integers n, into any finite set of patterns, can be proved.

for which the irreducibility

By working out a large number of N.p., one is led to make certain

conjectures. There also is some hope that the method used in the proof of Theorem i (e) and 1 (f) could he CONJECTURE

2.

perfected,

in order to prove

If p' # p" and p'p"In,

lead to incompatible

then the N.p. of Zn(X) w.r. to p' and to p"

schemes of factorization.

It is clear that the proof of Conjecture proof of Conjecture

i.

2 would go a long way towards the

In fact, the only integers not covered would be those of

the form n = 2mp k and which also cannot he represented priateiy

small q.

as n = qp"' ~ i, with appro-

CHAPTER 12 THE GALOIS GROUP OF B.P. I.

The Galois group of the BP of degree n will be denoted by Gn, the symmetric

group on n s}~bols by S n and the corresponding

alternating

group by A n .

A group is said to be of degree n, if it is isomorphic to a permutation n symbols.The

terms "transitive",

group theory as defined, over the rationals, The material

e.g.,

"primitive",

in [9].

group on

etc., have their usual meaning from

The term irreducible will mean irreducible

as in Chapter ii.

of this chapter is essentially contained

in [53, Section 5] and in

[54], but several proofs have been reworked. The main result is THEOREM i.

For every n, if G n is transitive,

This theorem can be rephrased. transitive, THEOREM i'

then G n = S n.

Indeed, the Galois group of a polynomial

if and only if the polynomial

is irreducible.

If yn(Z) i8 irreducible, then G n = S n.

We saw in Chapter ii, that a large class of BP consists polynomials

and we formulated there Conjecture

reducible.

This leads us to state

THEOREM

2.

i".

in fact

of ireducible

i, according to which all BP are ir-

Cor~ecture 1 of Chapter ii ~plies tb~t, for every n, G n = S n.

Theorem i' is equivalent

sufficient

is

Hence, we may state

to Theorem

to prove Theorem i'.

1 and implies Theorem I".

Hence, it is

For that we shall need several known results.

in Chapter ii, we shall have to content ourselves with the statements

As

only and refer

the reader to easily available proofs in the literature. THEOREM A.

(I. Schur

[SZ], Theorem V). f(x) =

n ~

j=o

Let

a.x n-3 , a.

J

rational integers,

J

have the discriminant D and be irreducible, and let G n be its GaZois group.

Let us

assume that there exists a prime p, which satisfies the following conditions:

pm]D with m > n, P]an, p2~a n. Assume also that if f(x) ~ xkg(x) p~d.

(mod p) and d is the discriminant of g(x), then

Any such prime is a divisor of the order of GnIf, furthermore, n/2 < p < n-2, then G n = A n in case D is a perfect square,

Gn = Sn, otherwise.

117

THEOREM B.

(Dedekind;

see [$2]).

Let us suppose that the polynomial f(x) splits

modulo p into a product of r irreducible polynomials fi(x) (i = 1,2 ..... r), incongruent in pairs modulo p.

Then the Galois group of f(x) contains at least one per-

mutation of r cycles, such that each cycle corresponds to one of the factors fi(x), and the order of each cycle is equal to the degree of the polynomial to which it corresponds. THEOREM C. (Burnside;

see [9],XI).

Ef n (> 3) and 2n+l are both primes, then

(i)

any triply transitive group of degree 2n+3 contains the alternating group (§ 165, page 214, Exercise). (2)

For n = 4, there are no primitive groups, except A 4 a n d s 4 (§ 166, page 214,

(ii)). [3)

Every triply transitive group of degree seven contains A 7 (§ 166, pages 216-218,

(v); see also [52(a)], p. 449, or [S2(b)], p. 197).

(4)

If pe is the highest power of p contained in n! and if p < 2n/3, then pe-I

is the highest power of p that divides the order of any primitive group of degree n, which does not contain A n 6§ 160, pages 207-208). THEOREM D. (Cauchy; see [30], p. 74).

If p divides the order of a group G, then G

contains an element of order p. THEOREM E.

(Jordan

[S2(b)], p. 196).

[39], Note C; see also [9], p. 214, Theorem I and [5Z](a), p.448;

If G is a primitive transitive ~arouv~ of dearee~ n and p > n[ ~s" a

divisor of the order of G, then the degree of transitivity of G is at least n-p+l. 3.

The method of proof follows, in general outline that of Schur ([52]), and it is

largely based on Theorem A. LS~MAI.For Proof.

(n+l)/2 ~ p < n,

We also shall need the following 0n(X ) ~ xPOn_p(X ) (mod p).

Trivially n z (n-p) (mod p); consequently,

for m < n-p+l, one has

m < n - (n+l)/2+l = (n+l)/2 < p, so that a(n) = (n-m+l)...n(n+l)...(n+m) m 2mm! ((n-p)-m+l)...(n-p)(n-p+l)...((n-p)+m) 2mm!

= a(n-p) m

(rood p).

On the other hand, for n-p ~ m < p, a (n) contains the factor p in the numerator, m but not in the denominator and a (n) E 0 (mod p). m

For p ~ m ~ n, the denominator

contains the factor p to exactly the first power (because n < 2p-i < 2p), while the numerator contains at least two multiples of p (p itself and 2p, because n ~ p,

118

m ~ p imply n+m ~ 2p). OnCX) =

It follows that

n [ a Cn) xn-m m m=0

- xp

_

n-p [ m=0

aCn)x n-m

nip a ( n - P ) x ( n - p ) - m m:0

_

m

n-p [ m=0

= xP0n_p(X )

aCn-P)x

(n-p) -m+p

m

(mod p ) .

m

We now verify that the conditions of Theorem A hold for 0n(X ) .

Specifically, we

shall do the following: (a)

compute the discriminant Dn of On(X);

(b)

show that for n > 14,and also for n = i0, there exists a prime p such that

(2n-1)/3

< p < n-2,

(c)

show t h a t

(d)

verify

@n(X) ~ xPOn_p(X ) (mod p) and P2Dn_p;

that

Dn i s n o t a p e r f e c t

Theorem A can be a p p l i e d be v e r i f i e d

(in fact,

ing condition follows,

Theorem D i t

condition

of n,

Gn # An, so t h a t

still

conditions

can

than a correspond-

f o r t h o s e n, G n ~ A n .

From (d) i t

i n d e e d Gn = S n.

h o l d f o r some p t h a t

follows

for which these

(h) i s more s t r i n g e n t

o f Theorem A) and we c o n c l u d e t h a t

however, that

then it

square.

to those values

the present

If those conditions (b),

a (n) , p 2 ~ a ~ n ) ; n

and a l s o pn]D n , p

falls

from Theorem A, t h a t

t h e n f o l l o w s t h a t Gn c o n t a i n s

outside

p divides

the range required

t h e o r d e r o f Gn.

an e l e m e n t o f o r d e r p.

by

By

I f a l s o p > n/2,

then the group G n (transitive by assumption) is obviously primitive.

Theorem E then

insures a rather high degree of transitivity, which, in some cases, suffices to show that G n ~

An, so that, by (d) above,

G n = S n-

In the few remaining cases we shall succeed to reach that conclusion, by appealing to Theorem B.

4.

(a)

We r e c a l l

mials f(x) =

(see,

e.g.

[Sg],

§ 28) t h a t

R(f,g),

the resultant

o f two p o l y n o -

n n-i m . ~ c.x of degree n and g[x) = [ d.x m-3 of degree m, is a polynoi=0 I j=O 3

mial function of the coefficients ci, dj (i = 1,2 ..... n; j = 1,2 ..... m) that vanishes n m if and only if f and g have a common zero. In fact, R(f,g) = Cod m n0i _ ~ 1 I I (xi-Yj), "= j=l where the x i range over the zeros of f(x) and the yj over those of g(x). n m R(f,g) = c~ ~.= g(xi) = (-l)nmd~ ~.= f(yi) = (-l)nmR(g,f).

Clearly,

We shall be concerned

here with monic polynomials, so that c O = d o = I, and nm will always be even, so

119 that these formulae simplify accordingly. The discriminant and f'(x).

D of a polynomial

f(x) is, essentially,

More precisely,

= (-1)n(n-1)/2Co D.

R(f,f')

(i) n

It follows that D = c~ -2 ~ - [ f ' ( x i )

: (-1) n(n-1)/2^2n-2 ~0

i=l c 2n-2 o

the resultant of f(x)

I 1I i<j

(xi_xj)2 '

where the x.'s ~ d

z

[ I (x i-xj) i#j

x.'s run through the zeros of f(x).

The

J

last expression is often taken as definition of D. In order to compute Dn, the discriminant of @n(X), we first compute R(@n_l,@n). By (3.5), @n+l(Xi) = X~@n_l(Xi)

for each zero x i of 0n(X).

over all these zeros and recalling also that O R(On'0n+l)

inductive argument, R(@I,eO)

is monic, we obtain

n 2 ~n • (a (n) = i=l] i On+l(X i) = (xlx 2. -xn) i:i en-l(Xi) : - n )2R(0n'en-i )"

Either n, or n-I is even; hence R(en,0n_l)

and

n

By taking the product

R(On'@n+l)

=

= R(0n_l,0n)

and we obtain by an obvious

{a(n)a (n-l) .a~l)}ZR(Ol or, by using (2 8) n n-i "" 'O0)' "

= I, n-i R(0n,@n_ I) = {]--F -(2k): }2. k=l 2kk!

(2)

On the other hand, by (3.8), On(Xi) = -XiOn_l(X i) and, if we take the product over all zeros x i of en(X ), we obtain n

n

R(@n,@n) = [ I @n(Xi) : ( - l ) n X l . . . X n i=l

@n_l(Xi) = (-l)na~n)R(@n,@n_l).

"=

By (i), the first member equals (-l)n(n-l)/2D D

n

= (-l)n(n+l)/2a(n)n

(3)

Rn(@n,@n_l),

D n = (-I)

n(n+l) 2

n

and it follows that

so that, by (2),

n-i (2n)! {T-T 2nn! k=l

(2k)! - 2kk!

~2 ~ •

The denominator in (3) may be written as (2.4.6...2n)

n-i ! I (2.4...2k) 2 and a k=l

fairly easy counting atgument shows that each even integer 2m occurs exactly to the power 2n-2m+l. so that ]Dn!

This is exactly the same power to which it occurs in the numerator, reduces to the product of the odd integers in the numerator.

The odd

factor 2m-l (1 ~ m ~ n) occurs exactly once in (2n)!, and also once in each factorial (2k)! for m < k < n-l, so that 2(n-l-(m-l))+l=

2n-2m+l is the exact power to which

120

2m-i occurs

in [3).

n + Dn = ~ T m=l

(4)

(b)

It follows

[2m-l) 2n-2m+l

9 Set n-3 = ~ x and recall

the interval

9 x 48, there

1

is a prime

is a prime p that satisfies

in

the in-

equalities 2n-i 8n 8 9 3 < ~-- - ]- = x 3 + ~ • 48 = 57.

provided

14 < n < 56 there also exists

It is an easy exercise

such a prime

(e.g.

for n = 20,

2n-i 19 Also for n = I0 we find ~ = - ~ < 7 < 8 = n-2. such prime divides

D n to the exact power

that Pla(n)n , p2~an(n).-

Indeed,

so that the last inequality (e)

By the Lemma, 2n-2p-l,

(d)

All primes

except

Next, by ~4),

that divide

IDn_pl

With this,

5.

2n is even,

<_ p < n, a f o r t i o r i

for

= 12n-2p-132n-2p-3...(2n-2p-3)3(2n-2p-l) 3p > 2n-l,

I"

and P2Dn_ p.

D n to an odd power,

of Theorem A are verified group

For n = 13, 12, ii and n ~ 9, no primes

irreducible

to the identity.

and the discriminants

For the remaining least the weaker all 4 < n < iS,

and it follows

G n of the irreducible

so that,

that for

BP @n(X)

values

inequalities

(2n-i)/3

is the

 n/2, the respective

the G ' s n

so that

one such prime

Specifically,

are transitive

and, as also in

By Theorem

for

P4 = S, P6 =

ii, that all BP

Gn'S are primitive.

£n ~ n-Pn+l'

at

from Chapter

We may recall

hence,

are

hence G 2 = S 2 and G 3 = S 3.

for n = 5 and n = II.

are in fact irreducible,

£12 ~ 2, and £13 ~ S.

in C2n-i)/3

of n, we try to find primes p = Pn that satisfy

(n ~ 10), except

of transitivity

exist

For n = 2 and n = 3, the polynomials

are not squares;

P7 = 5, P8 = P9 = 7, P12 = PlS = ii.

degrees

(2.8),

group S n.

G 1 = SI, both reducing

involved

from

square.

n > 14 and also for n = I0, the Galois symmetric

(n+l)/2

(2n)'. only once, divide

all conditions

however,

that any

to 2n < 3p.

by assumption,

for n = I,D n is not a perfect

for all

_ 13 < 17 < 20-2)

It also follows

p < n-2 < n+l < 2p < 2n < 3p+l;

is equivalent

because,

2.20-I 3

By (4), it follows

2n-p _> n+2 > n.

8nCX ) - xPen_p(x ) (mod p) for

(2n-i)/3

to verify that

E, the Gn's have

£4 ~ 2, £6 ~ 2, £7 ~ 3, £8 ~ 2, £9 ~ 3,

121

By Theorem C (2) there are no primitive groups of degree 4, except for A 4 and $4, so that G 4 ~

A4

and (recall (d), whence

G 4 # A4), G 4 =

S 4.

By Theorem C (3), the only triply transitive groups of degree seven are A 7 and $7, whence, as before, G 7 = S 7.

For n = 13 we invoke Theorem C (I), observe that

Pl = 5 and P2 = 2PI+I = ii are both primes and conclude that G13, with n = 2Pi+3 = 13, contains 6.

AI3, so that GI3 = S13.

The only values of n for which Theorem I' has not yet been proved are

n = 5,6,8,9,11,

and 12.

In these six cases we shall use Theorem B and observe that we have O5(x ) ~ (x3+x2+4x+5)(x-2)(x-l)

(mod 17)

O6(x ) ~ (x3+4x2-3x+6)(x2+x-l)(x+3)

(mod 13)

O8(x ) ~ (xS+9x4-7x3+8x2-7x+4)(x2+2x+2)(x+6)

(mod 19)

@9(x) ~ (xS+x4-Sx3-x2+Tx+13)(x3+12x2-13x+3)(x+3)

(mod 29)

(s)

@ll(X) ~ (x

7+

39x

6+

35x

5+ ~ 4+ 3 2 l.lx 148x 60x +25x+S3)(x4+27x3+14x2+74x+23)

@12(x) ~ (x7÷2x6+27xS+28x4+30x3+38x2+lSx+3)(xS+76x4+65x3+63x2+29x+lS) In each of these six factorizations,

(mod 89)

the polynomial factors on the right are mutually

incongruent and irreducible modulo the respective prime modulus. factorizations

(mod 149)

The first three

were done by hand, by the present author, in 1949; O9(x) has been

factored modulo 29 by M.Ne~Ean and K. Kloss on the IBM 704 at the National Bureau of Standards,

in October 1961, while @ll(X) and Ol2(X ) have been factored by J.D. Brill-

hart and R. Stauduhar,

on the CDC 6400 of the University of Arizona, in February 1969,

by using an algorithm of E. Berlekamp. It is, of course, quite easy to verify these factorizations by hand) once they are found.

The discovery of a prime that leads to a usable factorization and the

determination of the factors is perhaps less trivial.

The verification that the

factors are indeed irreducible modulo the respective prime is an operation of intermediate difficulty.

For whatever interest it may present, the factorization of

@8(x) will be discussed in detail in Section 7 and is fairly typical for all similar factorizations. By Theorem B we know that for n = 5,6,8,9)ii, tain at least one permutation

and 12 the corresponding G n con-

Pn' of the following structures,

respectively:

122

P5 = (a,b,c)

P9

P6 = (a,b,c)(d,e)

PII = (a,b,c,d,e,f,g)(h,i,j,k)

P8 = (a,b,c,d,e)(f,g)

PI2 = (a,b,c,d,e,f,g)(h,i,j,k,£).

P5 leaves two symbols invariant;

= (a,b,c,d,e)(f,g,h)

P6' P8' and P9 each leave one symbol invariant,

while PII and Pl2 move all symbols. G 5 contains a cycle of order three, hence the order of @5 is divisible by 3; as 3 < (~) • 5, 3[5:, and 32~5: , Theorem C (4) with e = 1 shows that G S ~

AS, whence

G 5 = S~. D With

P6' G6 contains also P63 = (e,d), a transposition.

that contains transpositions,

G 6 = S 6.

With P8' G8 also contains P82 = (a,c,e,b,d), of G 8 is, therefore,

As a primitive group

an element of order 5

The order

divisible by 5 and, by Theorem C (4) with e = l and the remark

that 5 < (32-) .8, it follows that

GS~A8,

i.e.

G 8 = S 8.

Similarly, G 9 contains P93 of order 5(< (~) • 9), 519!, 52~9 ' and we conclude as 4

before that G 9 = S 9.

For Gll we use Pll of order 7(< ( ) • ll) and for

P125 also of order v, to reach the same conclusion.

Gl2 we use

With this the proof of Theorem l'

hence also that of Theorem 1 and of Theorem I", is complete. 7.

In

order to prove that G n ~ A

desired factorization.

n, the first step is to decide on the type of a

For G 6 we could show that it contains a transposition,

in general, this cannot be achieved.

In that case we try to find a Pl < 2n/3, such

2, that Plln!, Pl~n., in order to apply Theorem C (4).

For n = 8, there is little

choice; only Pl = 5 can be used, so that we aim at a factorization existence of an element of order 5 can be inferred. p as a modulus in Theorem B.

which makes Theorem B inapplicable.

because

Next, we have to choose a prime

(mod p) already exhibits a repeated factor,

For n = 8, the smallest prime to be tried is,

We first look for linear factors modulo 17, but there are none,

88(k) ~ 0 (mod 17) for all integers k.

decomposition

from which the

No primes less than 2n can he used, because for them

a~n)~ a~ n) ~ 0 (mod p) and en(X ) ~ xZg(x)

therefore p = 17.

but

e8(x) ~ f2(x)f6(x)

factor f6(x) of degree 6.

Next one may want to consider a

(mod 17), with a quadratic factor f2(x) and a

This, however, would not be useful.

Indeed, in the

123

absence of linear factors one cannot obtain a factor of degree 5, by further factoring of f6(x).

Hence, we attempt to split @8(x) as a product f3(x)fs(X)

If we write monic polynomials determined coefficients,

(mod 17).

f3 and f5' of degrees 3 and 5 respectively,

we are led to a system of congruences modulo 17.

out that this system has no solution.

As a factorization

with unIt turns

04(x ) ~ f4(x)f4(x)

would not be useful, we pass to the next higher prime p = 19.

(mod 17)

Here we find that

e8(-6 ) ~ 0 (mod 19) and, by factoring out x+6, we obtain (6)

@8(x) ~ (x+6)(x?-8X6-6xS-7x4-Sx3+6x2-6x+8)

(mod 19).

By direct substitution of integers k (k = 0,1,...,18)

into the factor of ?-th degree

we verify that there exist

Hence, we look for a congruence

no other linear factors.

of the form (7)

xT-8x6-6xS-Tx4-Sx3+6x2-6x+8

~ (x2+~'x+B')(xS+~x4+Bx3+~x2+~x+E)

This leads to the following system of simultaneous

congruences,

a+a' ~ -8

y+~'B+~'~ ~ -7

E+~'~+B'y ~ +6

B+~'~+~' ~ -6

6+~'y+8'B ~ -5

~'~+B'6 ~ -6

(mod 19). all modulo 19: ~'~ ~ 8.

This system has a solution modulo 19, which, when substituted in (6), leads to the third congruence of (S). Before we can use Theorem B, we must still verify that all factors are irreducible [mod 19).

As there are no linear factors, the only further possible decomposi-

tion would be xS+gx4-Tx3+Sx2-Tx+4

~ (x2+~'x+~')(x3+~x2+Bx+y)

We are again led to a system of congruences

(mod 19) in the undetermined coefficients

and it turns out that this system has no solution. polynomial of

(mod 19).

It follows that the fifth degree

is irreducible modulo 19 and Theorem B may be applied to the factorization

88(x) in (5).

CHAPTER 13 ASYMPTOTIC PROPERTIES OF THE BP i.

Asymptotic properties of BP were considered already in [53].

It was sho~m there

that, for constant z # 0 and n ÷ ~,

(i) Moreover,

(2)

(2n)! znel/Z. 2nn!

yn(Z) ~

[yn(Z)

for n > i,

(2n)! znel/Z[ _< (2n)! [zln-Zel/[z [ = kn(Z )[(~n)! znel/Z[, 2nn~ 2n+2n!(n-1) 2 n!

1 where kn(Z ) - 4(n-l)

[z

-2

exp(~

i

i - ~)[ and kn(Z ) + 0 for fixed z # 0 and n + ~.

If one uses Stirling's formula for the factorials,

(i) is seen to be equivalent

to

.2nz. n ~ yn(Z) ~ (--~--)

(3)

Formula (3) has been generalized by Obreshkov b = -i.

If in Obreshkov's

el/z [82] to the polynomials Yn(X;a,b) with

formula one makes the change of variable -2x = z needed

to reduce b to its standard value b = 2, and recalls that in his notation m = a-2, the formula of [82] reads (4)

yn(Z) ~

.2nz.n~a-3/2 1/z (--~-) z e .

Formula (4) has been improved further by K. DoPey [48], who proved that (S)

yn(Z) = (--e--)'2nz'n~a-3/2z el/Z''ll- l+6(a-2)(a-l+2z-l)+6z -2 + 0(I a2j }. 24n n It is clear from Do~ev's work that the terms of the large bracket are only the

first few terms of an asymptotic series, which, in principle, can be computed to any desired number of terms. Formulae

(I) to (5) are meaningful only for z # 0.

In [53] the remark is made

that close to z = 0, yn(Z) behaves essentially like en2z/2, and even an error term is computed.

In fact, it is clear from (2.10) and (2.8) that if one wants to approxi

mate ynCZ) in the vicinity of z = 0, then the natural candidate is en(n+l)z/2

In-

deed, ynCZ) = l+(nCn+l)/2)z+((n-l)n(n+l)(n+2)/2.4)z 2 + 0(z 3) and e n(n+l)z/2 =

l+(n[n+l)/2)z+(n2Cn+l)2/2.4)z 2 + 0(z3), so that an elementary computation leads to THEOREM i.

For fixed n a ~

[z] ÷

0,

yn(Z) _ en(n+l)z/2 = _ 1 n(n+l)z 2 + 0(z3). 4

125

in particular,

yn(Z) r - j e n ( n + l ) z / 2 .

This result is better than that of [53], but it seems that neither of them has any particular significance.

Therefore we shall not consider them in what follows

2nn~ z-n and turn to discuss the approximation of the exponential e I/z by (2-~., yn(Z) (for 2nn! z ~ 0), or, equivalently, that of e z by ~ formulae (i) to (5).

en(Z ) (for z finite), as expressed by

This approximation is precisely the property of BP used in the

proof of the irrationality of e r for r rational (see the Introduction, Problem I, and Chapter 14). 2.

It is, of course, sufficient to prove (5), from which the other formulae readily

follow

This proof, which is the object of Section 5, uses rather deep results,

such as Docev's estimate of the zeros of yn(Z,a)

(see Theorem 10.6).

For this rea-

son we present in sections 2 to 4 an independent, elementary, although somewhat lengthy , proof of the following slightly stronger version of (i) and (3).

THEOREM 2.

For constant z ~ O,

yn(Z) = c 0 z n e l / Z + Rn(Z),

(2n)! with c o = 2nn! 0 < k(n,z) <

~a(n) =

n

,

and

where, for

1=l-2el/Izl 2(2n-i)

lim n

,~.

n-3/2" , with

lim ->

with

and, for n ~ 00,

an = 0. ¢~

From Theorem 2 one obtains (i), by the observation that

IRn(z)/yn(Z)l = 0.

Also the error term estimate in (2) is an immediate conse-

~

quence of t h e bound on IRn(Z) l (2n) ! c O = 2nn, so t h a t 3.

IRn(Z) I < c01z] n ~(n,z)

(l+~(n)); here 0 < ~(n) ~ 2

n

REMARKS.

Iz[ e 2/n, one ~ s

in Theorem 2. ~

Finally, by Stirling's formula,

( 2 n ) 2 n + l / 2 e -2n 2nnn÷l/2e-n

2n+l/2e-n =

,

(3) f o l l o w s from (1).

Proof of Theorem 2.

Yn (z) = c o

In Chapter 2, by u s e of ( 2 . 1 0 ) , we o b t a i n e d t h e r e s u l t

n (2n)' X h(n)z n-m, with c o = " m= 0 m 2nn.,

and

b(n) m

2m = m'

(2n-m)! (n-m) '

n! (2n) !

126

Hence,

n yn(Z) = c0zn{ ~ m=0

z m'

-m

= c0zn{el/Z

_[

n

2m (2nzm_) !n,.~ Z -m

X (i- (n-m) : (2n) ," F-T- }

m=0 ao

= c0znel/Z

IRlCz) I <-

Iz[ -n-1 (n+l) !

z (l-f(m,n)) ~

~ (i+

)]

, say,

Also,

]zl -n-i

Iz] -2

n+2

-m

m=0

- c0z n ( R l ( Z ) + R 2 ( z ) )

2mn~ (2n-m) : (2n) ] (n-m) ' "

where f(m,n)

n

-m

~ z m.-~- + m=n+ 1

+ (n+2) (n+3)

"'')

i [z[-1

-< (n+l)' l-

eor

I zl

< 2[zl -n-I (n+l)2

n+2

> 2/~.

In o r d e r LEb£%L~ i.

For' 0

m

<

n,

<

(6)

0

with equality ~br

m

=

<

1 -

f(m,n)

<

'

!e2(z) l IZl-2 n-2 ~

3

< m

<_ n ,

(6), it follows that, for z ~ 0, n

Izl -2 ml

m(m-l) < m~2 2 ( 2 n - 1 )

izl-2 .

2 (2n-l)

m <_ n.

If we accept for a moment

2(2n-i) m! 0

re(m-l)

<

and In = 1; the inequality on the left is strict for

0

and on ~he right for 4

n

-m z we use the following = ! y (l-f(m,n)) m.--]--[, m=O

la2(z) l

to estimate

=

1=!-2 el/,.= I

m~0 i~l-m

:< 2(2n-i)

m~

= 2(2n-i)

If we set R3(z ) = Rl(Z ) + R2(z), then

iR3(z) I

21~l -n-1 -<

(n+l)! +

[zl-2el/]:! 2(2n-i)

=

[zl-2el/Izl 2(2n-i)

4]zl-n+l(2n_l) e-l/Izl} {i+

(n+l)!

"

For positive u = Izl -I, the function eUu l-n increases to infinity for u ÷ ~ for u ÷ 0 (n > i) and it attains its minimum for u = n-l, when it equals Consequently,

One e a s i l y

]R3(z)l < [z1-Zel/Izl 2(2n-1) (l+~(n)),

verifies

that,

(2n+l)n ~(n+l) = ~(n) (2n-l)(n+2) p(n+l)

< p(n).

with

~(n)

=

4(2n-i) (n+l):

.

and also e

.n-1

[n_--~)

.n-l.n-i IT ) "

f o r n > 1, . n .n-1 (n---~)

1 2n+l e < 2n+3-2/n

Also, by direct computation p(1)

~(n), so that, for n > 2, (=

lira 4(2t-i) F(t+2) t÷l

(t-l.t-l) --~-)

=

2,

.

127

~(2) = 2/e, ~(3) = I0/3e2,...,"

hence,

8 ~ n-3/2 f o r n + ~. 2~7~-

mula, ~(n)~

~(n) _< 2.

It

follows

Finally, by use of Stirling's

for-

yn(Z) = c0znel/Z÷Rn(Z) ' with

that

Rn(Z) = c 0 z n R 3 ( z ) a n d

IRn(Z) I

(7)

Ill n

< c0

-

Lzl2 (-Net/Ill 2n-1)

(l+~(n))

I f we now c o m p a r e t h e s e c o n d member o f (2) w i t h consequence

1 of Theorem 2, provided that 4(-~_I) ~

"

(7),

l+~(n) 2(2n-i)

This obviously holds for large n, because ~(n) = 0(n-3/2); any n, provided that formula for ~(n). for all n > i.

8(2n-l)(n-l) (n+l).,

n-I n-I (~) (n) < 2(n-l) in fact, this holds for

as follows by using the explicit

one may check that this is indeed the case

The proof of (I), (2) and of Theorem 2 are complete,

except for the

proof of Lemma i.

4. LE~

I n tile p r o o f 2.

(8)

o f Len~a 1 we s h a l l

need

The inequality

(2n-l-

~)(2n-2)(2n-S)...(2n-m+l)

f (2n-2)(2n-4)...(2n-2m+2)

holds f o r 3 < m < n a n d is strict f o r 4 < m 5 n.

Proof of Lemma 2.

For m(m-l)

left hand side is non-positive, (8) reads

(2n-4)(2n-2)

~ 4n-2 the statement

is trivial, because then the

while the right hand side is positive.

f (2n-2)(2n-4)

For m = 3,

and reduces to an equality.

We now assume that (8) holds for some m with 3 < m < n and show that it also holds for m+l.

If we replace m by m+l, the first factor in (8) becomes

2n-l- (m+l)m = 2n-l- m(m-l) 2 2 (2n-l- m(m-l) 2

-m)(2n-2)(2n-3)

This is equivalent (2n-l- m(m-l) 2

-m, and we have to verify that

"'"

(2n-m+l).(2n-m)

< (2n-2)(2n-4) .

(2n-2m+2)(2n-2m) . . .

to

(2n-2)(2n-3)

.

(2n-m+l).(2n-m)-m(2n-2)(2n-3) . . . . .

(2n-m+l)(2n-m)

< (2n-2)(2n-4)...(2n-2m+2)(2n-2m). By the induction assumption

(8), it is sufficient

to prove that

(2n-2)(2n-4)...(2n-2m+2).(2n-m)-m(2n-2)(2n-3)...(2n-m)

( 2 n - 2 ) ( 2 n - 4 ) . . . (2n-2m+2) • ( 2 n - 2 m ) , or

128

(2n-2)(2n-4)...(2n-2m+2).m

< (2n-2)(2n-3)...(2n-m)'m.

This clearly holds, because the number of factors is the same on both sides of the inequality and corresponding factors are either equal (namely the first and last), or are larger on the right, than on the left. m(m-l) For m = 0 and m = i, f(m,n) = 1 and 2(2n-i) = O,

Proof of Lemma i. equalities hold

•

For m = 3, (6) becomes 0 < l-f(3,n) =

To complete the proof by induction, we assume that for m-l.

so that both

3 = m(m-l) 2n-i 2(2n-i)

"

3 < m-i < n and that (6) holds

We show now that (6) also holds if m-i is replaced by m.

on the left is immediate, because f(m,n) = 2(n-m+l) 2n-m+l

The inequality

f(m-l,n) < f(m-l,n)

The

inequality on the right is equivalent to 2m-l(n-l)(n-2) 1

-

(n-m+l) "'"

=

1

-

(2n-l)(2n-2)...(2n-m+l)

(2n-2)(2n-4)..(2n-2m+2) " (2n-l)(2n-2)...(2n-m+l)

~

m(m-l) 2(2n-i)

-

"

By clearing of denominators we obtain (2n-l)(2n-2)...(2n-m+l)-(2n-2)(2n-4)...(2n-2m+2) or (2n-l- ~ ) ( 2 n - 2 ) ( 2 n - 3 ) . . . ( 2 n - m + l ) on account of Lemma 2. (i), (2), (3) and S.

< m(~-l____~)(2n-2)(2n-3)...(2n-m+l),

< (2n-2)(2n-4)...(2n-2m+2),

which holds

This completes the proof of Lei~la I, and with it that of

of Theorem 2.

In this section we shall give the proof of

THEOREM 3. (9)

(Do~ev [48])•

yn(Z'a) '

=

For oonstant

.2nz,n~a-3/2 L--~) z e I/z~. tl-

1 24n

z ~ 0 ~a~d n + ~,

z -2 4n

f"a-2"'a-l+2z-l" ~f ~ + 0(-Tj 1 ~} 4n " n

In fact, in order to illustrate the remark made after (5) we shall actually improve (9) to (14) with the better error term 0(n-3). Proof of Theorem 3.

From (2.27) and with the notations of Chapter 2,

Yn (z;a) = d(n) n

~ {d(n)Id(n)}zk d(n) i c(n) k k "n = n n-k z ' k=O k=O

where (i0)

d(n)( = d(n)(a)) = 2-n(2n+a-2) (n) = 2 -n F(2n+a-l)/r(n+a-l) n n

c(n):_ n=k k- d~n)/d~n)) If (ii)

= 2n-k(~)F(n+k+a_l)/F(2n+a_l).

~j(= ~(n)) (j = 1 '2, • ..,n) are the zeros of yn(Z;a), then j yn(Z;a) = d(n) znp(z) n )

129 where n

P(~)

(12)

I] (i-~j/z). j=l

=

By (I0) and Stirling's formula d(n)= 2 -n (2n+a-l) 2n+a-3/2 n (n+a_l)n+a-3/2

e-2n-a+l e-n-a+l

l+(12(2n+a-l))-l+(288(2n+a-l)2)-l+0(n -3)

l+(12(n+a-1))-l+(288(n+a-1)2)-l+o(n -3)

By routine (but lengthy) computations we obtain the following development, with error term O(n-3): d(n) = 2n+a-3/2nne-n{l_ 6(a-l)(a-2)+l + h(a) + O(n-3)} n 24n i152n 2 where h(a) = 12(a-l)(a-2)(3a+7)+l.

Next, by (12),

e.

n

n

~

r

logP( ):logl I

~ °

j=l

j=l

or

--v,

-

r=l rz

r=l rz

n

=

where o r

[ 3

~i. 3

The values of the first few o ' s r

-2n Specifically, o I = 2n+a--------~ '

i0.

[ ~

4n(n+a-2)

(2n+a_2)2(2n+a_3)

= O(n -r) and or = O(n l-r)

r

~

n

'

2 Also, by Theorem 6 of Chapter i0, [ej[ -< n-l+Re a

-8(a-2)(n+a-2)n 03 =(2n+a-2)3(2n+a-3) (2n+a-4) r so that [~jl

°2 =

have been computed in Chapter

cn

- ~ I _< c r=4 ~?zl r <- --4 r=4 -rz

Consequently,

~

1

r=4

(n[z I) r

c = ~

1 3] 4 n z[ (l-(n ] z [)-1 )

'

with a constant c, that depends only on a. For fixed z and n sufficiently large, oo

(7

1-1nz] -1 > 1/2, SO that ] ~

~[r

r=4

rz

< -

~C

1

= O(n-3)

n 3 ]z[ 4

In fact, this estimate holds uniformly in z, provided that [z[ _> n, for some fixed n > O.

We conclude that log P(z) = 2n 2n+a-2

=z

-I

1 z

-

2n(n+a-2)

(2n+a-2)2(2n+a-3)

(.(a-2)z-I 2

--

1

z2

+

8(z-2) (n+a-2)n

3(2n+a-2)3(2n+a-3) (2n+a-4)

zi2)n-i (a-2)2z -I zi2)n-2 + 0(n-3) +-+ ( 4 ---

1 + 0(n-3) -y

130

By exponentiation we obtain (13)

P(z) =

el/Z{l_((a-})z-I

+ ~)n

( a - 2 ) z -3 z -4. -2 ~ . - 3 . -I + (.(a-2)2z-I + [ ( a - 2 ) 2 - 1 ] z -2 8 + 8 + 3-T--Jn +urn )}. 4

Finally, we substitute (13) in (Ii) and obtain .2nz.n~a-3/2 i/z . Yn(Z; a) = L--~-J z e gLz,a),

(14) where

g(z,a) = l-[(a-2)(a-l+2z-l)+(l + z-2)](4n) -I + [(a-2) ((a-l) (3a+7)+(12(a-2) (a+l)+2) z-l+6(3a-5) z-2+12z -3) + (1~

+ llz -2 + 3z-4)] (96n2) -I + O(n-3).

If we combine in (14) the terms of order n -2 with the term obtain precisely Do~ev's formula (9).

0(n-3) ,

then we

In the particular case a = 2, (14) reduces to

. 2 n z . nVZ ~ e l / Z { 1 - ( 1 +z -2) 1 yn(Z) = I.---~-) 1 + llz-2 + 3z -4) - -1 4~-- +(T~ 96n 2 from which (i) and (3) immediately follow.

+ 0 (.~_)}, n

PART IV CHAPTER 14 APPLICATIONS i.

In the present chapter we shall discuss some of the problems, in which BP have

found applications and which, in fact, led to their study.

Three of these have al-

ready been mentioned in the Introduction and sections 2-S of the present chapter are mainly an expanded version of the earlier presentation.

Somewhat unfortunately for

this exposition, those problems present their own difficulties; while these can be overcome by classical, well established methods, even the short sketches of those procedures given here, may well divert the attention of the reader from the role played by the BP themselves in the solutions of the respective problems. 2.

The irrationality of the exponential function e c for rational c.

While the

irrationality" of e is an almost immediate consequence of its usual series expansion, the irrationality of e c for rational c is less obvious.

We shall prove it here, by

using results of Chapter 8. We recall that the polynomials Pn(X) collect the terms of Yn(X) with powers of x of the same parity as n, while Qn[X) is the sum of the other terms of Yn(X).

Let

x be a fixed, positive, rational number and set x/2 = s/r, with coprlme integers s and r, so that e 2/x = e r/s. are integers.

Also, ~ C x ) = an-I [n) xn-I + an-3 (n)x n-3 + "'" > an-lX (n) n-i , so that, in par-

ticular, Qn[2S/r) > a ~ Bn

It is clear that rnPn(2S/r) = An and rn-l~[2s/r) = Bn

2n-l(s/r)n-I = 2(n~)(2n)![~)'s'n-iand

= rn-l%(2s/r) > ~(2n)! sn-l.

If er/S would be rational, then er/S+l er/S_i

a,b coprime positive integers would follow. Pn [2s/r)

a

A

0 < Iqn(2S/r ) - ~ I = Ir-~nn

a = ~ , with

By Corollary 8.4 1

r 2n-I

- ~1 < Qn(2S/r)Qn+l(2s/r) - BnBn+l ,

or 0 < [bAn-arBn[ < r2nb Bn+ 1

< r2n(2b) sn

(n+l)'. < 2br2n (2n+2)!

By Stirling's formula, the right hand term is less than er

(n+l)'. (2n+2)! 2 .r e.n+l.. (--~) [l+en)(lim en=0); n-~=

hence, for sufficiently large n, the positive integer IbAn-arBn[ is less than one, which is impossible and shows that er/s cannot be rational.

In particular e itself

and all its integral positive and negative powers are irrational. S.

The irrationality of ~.

We shall use again results of Chapter 8 and follow, in

the main, the presentation of Siegel [55].

132

First we obtain a new representation dn+l tiate (8.6) n+l times, observe that ~dx the derivative

for Rn(X), defined by (8.6). @n(X/2)

= 0, use Leibniz's

We differen-

formula for

of a product and obtain

dn+ 1 (i) - Rn(X ) dx n + 1

dn+l dx n+l {@n(- x/2)eX}

n+l r+l n+l -r (r) ex = r=0~(-I) ( r )2 en (- x/2) = eXs(x);

n

h e r e S(x) =

[ a.x j is a polynomial in x of degree at most n. We now observe that, j=O J dn+l on t h e one h a n d , by ( 8 . 6 ) , d x - ~ Rn(X) contains only powers of x with exponent at least

n; on t h e o t h e r h a n d ,

eXs(x) = ao+(al*aO)x+...+(aj

+aj_l+...+ao/J!)xJ+...+(an+an_l+...+ao/n!)xn+ ....

It follows that a 0 = a I = . . . r = 0 contributes

= an_ 1 = 0.

to it and it is immediate

S(x) = (-l)n+12-nx n and ~ddEn+l

As for an, only the term of (i) with from (i) that a n = (-l)n+12 -n, so that

Rn(X ) = (- l)n+12-nxneX .

By an (n+l)-fold

eX (x-t)ntnetdt" Jo

from 0 to x it now follows that Rn(X ) = (-l)n+l 2nn!

integration

Finally,

if we

replace t by tx, we obtain the desired new representation 2n+l (2]

Rn(X ) = (_l)n+l x

ii (l_t)ntnetXdt. o

2nn! For x = iT, in particular, 2n+l ,i (l-t)ntndt ~o

IRn [iw) i f ~ ,

2nn!

r (u) r (v) F (u+v)

where B( u , v )

2n+l = ~ n , 2 n.

B(n+l,

is Euler's Beta function.

n+l),

Consequently,

2n+l IIPRn(iW)l < ~ -

(S)

n!

2n

(2n+l)!

IRn(i~)I

Let us assume gers.

2n+l <_ ~ 2 n

that

, or, hy Stirling's

re__~n+l l+c n <4n, ~

2

= (~)n+l

w2/4 = a/b is rational,with

B Z setting x = wi in (8.6), we obtain,

Rn(i~)

formula,

/f

( lim n ÷

e n = 0).

a,b coprime positive,

rational

~ k=0

(-l)ka n

= 2

inte-

with ~ = [n/2],

= @n(Wi/2)-@n(-~i/2)e wi = @n(Wi/2)+en(-~i/2) 2

(l+en) ~e

~ (-i) k=0

= 2

~ a (n) n-2k k=0

(wi/2) 2k =

133

with a,n,r~ n-2k 12

given by ( 2 . 8 ) .

By Theorem 3.1 i t follows t h a t

Z (-I) ka(n) n-k akb~-k I = N is a non-negative integer. k=0

Ib~Rn(i~)[ =

In fact, N > 0, because (2)

shows that for x = ~i the integrand has the positive imaginary part (l-t)nt n sin ~t (0 < t < i).

On the other hand, by [3), n/2 ~2e n+l /8 (l+en) < .w2ebl/2 . n + l ~ _

0 < N = ]b~RnfiW) l < b

~e

(~)

l+en

- ~--~)

for sufficiently large n and this is not possible for any integer N. that the assumption of the rationality of ~

< 1

~e

This shows

2 . 2 . is not tenable, ~ is irrational and

this holds, of course, for ~ itself. 4.

Solution of the wave equation in spherical coordinates.

course, classical and the reader may want to consult, e.g. solution.

This problem is, of [55] for a traditional

The equation to be solved is 2 Au ~ 1___ 3__~u

(4)

c2

~t 2

where A is the Laplacian and c is the speed of propagation of the waves.

In spheri-

cal coordinates A = ---~- + ?

3r

~-~ +

r

--

+ cot e ~ )

+

r

2

sin2O

8¢2

It is sufficient to consider only monochromatic waves, i.e., waves of a single frequency; indeed, the general case is reducible to the monochromatic one by ordinary Fourier analysis, that means by superposition of a countable number of monochromatic waves. In order to solve (4), we use the method of separation of variables, i.e., we set

(s)

u = R(r)O(O)¢(¢)T(t),

where each factor depends only on a single independent variable.

Specifically, r

is the distance of a point P from the origin O, 8 is the angle of OP

with a fixed

"polar" axis [0 < 8 ~ =), $ is the angle that a plane through P and the polar axis makes with a fixed plane through the polar axis, and t is the time variable. If we substitute this in (4) and divide by u, we obtain (6)

R" 2 R' 1 @' 1 ~-- + ~ ~-- + -~-~-- cot 8 + ~ r r

@" @- +

2 r

1 . 2 sln @

@" @

1 2

T" T

c

Here the right hand member depends only on the time t, while the left hand member stays unchanged when t varies; hence, both sides of (6) are constant.

In order to

obtain a periodic solution, this constant must be negative and we denote it by -k 2.

134

Hence, T" + k2c2T = 0 and, with proper choice of the origin for time, and after the suppression of a multiplicative constant (that would anyhow be absorbed into a product of similar constants - see (S)) we obtain T = cos kct. way with the separation of the other variables. 2 . 2A.R" 2 R' O'-1 r sln ~t~-- + ~ ~-- + O + ~ r

(7)

O'

cot O. 7

We proceed in the same

First, we obtain + k2}

¢" = - ¢-- = ~"

In order to obtain a solution $(¢) that is single valued and periodic with period 2 2~ (as it has to be), it is necessary to take ~ = q , with q an integer, so that ¢(~) = a [8)

q

cos q~ + b

q

sin q¢.

Next, (7) with ~ = q2 leads to

2 r 2 R" R' k2r 2 e" C' q } ~-- + 2r ~-- + = -{~-- + cot e ~-- + sin2 @

= X.

To solve the last (ordinary) differential equation of (8), set cos e = x, @(0) = F[cos e) = F [ x ) (9)

= (l-x2)q/2v(x).

Then V(x) satisfies the differential equation

(l-x2)V '' - 2(q+l)xV' + (~-q2-q)V = 0.

Equation (9) has, in general, only solutions by (often divergent) series; it admits, however, pol)momial solutions if and only if X = n(n+l), for some integer n. that case, its solution is V(x) =

dq

In

Ln(X), where Ln(X ) is the Legendre polynomial

dx q of degree n.

The proof is immediate, by q-fold differentiation of the differential

equation for Legendre polynomials: (l-x2)L~-2xL~+n(n+l)en = 0.

It follows that @(O) = sinqo{ dq Ln(X)} x cos e dx q = ,

a function usually denoted by Pq(cos e) n

the associated Legendre function.

clear that for q > n, Pq (cos e) = 0 identically. n nary differential equation. (I0]

We now solve the remaining ordi-

By (8) with ~ = n(n+l), this is

r2R '' + 2rR' + (k2r2-n(n+l))R = 0. Equation (i0) may be transformed into Bessel's equation.

to proceed differently.

We prefer, however,

In order to obtain polynomial solutions we set

x = (ikr) -I and R(r) = (kr)-le-ikry(i/ikr) (ii)

It is

= ixe-Xy(x), so that (i0) becomes

x2y '' + 2(x+l)y' - n(n+l)y = 0.

In (ii) we recognize equation (2.11) of the BP Yn(X). admits solutions of the type

The result so far is that (4)

135

(12) u(r,¢,8,t) = (ikr)-le-ikryn(I/ikr)

(aq cos q¢+bq sin q¢)P~(cos @)cos(kct),

a fact that can be verified by direct substitution of (12) in (4), or (5). With (12) also n

U(r,O,¢,t) =

~ (ikr)-le-ikryn(i/ikr) ~ (an,qCOS q¢+bn,qSin q¢)P~(cos O)cos(ckt) n=O q=O

are solutions,for any choice of the constants an, q and bn, q.

One can use this fact,

in order to choose the constants so as to satisfy initial and boundary conditions. Let us assume that at t = O, U(r,@,¢,O) = F(r,@,¢) is given. Then, if we set (irk) -I = x and x-iF((ikx)-l,o,¢) = Fl(X,@,¢), we obtain n

~ e-i/Xyn(X) X (an,qCOS q¢+bn,qSin q¢)P~(cos O) = F(x,O,¢). n=O q=O

(13)

We now multiply both sides of (13) by Ym(X)e"I/x and integrate around the circle.

unit

By taking into account Corollaries 4.S and 4.7 we obtain m

(-l)m+l 2m+12

~mCO'¢) = ~

i

~ (am,qCOS q~+bm,qSin q¢)P~(cos 0) = ~m(O,¢), where q=O /

l~l--i

Fl(X,O,¢)Ym(X)e-I/Xdx.

We rewrite this as

m

(am,qCOS q¢+bm,qSin q¢)Pq(cos 0) = Km(O,¢), where Km(O,¢) = q=O (-l)m~l(m + i/2)~m(@,¢ ).

By using the orthogonality of the trigonometric functions

and of the associate Legendre functions, one obtains (see, e.g. [SS]) explicit values for am,q and bm,q : 4~am, 0 = (2m+l) #2~ /=0 0 2~am, q

=

Km(e,~)Lm(COS O)sin Oded¢; and for q # O,

(m-cO : (2re+l) (re+q)! /o2~ ivo Km(e'¢)Pq(c°s O)cos q¢ sin 0 dOd¢,

(m-q) : 2~bm, q = (2m+l) (m+q)' fo2~ ITO Km(O'¢)Pq(c°s O)sin q¢ sin @ dOd¢. Here Lm(COS 8) = PmO(cos @) is the Legender polynomial of degree m.

Replacing again

m by n, the function U(r,e,¢,t)

(14)

X n=O

(ikr)

-i -ikr e Yn(i/ikr)

n

X (an,qC°S q¢+bn,qSin q~)P~(cos @)cos(kct) q=O

136

is the required solution for the wave component corresponding to the given k.

If

there are several frequencies present, then the superposition of solutions of the type [14) will give the complete solution. 5.

The infinite divisibility of certain probability distributions. Let f(x) be a probability density function.

theory to call its Fourier transform ¢(t) = I ~ function" of f(x).

Then it is customary in probability f(x)eitXdx,

the "characteristic

Given a probability density f(x) and its characteristic

¢[t), let us set c m [ t ) =

{¢(t)} I/m.

function

The question arises: is it true that for every

integer m, ~m(t) is the characteristic

function of some probability distribution?

In other words, given any integer m, is it true that there exists a non-negative, Lebesgue integrable function

g(x)

(= gm[X)), such that ~m(t) = f ~

g(x)eltXdx?

If such a function g(x) exists for every m, we say that the probability density function f(x) is infinitely divisible. a [not necessarily differentiahle) discussion of this topic see [~7].

The same terminology is used in the case of

distribution

conditions that insure infinite divisibility Recently, Thorin [SS] and Bondesson

function F(x).

For a more detailed

Much work has been done on the investigation of (see, in particular,

[45] and [19]).

[7] have determined large classes of distribution

functions that are infinitely divisible.

In fact, the results of [7] contain as

particular cases those presented here; however, due to the generality of the problem considered in [7], the methods used there are rather deep.

Here we shall restrict

ourselves to some simple, specific cases, namely the Student t-dlstributions of an odd number of degrees of freedom. (15)

The corresponding density function is

f2k_l(X) = Ck(l+

x

2

)-k

r(k)

with the normalizing constant c k = ¢[2k-I)~ F(k - 1/2) REMARK.

Let us observe parenthetically

2

that if Xm is a chi-square random variable

with m degrees of freedom, then the infinite divisibility of fm(X) implies also 2-i

that of (Xm)

m = 2k-l. [56].

; we shall prove this infinite divisibility,

as stated, for all odd

A proof valid for all m (in fact, for all real m > 0) can be found in

Still more general results are obtained in [36]. The present proof is based on the following Theorem A of Kelker [65].

We re-

call that a function ¢(t) is said to be completely monotonic on an interval I, if on I the function and all its derivatives satisfy the inequality [n = 0,i,2 .... ).

(-l)n¢(n)(t)

> 0

137

THEOREM A.

The Student t-distribution of 2m degrees of j~eedom is infinitely divi-

Km_ 1 (,~-) sible, if and only if the function Cm(t) = #t Km(/t) is completely monotonic on

[0,~). Here Km(U ) is the Bessel (or Macdonald) function discussed in Chapter 2. Kelker [65] and Ismail and Kelker [62] proved the complete monotonicity of Cm(t) for some small half-odd integral values of m, from which the infinite divisibility of f2k_l(X) as given by (15) followed for the first few integral values of k.

They also conjectured that ~m(t) is completely monotonic for all real m, from

which follows, in particular, the infinite divisibility of f2k_l(X) for all integers k.

For further results on this problem, see besides [55] and [56], also [35], [36]. It is clear that if ¢(f) = I~ g(u)e-tUdu, with g(u) > 0 for 0 < u

< ~,

0

then ~(t) is completely monotonic.

That this sufficient condition is (essentially)

also necessary is less obvious and is contained in the following theorem of S. Bernstein: THEOREM B. (S. Bernstein; see [6S]). The function ¢(t) i8 completely monotonic, if and only if it is the Laplace transform of a (not necessarily j%nite} measure, i.e. if are onlu if it is of the Form ¢(t) = f~ e-tUdF(u) In particular, if F(u) is differentiable, so that F'(u) = g(u) exists, then one has ¢(t) = Ie g(u) e-tUdu, with g(u) > 0

on 0 < u < ~.

We shall denote the

Laplace transform of g(u) by i(g)(t) and the inverse transform of ¢(t) by i-l(¢)(u). We shall prove THEOREM i.

For every odd integer n ~ i, set ¢(t) = On_l(tl/2)/On(tl/2 ) and let

8j (= 8~n)) (j = 1,2 ..... n) be the zeros of the BP @n(Z); then, if g(u) = L-l(¢)(u), we have (16)

g(u) = (~u)-I/2- 2~ -I/2 i

2 e ~.u J I~

j=l

e _v2dv,

-~j~-

and g(u) is completely monotonic; in particular, g(u) ~ 0 COROLLARY I. and (X~) -I

For every odd integer n, fn(X)

(as defined by (iS) with n = 2k-l)

are infinitely divisible.

Proof of Corollary i. Kn_i/2(z)

@n_l (z) =

ZKn+I/2 (z)

on 0 < u < ~.

0n(Z)

By (S.l), Kn+i/2(z) = (~/2z)i/2z-n-i/2e-ZOn(Z); hence

138 and

On_l(t 1/2) en(t 1/2)

Kn_l/2(tl/2)

¢n+l/2(t) = tl/2Kn(tl/2)

Now the Corollary follows from Theorem 1 on account of Theorem A and the Remark preceding it.

It remains to prove Theorem i.

On_lCtl/2) By ( i 0 . 1 4 ) ,

Proof of Theorem i.

en(tl/2 )

n =

n I

g(u) = /-ic¢)(u)

n

=

i-i

j!l ~j

1

-

8j-t I/2

-

Z

1

j=l 8j (Sj-t 1/2) I

i-i

; hence,

I

j=l ~ j

tl/2+C-Bj )

By Theorem i0.5 and Corollary 10.2, Re 8j < O; hence, Re(-8j) > 0 and the last inverse Laplace transform may be computed.

In fact, it is knowm (see e.g.,

(29.3.37)

in [~]) and we find n

g Cu) : -

l

2

1 + 8je 8.u ] erfc(-Bjul/2)}, where erfc(-Bju I/2) =

1

j=l ~ { ~ 2

2 - 1 / 2 /-Sj~

e-v dv.

By using also (10.2') we obtain (16).

the complete monotonicity of g(u).

It remains to show

In fact, we only need the positlvity of g(u),

in order to apply Theorem B, but it is easier to prove the stronger statement. Indeed, let @(x) = w-l{x-i/2 +

n -1/2 2 -i [ 8ix (x+Bj) }. j=l

We verify by direct computation, or on hand of tables of Laplace transforms (see, e.g., [I], (29.3.4) and (29.3.114))

that i(@)(u) = g(u).

If we set ~[x) =

~xl/2@(x), then by Theorem B,g(u) is completely monotonic, provided that n

,(x) o l÷

n

Z

o.

j:l

, o . . ,cx) o p(x)/qc.),

with qCx) - I

I (.÷5)."

j=Z

polynomial with real coefficients. As just recalled, we know from Chapter I0 that none of the 8j's is purely imaginary; hence, q(x) has no positive zeros and, consequently, does not change n sign

on

[0,~). As q(0) = ( ~

8j) 2 = {(2n)'./2nn!} 2 > 0, one has q(x) > 0 for

j=l 0 < x

< ~.

Also n

(17)

n

p(x) = q(x)+ j=IZ 8j [k~j [ (x+6k2) = xn + j=l ~" Yjxn-"J

a

139

is a polynomial tric functions

of exact degree n; it also has only real coefficients of the 8j's) and so is real for real x.

p(x)/q(x) ÷ i, so that, by q(x) > O, also p(x) > 0

(namely symme-

For x + ~, ~(x) =

for large x.

We now claim the

validity of LEM~

i.

p(x) =

x

n

.

Assuming the Lemma for a moment, 0 < x

p(x) > 0 for 0 < x

< ~, so that g(u) is indeed completely monotonic

is complete.

and the proof of Theorem

From q(O) / 0

the same order at x = O.

follows that ~(x) and p(x) vanish

However,

by (10.2') 4(0) = l+

n = ~ 8j (2m+l) = 0 (m = 1,2 ..... n-l), j=l

least n at x = 0.

n [

(if at all) to

B? 1 = 0 and

j=l l so that ~(x) has a zero of order at

This shows that p(x), of degree n, also has a zero of order n

at x = 0, so that in (17) all yj = 0 (j = 1,2 ..... n) and pox) = x n, as claimed. this the Lemma is proved and all assertions 6.

Electrical

work.

networks.

may contain resistors,

into a finite collection

coils

[self-inductances)

electrical

and condensers

[capacitors).

closed loop contains no source of electrical

potential,

around the loop vanishes.

of potential

that if the current i = i(t) is measured

the terminals

of a coil of self inductance

di (t) of L henrys equals L ~

of potential

volts,

and

of v(t) volts, then the total drop

around the loop equals v(t).

(voltage)

at all terminals

tial, usually the "ground". initial conditions,

(or "nodes"),

in all branches

and the

with respect to some fixed poten-

In order to do that it is necessary to know also the of

We assume that in each loop, R, L, and C are "lumped",

rather than distributed

in particular,

sources of energy.

the currents

i.e., the value of one such current and of such a difference

at, say, t = O.

i.e., concentrated,

("ports").

laws

We recall

of a condenser of capacity C equals C -I I i(t)dt volts.

To "solve" the network means to determine

consider,

then the Ohm-Kirchhoff

in amperes then the difference

If the loop is closed by a source of potential

potential

If a

of a resistor of R ohms equals Ri(t) volts, while that between

that between the terminals

potential

net-

of loops, each one of which

state that the total difference

between the terminals

With

of this sections are justified.

Let us consider a [perhaps very complex)

It can be decomposed

of potential

1

It only remains to justify the Lemma.

Proof of Lemma i.

~(m)(0)

< ~; hence, ~(x) > 0 on

and that they are constant in time.

"passive" networks,

They will have,in general

that is networks without (see Fig.

To the first one we may connect an outside,

there a given, variable potential

We

internal

i) two pairs of terminals known source, that applies

el(t), the input signal.

The second port may be

140 connected, e.g., to a large resistor, say p, and we are interested in the output potential e2(t) between its terminals, or, equivalently, in the output current i2(t ) = e2(t)/p.

lnl

il(t)

i2(t)

Rn3~~l :

C , ~ C n 2

I ÷

e~(t

2(

p

A

Cn3 ~ / ~ Fig,

%2 Fig. 2

1

A typical loop, say, the n-th loop may look like Fig. 2. If we write the Ohm-Kirchhoff equations for the n-th loop, this has the form of an integro-differential equation •

(18)

d

J[ {Rnjlnj(t)+Lnj ~

.

Znj(t) + ~Cn.

I inj(t)dt} = en(t)

(en(t) = 0 if the loop is passive). In order to solve this system, we replace each equation by its Laplace transform. This means that we multiply each term by e-st (s = ~i~, a complex variable) and integrate with respect to t (= time) from t = 0 to t ~. In this way each integro-differential equation becomes an algebraic equation in the new variable s. =

We s e t I n j ( S ) = I~o Z" n j ( t ) e - S t d t

(whenever p o s s i b l e we s h a l l s u p p r e s s t h e s u b s c r i p t s )

for the transform of a given current i(t) loop and E(s) = I ° e ( t ) e - S t d t points

(or n o d e s , o r t e r m i n a l s )

becomes R I ( s ) ,

becomes s L I ( s ) ,

n-th loop, the transform of equation (19)

in a given branch j of the n-th

f o r t h e t r a n s f o r m of a d i f f e r e n c e

between two s p e c i f i e d Li'(t)

= inj(t)

and C-1 / i ( t ) d t

of potential

of t h e n e t w o r k .

A term Ri(t)

becomes ( C s ) - l I ( s ) .

s i n g one t o be p u t i n t o t h e " b l a c k box" o f Fig. given a certain

So, e . g . , potentials,

For t h e

(18) becomes

~ { R n j + S a n j + ( S C n j ) - l } I n j ( S ) = En(S ) (En(S) = 0 i f e n ( t ) = 0 ) . J The problem o f " s y n t h e s i s " o f a network t h a t we a d d r e s s now c o n s i s t s

so t h a t ,

e(t),

in divi-

1, c o n n e c t e d t o t h e f o u r t e r m i n a l s ,

t y p e o f i n p u t , we s h o u l d o b t a i n a d e s i r e d t y p e o f o u t p u t .

we may want t h a t ,

i f a t t h e i n p u t we a p p l y a l a r g e number of s u p e r p o s e d

each one o s c i l l a t i n g

with a different

frequency, all but those within

a g i v e n , n a r r o w band s h o u l d be s u p p r e s s e d , w h i l e t h o s e w i t h i n t h a t band s h o u l d be

141

collected at the output with a minimum of distortion.

For instance, at the input

we may collect through an antenna the e(t)'s due to many broadcasts,

but at the out-

put we want to eliminate all, except one, and this one with as little distortion as possible.

Such a network,

called a filter, would give us, ideally, an output like

that of Fig. 3, with the "cut-off frequencies" ~i and ~2"

Often, with no particular

loss of generality, we may set ~i = 0 and

l

1

~1

°~2

then speak of a low-pass filter.

Sometimes

the purpose is to reproduce the input as faithfully

Fig. 3

(distortionless)

the output.

as possible at

Then we speak of a time-delay

network. In view of the linearity of the Ohm-Kirchhoff equations and of the Laplace transforms,

also the relations between the transformed input and output currents and

potentials,

ll(S), I2(s), El(S ) and E2(s ) are linear.

output a large resistor, considered a one-port

In case one connects at the

in order only to collect there e2(t), the network may be

(two-terminal)

one.

The relation between ll(S ) and El(S )

being linear, there exists a function Z(s), called driving point impedance function, such that El[S) = Z(s)II(S ). Similarly, function).

E2(s ) and El(S ) are related hy E2(s) = T(S)El(S )

Both, Z(s) and T(s) are

(T(s) = transfer

obtained as solutions of linear equations of

the form (19), so that both are rational functions of s.

In many cases one is

interested in Z(s), or in T(s), for s = i~ (m = 2~f, f = frequency, T = f-I = period) purely imaginary;

in that case one speaks of Z(iu) simply as the (complex) impedance.

The following properties of Z(s) may be proved (see, e.g. work is passive and consists only of resistors, real for real s, Re Z(o+im) ~ 0 a = 0.

If the net-

for o ~ 0, with Re Z(c+im) = 0 possible only for

Any function with these properties is said to be a p.r.

function;

[60]):

coils and condensers, then Z(s) is

it maps the closed right half-plane

[positive, real)

into itself, so that any purely ima-

ginary boundary points of the image can have as inverse images only points also on the imaginary axis.

If Z(s) is p.r., then so is Z(s) -I and also the composition of

Z(s) with any other p.r. function.

Neither zeros, nor poles of Z(s) may belong to

the open right half-plane and if a pole is purely imaginary, with positive residue.

then it must be simple,

The difference between the degrees of numerator and denomina-

tor of Z(s) cannot exceed one.

Many of these properties are shared by T(s), hut not

the last restriction. If a polynomial has the property that it has no zeros in the right half-plane, it is called a Hurwitz polynomial. denominator of Z(s) Proposition holds:

By what precedes, both, the numerator and the

(and also of T(s)) must be Hurwitz polynomials.

The following

142

PROPOSITION I.

The ratio of the s~n of the even powers to the s~n. of the odd power8

of a Hurwitz polynomial is a p.r. function. The converse of this proposition is not quite true; indeed, if H(s) is a Hurwitz polynomial, then K(s) = (s2-a2)H(s) is not Hurwitz; nevertheless, the ratio of the sum of its even powers to that of its odd powers is the same as for H(s), and hence is a p.r. function.

We also remark that, after the elimination of factors that are

common to the sums of even and of odd powers, the even and odd "parts" of a Hurwitz polynomial are themselves Hurwitz polynomials, as numerator and denominator of a p.r. function. If we now consider the ideal output like the one in Fig. 3, it is rather clear

that continuous functions will not lead to it.

We may try to obtain an acceptable

approximation to it by continuous functions, by settling for a graph of one of the two shapes of Fig. 4(a), or (b).

(a)

(b) Fig. 4

We face, however, still another difficulty.

As seen, passive networks, consisting

only of resistors, coils and condensers lead only to rational functions for Z[s) and for T(s).

However, the realization of an output like those of Fig. 4., may require

for Z(s) (or T(s)) a transcendental function, say F(s).

In this case we may have

to approximate F(s) by a rational function of s, which, in addition, will have to be also p.r.; this then can be realized by an R-L-C network. Indeed, once Z(s), or T(s) have been determined, with the restrictions mentioned, some very simple rules tell us how to realize the corresponding network, i.e., how to find the geometry of the network and how to compute the numerical values of the resistors, coils, and condensers to be used. We shall discuss here only how to determine T[s) similar) and ignore the "hardware problem".

(the procedure for Z(s) is

The interested reader can find exhaus-

tive treatments of this problem, i.e., in [ST], or [60].

For the "Bessel case",

among others, an almost automatic procedure is outlined in [I01]; see also [64]. We start with the consideration of an ideal network that introduces no distortions, only a fixed delay.

This means that, given e I = el[t) for t > 0, e I = 0 for

t < 0, we want to obtain an output e2(t ) = el(t-t0).

By taking Laplace transforms

143

we obtain E2(s) = f~o e2(t)e-Stdt = i~o e I (t-to) e-Stdt =

e-st° fo

el(t-to)e

-s(t-t°)

d(t-to)

= e-St°

~

/-t

el(t)e-Stdt = e

-st o f~o

el(t)e-Stdt =

o -st e

o

El(S) • Let us take, for simplicity, t o as unit of time (i.e., set t o = i); we shall

indicate later the modifications needed in a different time scale. We have obtained E2(s)/El(S) = T(s) = e

-s

As e

-s

.

is not rational, one could

attempt to approximate it by partial sums of its Maclaurin series.

However, not only

the partial sums of e -s (l-s, l-s+s2/2 etc, have obviously zeros in the right hand plane), but also the partial sums of e s with more than 4 terms have zeros with positive real parts (see [37]), hence, are not Hurwitz polynomials. function cannot be realized by an R-L-C network.

Such a transfer

On the other hand, we know from

Chapter i0, that @n(Z) is a Hurwitz polynomial and from Chapter 13 that anl@n(S ) (an = (2n)!/2nn:, as in (2.8)) approximates e s.

This leads us to consider s

a succession of transfer functions Tn(S ) , that approximate e : (20)

Tn(S ) = anenl(s) = an(@n(S)e-S)-le -s = An(S )'e -s

Formula (20) puts into evidence the "distortion factor" An(S) = anSnl(s)e s.

This

approach seems to have been discovered by W.E. Thomson around 1949 (see [107] and [108]), by the study of certain multistage amplifiers that gave desirable outputs. The corresponding transfer functions led him to the BP (at that time not yet so named) and, specifically, to the recursion relation (3.5). seem to be also Burchnall's relations (10.2), (10.2').

Implicit in his work

Thomson also tabulates the

zeros of @n(S) for n = 1,2,...,9 to four decimal places (Table 1 in [108]). If we had A(s) = i, then T(s) = e -s would be the ideal transfer function, that reproduces the input function faithfully at the output with only the fixed delay 1 (or t o in the general case).

In fact, An(S ) ~ I; specifically, for s = i~, or imt

rather s = i~t ° if t o ~ i, An(i~to) = an@ ~ ici~to) e IAn(imto) I.

o = An e iE , say, with A n =

For s u f f i c i e n t l y large n, ]i~tol is negligible with respect to an, so i~t

that indeed, An(i~to) ~ e

o, IAnl ~ 1 and c = mt o.

In other words, the filter re-

produces faithfully at the output the variation of the applied potential el(t), with only a fixed time delay t o .

144

On the contrary, if ~ is so large that On(imt o) = z.nwn tno , then An

an/~nt ~ and decreases to zero as ~ -* =.

In other words, high frequencies (and

what here "high" means depends on n) are practically not transmitted, so that the filter can be made to work (by proper choice of n and by the realization of the network corresponding to that Tn(S) ) as a low-pass filter. What happens for intermediate values of ~?

At this point it is perhaps worth-

while to remember that T(s) is the result of a Laplace transform so that it should not be considered as the real value of the ratio e2(t)/el(t).

Nevertheless, as seen

in the two extrema cases, it does convey much information on the dependence of e2(t) on el(t ) . With this caveat in mind, we now proceed to determine the exact value of An(S)By (S.I), On(S)e -s = /2~

sn+i/2Kn+I/2(s).

By (9.6.4) and (9.1.4) of [3] we

obtain On(S)e-s = A ~

sn+i/2(-

~i~i.i-(n+I/2))H~/2(-si )

= _i-(n-1/2) ~

sn+l/21I(~!.~(-si) n l/z

= _i-(n-i/2) ~

sn+i/2

i sin (n~+~/2)

j

•

.

_(n+i/2)(-sz)-12n+iJn+i/2(-si)).

For s = i~, in particular, 0n(i~) e-i~ = _ ~

(-])n(i~)n+I/2i-(n-3/2) ( J _ ( n + i / 2 ) ( ~ ) + ( - l ) n ( - i ) Jn÷I/2 (~))

~-~

n+i/2((_l)n J

=

~

-(n+i/2)(~)-iJn+i/2(~))

and a

(21)

Tn(im ) =

n mn+l /~___ 2~ {(-l)nJ-(n+i/2)(~)-iJn+i/2 (~)}

a result already obtained by L. Storch (see [106]).

e

-im

,

If to ~ I, we have to replace

everywhere m by ~t . o

From (21) immediately follows a

(22)

A = n n+l{

n

~

2 ~-~ (J_(n+I/2)(~) +J +i/2(~)} I/2

'

also found in [I06]. Formulae (21) and (22) permit one to compute the loss (usually in decibels), the phase delay and other characteristics of the network. For these topics we have to refer the reader to [106], [64], [i01], [I15] and [116].

145

It is of some interest to observe here that the vanishing of many of the sums of odd powers of zeros of the BP (explicitly quoted in [106] and in [108]) is the fundamental reason for the favorable characteristic of the networks based on BP, that earned it the name of maximally flat delay network. For whatever interest it may have, the general t~)e of network that realizes a transfer function Tn(S ) = (an@nl(s)e s) is:

Fig. 5 7.

Inversion of Laplace transforms.

As already pointed out by Krall and Frink

(see, e.g., their remark on p. 106, after (23) in [68]), there seems to exist surprizing analogies between the BP and the Legendre Polynomials.

One more instance

of such similarity is the following application, that seems to attract increasing attention. We recall that the Laplace transform F(s) = f= f(x)e-SXdx is inverted by the o formula f(x) = ~ 1

(23)

fc+i~ c-i~ F(x)eZXdz

valid for Re c sufficiently large. Often the function F(s) is of a nature that precludes an integration of (23) in closed form.

In these cases one is led to m m m r i c a l

integrations.

In the case of a path of integration along the real axis, one favorite method is that of Gaussian quadratures based on n nodes.

These formulae use the zeros of

the Legendre Polynomials and are exact for polynomials of degrees up to 2n-l.

In

an analogous way, in the present case, a method of Gaussian quadrature has been devised, with the zeros of the Legendre Polynomials replaced by the zeros of generalized BP.

The method is due to H.E. Salzer (see [99] and [i00]) and has later been

elaborated also by other mathematicians we can papers. (24)

(see [70], [102],

[103], [69], [87]).

Here

only sketch the basic ideas and refer the interested reader to the original First, following [99], we simplify (23), by setting zx = s, and obtain xf(x) = ~ 1

icx+i~ cx-l~ eSF(s/x)ds = ~ 1

fc'#i~ c'-i~ eSG(s)ds.

146

Here G(s) depends, of course, also on x, but, for simplicity this dependence will not be emphasized by the notation.

Also, we shall write again c, rather than c'

for the (rather arbitrary) abscissa of integration and denote an integral taken along a

parallel to the imaginary axis, at the abscissa c, simply by f(c)"

Next,

we recall (see, e.g. [15], p.128) that F(z), being a Laplace transform, has to vanish for z ÷ ~ and satisfies also some other conditions. In many cases of inte-i rest it is either a polynomial in z , without constant term, or at least can be well approximated by such a polynomial.

In analogy with the real case, the quadra-

ture formula to be obtained will be exact, if G(s) is a polynomial without constant term in s -I , say P2n(s-l), of a degree not in excess of 2n, where n is the number of terms in the formula.

We t h e r e f o r e start

with,

assume t h a t

arbitrarily)

G(s) = P2n(S -1) i s such a p o l y n o m i a l

n distinct

points

(to

s 1, s 2 . . . . ,s n, t o which we a d j o i n a l s o

n+l i)/ We now observe that if L!n+l)(s-l)1 = I I (s-l-sk k=l k#i

Sn+ 1 = ~.

and s e l e c t

n+l ~l_s 1), ! I (s k then k=l k~i

L~ n+l)(skl ) = 6ik (Kronecker delta), so that the Lagrange interpolation polynomial L(n+l)(s -I) of degree n+l, that coincides with P2n(S -I) at all points sk (also at Sn+ 1 = ~, where both vanish) is

L(n+l)(s-1)

=

n+l [ L!n+l) -1 -1 1 (s )P2n(Si ). i=l

It follows that the polynomial P2n(s-l)-L (n+l)(s-1), of degree 2n, contains the

factor

s-lpn(S-1),

s -lpn(S-1)rn-l(S-1)

where Pn(S -1) :

$c es

•

, w i t h a p o l y n o m i a l rn_ 1 ( s - l )

1 f(c) e s P2n (s-l)ds = ~ 1 2~i

1 2~i

n I I (s-l-ski) k=l

te~

function

s-1 e s

Indeed,

the Pn(S-l)'s if

In the

(corresponding to Sn+ 1 = =) has been left out, because

if the polynomials Pn(S-1)

t e d i n such a way, t h a t

Consequently,

f(c) e s s - ipn(s-l)rn_ I (s-l)ds.

-i it vanishes on accoant of P2n(Sn+l) = P2n(0) = 0.

identically,

o f d e g r e e at most n - 1 .

I(c) eS{L (n+l) (s-l)+s-lpn(s-l)rn_l(s-l) }ds =

n ~ L~ n+l)(s - l)P2n(sil)ds + ~ -1 i=]

first integral, the last

Hence, P2n(S -1) = L ( n + l ) ( s -1) +

(that

is,

The second integral vanishes

^ ( n ) ,j have been s e l e c i f t h e s k = ~k

are orthogonal

t o each o t h e r ,

with the weight

147

(25)

I(c)eSs-lpn(S-1)pla(s-1)ds

for all

0 ~ m < n, then it

i s i~mncdiate t h a t

also

l ( c ) e S s - 1 Pn(S -1 ) s - 3 d" s

(26) for j = O,l,2,...,n-l,

= 0

= 0

whence

(27)

2~ii /(c)eSs-lpn(s-l)rn_l(s-l)ds

= 0

follows. We shall

see that such a choice

Assuming

is possible.

this for a moment

, (24)

becomes n

(28)

x f(x)

=

[

A~n)P2n(sil)

i=l

eSs-lL(n+l)(s-1)ds. i

w i t h A! n ) 1 t - --Y(c) 2~i

'

It is clear that the A~njf~, the so called Christoffel i of the functions

(1 5 k S n ) .

o f P2n(S -1) d e p e n d ,

quadrature

term and of degree

are independent

as t h e L!n+l)l d e p e n d o n l y on t h e c h o i c e o f t h e s~ n)

The c o e f f i c i e n t s

The Gaussian constant

involved,

constants,

formula at most

(28), exact

of course,

a l s o on x.

for polynomials

02n(S -1) without

2n will have been established,

show how one can select the constants

s k = s~n)-

so that

as soon as we

(25) should hold.

n-I In g e n e r a l ,

pn(Z) = z n + n ~ b ~=1 u

(29)

~

b z u, so t h a t

1 2--~T l ( c )

s -i e s

(26) becomes

• s-~-Jds

= 0, b n = 1. m

The i n t e g r a l vanish,

except

1 (~+j)!

and

i n (29) e q u a l s

~

1 m=0 m!

that for m = u+j, when ~

n ~ ~=0

b (u+j),

= 0

One may verify that the determinant

from z e r o f o r a l l

0 ~ n e Z, so t h a t

Once we know t h a t

tile b ' s

results

2~i 1

s

l(C)

ds.

Here a l l

integrals

s~+J+l

ds l(c) --~

= i.

The sum over m becomes

(29) y i e l d s

(30)

determined,

1

(hence,

for j = 0,i, . ..,n-l. of the b ' s

t h e b 's ( b

(0 ~ ~ S n-l)

= b ~( n ) )_ a r e u n i q u e l y

t h e pn(Z) a n d , t h e r e f o r e ,

we can a v o i d t h e l a b o r o f a c t u a l l y

is different

solving

(30),

the Sk'S)

determined. are uniquely

by u s i n g some o f t h e

o f C h a p t e r 4.

We f i r s t o b s e r v e t h a t a b s c i s s a c may be r e p l a c e d

the infinite vertical by t h e open p o l y g o n a l

path of integration along the c o n t o u r ( - ~ - i Y , c - i Y , c+iY, -~+iY)

148

(as Y ÷ ~, the contributions to the integral in the original and in the modified path, essentially reduce

to that from c-iY to c+iY); next this open contour may

be replaced by the closed rectangle of vertices c±iY, -X±iY (observe that the integrand has no singularities with Re

s < -X, if X is sufficiently large); in a third

deformation, the rectangle may be replaced by a circle of diameter from -X to c; finally, we invert this circle in the unit circle and take as new variable z = i/s. In this way, it is seen that (25) is equivalent to (31)

for m = O,1,...,n-1, of the integrand We

= 0

C a circle

(-X - 1 , c - i ) .

is at the origin

now r e c a l l

any c o n s t a n t

with

I c el/Zz-lpn(Z)pm(z)dz

Corollary

to p(z;a,b),

r(a)

b n ( - ~) "

p(z;l,-l) =

one may a c t u a l l y

First,

it

without affecting

in Corollary

n=l F(a+n-1)

so t h a t

4.5.

t h e p r o d u c t o f two p o l y n o m i a i s may r e p l a c e

of diameter

is clear

N e x t , we c o n s i d e r

~ (r(n))-iz -n n=l

=

z- l e l / Z .

p(z;a,b),

the particular

circle.

f o r any a , one may add

of the corollary,

is holomorphic in the unit

5 the weight function

take for C the unit

that

the validity

The o n l y s i n g u l a r i t y

circle. by p ( z ; a , b )

because

We, t h e r e f o r e , =

c a s e a = 1, b = - 1 ,

so that

Corollary 5 now reads

I,lz I=I e l / Z y n ( Z ; l'-l)Ym(Z;l'-l) d-!z z = 0

2wi for0<m
It follows that if we take in (31) pn(Z) = CnYn(Z;l,-l), then (25), hence,

also

Here c n = {(-l)nn(n+l) . . . ( 2 n - i ) } - 1 ,

(26) and (27) h o l d .

the coefficients

o f z n on b o t h s i d e s

from t h e u n i q u e n e s s CnYn(Z;1,-1) reciprocals

of the equality.

of the b 's (i.e.,

are the only solution

of the zeros of yn(Z;1,-1),

@n(Z;1,-1).

This finishes

On t h e o t h e r h a n d , i t

of the Pn(S)'S) , that

of (30).

The s~ n)

i.e.,

as f o l l o w s by c o m p a r i n g

the coefficients

(k = 1,2 . . . . .

follows of

n) a r e t h e n t h e

they are the zeros of the polynomials

the proof of the exact quadrature

formula (28).

I n c a s e G(s -1) i s n o t a p o l y n o m i a l o f d e g r e e 5 2n, t h e n G[s -lj-- = L(n+l)(s-lj- + s

-i

Pn(S

-i

)g(s

-i

), where g(z) is not, in general, a polynomial

of degree < n-I but is holomorphic for Re s > c (see [13]).

Instead of (28) we now

obtain n

xf(x) 1 where R = 2 ~ i

ral,

l(c)

s-i e s

=

[ A~n) G ( s i l ) + i=l

CnYn(S 1 ) g ( s - l ) d s

is

R,

an e r r o r

term.

While this,

in gene-

is not identically zero, it may still be possible to obtain an upper bound for

149

IRI, in a way similar to the classical case of real paths of integration and (28), while not exact, gives a usable approximation to the inversion. In [99] Salzer lists the polynomials yn(z;l,-l) for n = 1(1) 12 (this notation means that n runs from 1 to 12, with the difference between consecutive entries indicated in the parentheses), the zeros of the corresponding @n(Z;l,-l) to 8 significant figures and coefficients A {i) to 7 or 8 figures. n In [I00] Salzer completes the theory of BP, by quoting Krall and Frink, Burchnall, Agarwal, A1 Salam, Ragab and Brahman.

He extends his list of polynomials to

n = 16, and tabulates the zeros of the On(Z;l,-l) and the Christoffel numbers for n = 1(1)16 to 15 significant figures. Already in 1959, Kublanowskaya and Smirnova

[70] had published a tabulation of

the zeros of On(Z ) (= On(Z;2,2)) and of hen(Z) + Z2On_l(Z) for n = 1(i)30.

Other

tables for the numerical inversion of the Laplace transform were published by Skoblja [102] for s and A k in the formula

~

1 /(c)eZz-S,(z)dz

n A~n)~(Zk)

~ ~ k=l

with

n = i(i) i0 and s = .i (.1)3, to about 8 and 7 significant figures for the zk and ~n),

respectively.

s and A k, as follows:

See also [103].

Next, Krylov

and Skoblja, in [69] tabulate

s = i(i)5 for n 1(1)15 to 20 significant digits and for

s = .01 (.01)3 and n = i(i)i0 to 7 or 8 significant digits. these procedures can be found in R. Piessens the books of Y.L. Luke [71], vol. II, p. 189

[87].

A generalization of

Further comments may be found in

et seg., and [72], p. 229 and 433.

CHAPTER 15 MISCELLANEA Many papers related to BP have not been considered chapters.

Sometimes,

rediscovery

in detail in the preceding

when such a work was related to a topic discussed,

of an older result,

with many papers on generating

it was at least quoted. functions

of BP.

or was a

This was the case, e.g.,

In other cases, however, when a

paper treated some isolated topic, not easily fitted into some broader category, paper often was not mentioned

at all.

the

Among these papers are some very valuable

ones, other of a more routine character and also some (fortunately very few) with erroneous

claims.

It also is not surprizing that among the so far ignored papers,

many are by authors some of whose other work on BP has already been prominently displayed. The purpose of the present chapter is to attempt to mention all these, so far neglected papers and their authors and to state at least briefly and without proofs the results obtained. A major

difficulty

could, perhaps,

is to make such a presentation

be considered desirable

In fact, when several papers discussed related topics, be fitted into one of the preceding chapters. consists of papers each of which discusses In spite of many advantages, not be very useful.

Indeed,

in a systematic way.

in general they could already

Most of the material here considered,

a more or less isolated subject.

also a strictly chronological

some authors worked on different

throughout many years and their names would be intermingled way.

It

to arrange the material by subject matter.

presentation

would

topics related to BP,

in a rather confusing

For these reasons it has appeared that the most convenient way is to list

these papers by authors. whose author is known,

In order to facilitate

the search for a specific paper,

the authors are listed in alphabetic

order.

any given author, however, will then be listed in chronological

The papers of

order, under the

heading of their author. With very few exceptions,

it has been possible to obtain the original papers.

Whenever this turned out to be impossible, time, the information journals,

or to require an excessive

investment

of

concerning its contents was obtained through the reviewing

especially the Mathematical

Reviews.

Even in the cases, when the original paper was accessible, sal of its contents appeared possible,

or even desirable.

when a result appeared to be plainly erroneous,

no critical apprai-

An exception was made

or valid only under restrictions

not

stated explicitly. i. e.g.

W.H. Abdi [6]).

[i].

We recall the definition of the basic hypergeometric

A new parameter q is introduced,

that occur in the definition

lql < i.

of the hypergeometric

series

(see,

a(a+l) ~:: Ca+n- l) The ratios c(c+l) Cc+n-l)

series,

are replaced by

151

(l-qa~(l-~a+l)'''~ (a-qa+n-l) One observes that the limit of the new ratio, for (l-q c) (l-q c+l) ... (l_q c+n-l) • q ÷ i, is the previous ratio.

In analogy to the definition of the basic hypergeome-

tric function, the author defines a basic analog of BP. 2 ~ o ( q - n ,q a+n-1 ; - , x )

For this he starts from

2Fo(-n,a÷n-1;-;-x/b),

(the basic analog of

(see [6]), but with

x i n s t e a d o f - x / b ) and d e f i n e s t h e b a s i c BP by (1)

J(q;a-1;n;x)

= { ( 1 - q a - 1 ) n / ( q ; n )} 2 ~ 0 ( q - n ; q a + n - 1 ; x ) , where ¢o

(l+x) n = I [ ( l + x q k ) / ( l + x q k + n ) , k=0 The a u t h o r s t u d i e s many p r o p e r t i e s tions,

recurrence relations,

of J(q;c,n;x), definition (a-l) n n!

Rainville's [90] normalization ~n(a-l,x) coincides with

"basic" function, J(q;a-1;n,x) 2.

(a-l) n n~

yn(-bx;a,b);

lira _ J ( q ; a - 1 ; n ; x ) q÷l

t o Yn(X;a,b) i s r a t h e r

R.R. Asarwal.

such as i n t e g r a l

(1) may, p r e s m n a b l y , be found i n 2F0(-n,a÷n-l;-;x).

= ~n(a-l,x).

(b)

Integral

Otherwise, the connection of

n~ Yn ( z ; a ' b )

= ~

(see Chap-

such a s :

recurrence relations

representations,

of a

tenuous.

t e r 5 ) , one f i n d s a l s o many o t h e r r e s u l t s , Some a d d i t i o n a l

On the one

on t h e o t h e r h a n d , i n t h e s p i r i t

In [ 2 ] , i n a d d i t i o n t o t h e m a t e r i a l m e n t i o n e d e a r l i e r

(a)

representa-

etc.

The m o t i v a t i o n f o r t h i s p a r t i c u l a r

hand, this

and ( l - q ) n = ( q ; n ) .

for yn(Z;a,b).

like z n (b)

¢]ul=l

2-a u ebU/Zdu ' u(l_u)n+l

and i f~ ta-2+n(l+tx/b)ne-tdt. Yn (z;a'b) = r(n+a-l) o (c)

Expansion formulae, of which the following is typical: Y2n ( 2 x ; a ' b )

3.

W.A. A1-Salam.

=

2n (-2n)2r(a+2n-l)r .x. 2r ~ r! (~) Y2n-2r ( x ; 2 a + 2 n + 4 r - l ' b ) " r=0

In addition to the contributions

already presented,

i n g s h o u l d be m e n t i o n e d . In [3] we f i n d : (a)

New i n t e g r a l

representations

and new g e n e r a t i n g f u n c t i o n s .

(b)

A connection with Jacobi Polynomials:

the follow-

152

Yn(X;a) = n!(-x/2) n

lim

p(-2n-a+l,8)(l_(x8)-l). n

(c)

Characterizations of BP, as follows: (i)

Let a sequence of polynomials fn(X,a), of degrees n = 1,2,..., depend

ing also on a parameter

a, and normalized by fn(0,a) = i, satisfy the recurrence

relation (due to Krall and Frink; see Section 19 of the present chapter) (2)

f~(x,a-2)

1 = ~ n(n+a-l)f_l(x,a);

then fn(X,a) = Yn(X;a)(ii)

Let AaYn(X,a ) = Yn(X;a+l)-Yn(X;a);

then

1 AaYn(X;a ) = ~ nXYn_l(x,a+2), and,more generally, A~Yn(X;a ) = n(k)(x/2)kyn_k(x;a+2k); in particular, A~Yn(X;a ) = n!(x/2) n. From this and Newton's formula f(a+v) = ~ (~)Arf(a) the author infers that the r

BP satisfy the identity

(31

Aaf(X,a ) = {x/(n+a-l)} f'(x,a). Let now fn(X,a) be any sequence of functions, not necessarily polynomials, de-

pending, besides the variable x, also on a parameter

a.

Furthermore, let fn(X,a)

be normalized by fn(O,a) = fo(x,a) = 1 and assume that the fn(X,a) satisfy (3) and the "initial condition" fn(X,2) = Yn(X); then fn(x,a) = Yn(X;a). (d)

Expansion formulae, such as

Yn(X) = l+x{(2n-l)Yn_l(X ) + (2n-3)Yn_3(x)+...}; k

Yn (x'k-n+2) = e- x

k=o

n~

(-n)s(-X/2)SL~ °)(~)

s=o (_l)kk2kyn(x,2k_n+2)

k=o (z/2) ~-leXZ2/8

=

~

1 f~ e-t(l+xt/2)n sin 2Xt =~i- o t

( L(o) s (~) = Laguerre Polynomial); dr;

(2n+8-i) F(n+B-l) n! Yn(X;S)J2n+B-i (z)

n=o

(6 ~ O, -i, -2 .... ; J (z) = Bessel function).

153

(e)

Product expansions (see also Brafman [15]):

n (-n)r(n+a-i) ~ r' r=O and the c o r r e s p o n d i n g i n v e r s i o n s Yn(U'a)Yn ( v ' a ) :

n ~

u+v~n . uv (- ~ J Yn[u-~-~v,a) =

(- ~ ) 2 y r ( u +~

(-n) s ( 2 s + a - 1 ) r ( s + a - 1 ) s'F(n+s+a)

,a)

Ys(U'a)Ys ( v ' a ) ;

S=O

also similar formulae involving @n(X,a). (f)

Product representation formulae, like the following two:

n! Yn(U'a)Yn ( v ' a ) _ r(n+a-1)

2))e-tdt I : ta-2p(a-2,0)(l+t(u+v+ n

and yn(~)yn( _ x

~)x = ~l (~)'x'2nf~o t2nL(-n-i/2)(-s2/2tx2)e-t/2dtn 2n÷2 s 11'

2 f~o tnyn (-8x2t' 2-3_n)e-tS dr.

In [4] the author considers the second (non-polynomial) solution of (2.11), which i s qn(X) = (-1)ne2/Xyn(X), and e s t a b l i s h e s r e c u r r e n c e s for qn(X), analogous

to (3.16) and (3.22) (for a = 2).

Ile also gives new characterizations of the BP,

of which the f o l l o w i n g i s t y p i c a l : Given a sequence of polynomials fn(X), of degrees n (n = 0 , 1 , 2 . . . . ) , n o r m a l i z ed by fo(x) = 1, then fn(X) = Yn(X) i f and only i f (4)

x2 d )2 ~" (fn(X)fn_l(X)) = (fn(X)-fn_l(X) • In [5] the a u t h o r o b t a i n s a second s o l u t i o n z ( a ) ( x ) to (2.26) and a second n

polynomial s o l u t i o n v n( a ) ( x ) to ( 3 . 2 0 ) .

z n( a ) ( x ) can be r e p r e s e n t e d by a d e f i n i t e

integral and also in terms of Whittaker functions.

For integral a > -n, z(a)(x) is -

n

an entire f u n c t i o n ; otherwise i t i s holomorphic only i n the p l a n e cut along the p o s i t i v e r e a l a x i s , where z ( a ) ( x ) has along x > 0, the d i s c o n t i n u i t y n -2i sin(a~)Yn(X;a).

For z ( a ) ( x ) , n

the author proves r e c u r r e n c e r e l a t i o n s ,

generating

f u n c t i o n s , m u i t i p I i c a t i o n theorems, e t c . The " a s s o c i a t e p o l y n o m i a l s " V(a) (x) are orthogonai on the u n i t c i r c l e with n

r e s p e c t to the weight f u n c t i o n ~(x,a) = x { 1 F l ( 1 ; a ; - z x - 1 ) } - I = m

the c o e f f i c i e n t s may be computed r e c u r s i v e l y by

~ r=o

~

B r ( a ) x l - r ; here

r=o

( - 2 ) r g m _ r ( a ) / ( a ÷ l ) = Bm(a) for

154

m ~ O, 6o(a ) = i; for a = 2, 6n(2 ) = (-l)n2nBn/n!, with B n the Bernoulli numbers. As particular cases, the author recovers several results of Ai-Salam and Carlitz [7].

Next, some very general identities of Christoffel-Darboux

Several properties of the BP yn(Z;a) are obtained, known (but often with new proofs), plicity

some of them new, others already

such as orthogonality

of zeros (see [53]), and others.

type are presented.

(see [68] and [93]), sim-

The author shows that, for a > 2, all

zeros of yn(Z;a) are inside the unit circle (for a = 2 the result had been proved in [53]) and those of v(a)(x) inside n yn(Z;a),

for

a

an integer,

W.A. AI-Salam and L. Carlitz.

which are, essentially characteristic

Also the irreducibility of

is proved for certain values of n.

In [6] the author generalizes 4.

Ix[ < J 3/5

some results of Chatterjea

[23].

In [7] the authors study polynomials Un(Z), Vn(Z),

, the polynomials Pn(Z), Qn(Z) of Chapter 8.

Here are some

results:

Define coefficients

gn and yn by 6n = 2nBn/n! (Bn = B e r n o u l l i numbers) and

Yn(X/2)n; then, for 0 < m < n, ym+lun(Y) = (-1) n ( 2 2nn! n+l~6mn

2(eX+l) -1 =

and

n=o 6m+2vn(6)

( l n 2n+l(n+l)" = ,-I,

(2n+3)!

6

mn

.

IIere the 6's are Kronecker deltas and the

)'powers" are symbolic: a f t e r ym+lun(Y) and Bm+2vn(6) are computed, the exponents are lowered to become i n d e c e s .

Also,

2~il f l z [ = l e2/Z_12 zmun(z)dz = (2n+3)(2n+2). ~ 1

2 f l z [ = l e2/Z 1 zmvn(Z)

( _ l ) n + l ( n + l ) ! 2 n+l 6 (2n÷1)! mn" In [8] the a u t h o r s study the r e l a t e d polynomials Un(X) = i - n u n ( i X ) , Vn(X) = i-nvn(ix) •

Let a(x) be a s t e p f u n c t i o n with s a l t i

J(Xk) = x k2 at Xk

2 2k+I 1 ~ (k E Z ) ;

then ,~ mn . f~ xmUn(x)da(x) = 2nn! and j_~ gn(x)d~(x ) = -~ (2n+l)-------~-6mn' Um(X) 2n+l S i m i l a r l y , i f 6(x) i s a step f u n c t i o n with s a l t i x~ = ~

1

(k = ±1, ~2 . . . .

j(x~)

=

3 2k2

at the p o i n t s

) , then f~-~ Vm(X)Vn(X)da(x) = 2n+33 ~mn.

S.

W.A. Ai-Salam and T.S. Chihara.

The classical polynomials of Jacobi, Hermite,

and Laguerre are the only orthogonal polynomial

solutions of

155

(5)

~(x)P~(x) = (~nX+Bn)Pn(X) + ¥nPn_l(X)

(n > i),

with ~(x) a polynomial and ~n' 6n' Yn constants. In [9] the authors show that, if the classical meaning of orthogonality is relaxed to admit weight functions of bounded variations, then the BP have to be added

to the solutions of (5) (indeed, see (3.10)).

This result solves affirmative

ly an older conjecture of Karlin and SzegS. 6.

I. Bailey [i0] studies the convergence and, especially the C-I summability of

series of 7.

BP.

D.P. Banerjee [ii] discusses some non-linear recursions; his "Turin inequality"

is incorrect as printed (see [28] for correct form; an equivalent result had been obtained earlier by L. Carlitz [19]). 8.

L. Carlitz defines (see [19]) the functions fn(X) = X@n_l(X) introduced in

Section 6.7.

In addition to the results discussed there, he proves recursion

identities for fn(X), the formula fn(U+V) =

n [ (~)fr(U)fn_r (v); r=o

also representations of Laguerre Polynomials and of BP as sums of fn(X)'S.

He

shows that n @n(X)en(Y ) = r=o ~ (n-r):r!2 (n+r): r (xy)n-r@r(X+Y)' and proves the inequalities of Tur&n type @n+l@n_l-e~ ~ 0 and en@~+l-@n+10~ ~ 0 for x > 0.

Finally, the author investigates some arithmetic properties of @ (z).

-

9.

n

S.K. Chatter~ea published at least 15 papers related to BP.

In addition to the

author's work quoted in Chapter 6, tile following deserves mention.

In [21] he gives

two representations of yn(Z) as a determinant (one of them essentially equivalent to that discussed in Chapter 3). (4n+S)x 2

In [22] he proves the formula Y2nY2n+2

2n ~ (2k+l)y~ + (4n+3)x(x+l). k=l

2 = Y2n +

From this it easily follows that Y2n(X) > 0

for x ~ -i, a result used by Cima [40] in his proof that Y2n(X) # 0 for all real x. In a sequence of three papers [23], [24], [25], the author establishes and then uses operational formulae. n ~-~ j=l

{x2D+(2j+a)x+b} =

Yn+m(X;a,b)

min~n,m) =

r=o

In

d [23] he proves (with D = ~x ) that

n ~ bn-rx2ryn_r(X;a+2r+2,b)Dr , which he uses to show that r=o

(~) (~)r!

x (m+2n+a-l)r(~)

2r Yn_r(X;a+2r,b)Ym_r(X;a+2n+2r,b).

156

In [24] he proves the formulae n

x 2n (D+2 (nx+l) x- 2) n =

( : ) 2 n - r x 2 r y n _ r (x; 2+2r, 2) Dr; r=o

and xn(D_(2x+n+l)x-l) n =

~

(-2)n-rxrOn_r(X;2+r,2)D r.

r=o

In [25] the author proves that

e from

bx

n

[ I j=l

(xD-a-2n+j+l)e

-bx

y =

n [

(~)(-b)n-rxr@n_r(X;a+r,b)Dry,

r=o

which follows that n

(-b)n@n(X;a,b) = ] [ j=l

(xD-bx-a-2n+j+l)'l.

In [26] the a u t h o r o b t a i n s a r e p r e s e n t a t i o n of the BP as a double i n t e g r a l . In [27] the author gives a new generalization of the BP. In [28] some of the Christoffel-Darboux identities contained in (or easily derived from) [5] are rediscovered. In [29] the author considers the polynomials defined by Mn(X,k) = x(2-k)nk-nek/XDn(xkne -k/x)

(2 < k ~ Z).

These polynomials generalize the BP, to which they reduce for k = 2 (see (7.2"), while Mn(X,3 ) = Yn(X;n+2,3). 2 In [32] the author proves Tur~n's inequality &n(X) = ynYn+2-Yn+l >_ 0

for x ~ 0

(which as already observed is actually an immediate consequence of Carlitz's result in [19]), by first obtaining an explicit formula for An(X).

He then proceeds to

use this formula to prove also

f]xl= 1

(x-2an(X)-X-1)e-2/Xdx = 8 ~ i ( - 1 ) n + l ( l + [ ~ ] ) .

In [33], in analogy with Burchnall's operator Q(~) (see [17] and Theorem 2.21 the author introduces the differential operator

Q(6) = 6(6-2)...(6-2n+2) and studies

the polynomials defined by cn(Z) = (-l)neZQ(6)e -z (n ~ i), $o(Z) = I. In [34] the author gives a proof, different from Carlitz's [19], for Tur~n's inequality @n_l(X)@n+l(X)-@~(x ) > 0 for n > i, Ixi < i.

10.

C.K. Chatter~ee [38] proves (a) t h a t f o r n > 1, y~(x) and y~_l(X) have no

common zero; (b) the r e l a t i o n

" " and (x 3 d~) k (xne 1/Xyn_l(X)) = xn+k e - 1 / x Yn+k_l[X)

s i m i l a r ones; and (c) some i n e q u a l i t i e s

-

for products of BP.

157

Ii.

J.A. Cima [40] gives a correct proof for the statement that Y2n(X) # 0 for

all real x, whose proof in [53] had been incorrect. 12.

M.K. Dan published at least 5 papers related to BP.

In addition to his work

mentioned in Chapter 6, we find in [41] that, if Y is a sufficiently differentiably d function and 6 = x ~-~x , as in Chapter 2, then [x(~+a+k_l)]n(xn+ke-b/Xy)

= e-b/x

~ (~)bn-Pxn+2p+kyn_p(X;a+2p+2,b)DPY, p=o

and the author makes some nice applications of this and related formulae. In [45] the author proves a very general operational identity. cases he obtains, among others, 13.

As particular

again the formula of [41].

D. Dickinson's paper [47] contains, in addition to the work discussed in

Chapters 5, 9,and 10, and material not directly related to BP, also the following result: Let D be an arbitrary finite set of integers and let s be a fixed integer. Also, let fn(X) be a sequence of algebraic functions such that fn(X)Jn+s+i/2(s ) vanishes identially (J (s) = Bessel function); then neD both sums,

[ neD

fn(i/ix)inyn+s[X)

and

[ neD

fn(-I/ix)i-nyn+s(X)

also vanish identi-

cally (observe the misprint s for x in the theorem of [47]). 14.

K. DoPey [48] completes and refines work of Obreshkov [82], both, on the zeros

of BP and on the expansions of functions in series of BP. to the first topic was presented in Chapter 10. author proves the following Abelian theorem:

The author's contribution

In connection with the second, the

Let us assume that the series

anYn(X;a,-I ) converges in a subset D of Izl < IXol to the function f(x), that n=o x ° belongs to the closure of D,and that the series converges also at x = x ° to s; then

15.

lim x+x

f(x) = s, provided that the path along which x approaches x ° is inside D. o

M. Durra S.K. Chatter~ea, M.L. Moore study in [49] a class of orthogonal poly-

nomials, that can be represented by H~(x) = x n 2F 0(_ ~, _ ~i (n+~_l);_;_i/x 2)

and

note the similarity of these polynomials to the BP.

16.

M.T. Eweida [51] indicates connections of BP with Meijer's G-functions,

Laguerre Polynomials and Bessel Functions and gives integral representations, of which the following are characteristic:

158

@n (x) =

-12n+In!eXe2n+l f~ o

cos t (t2+n2)n

dt

@n(X;a) = (_l)n+a-12(a-l)/2e2Xx(2n+a-1)/2 io ~ e-tt-(a-l)/2J2n+a_l(2 2/~x)dt @n(U)@n(V)

=

(2~)-I/2eU+V I~ e -(t+(u+v) 2/t)/2~n-I/2~ rU+V~j. L ~ n L'--~J UL.

In [52] the author rephrases some (known) ~nfin~te integrals that involve Kn+i/2(x), by replacing the Bessel function by its value (see Section 3.1)

2-1/2~l/2e-Xx-n-i/2On(X). 17.

A.M. Hamza [59] e v a l u a t e s

18.

M.E.H.

Ismail

some i n t e g r a l s

[61] g e n e r a l i z e s

that

and s o l v e s

involve

a problem that

a s k e d b y R. A s k e y [ Z ] .

The p r o b l e m , i n i t s

following:

be a s e q u e n c e o f p o l y n o m i a l s ,

Let Pn(x,a)

with respect function

to a weight

depend

q u e n c e a n (n =

function

w(x,a),

besides

the variable

0,1,2,...),

integers

generalized

where both,

x aiso

BP. had previously

formulation,

orthogonal

is

o v e r an i n t e r v a l

the polynomials

on a p a r a m e t e r

N a n d M, and c o n s t a n t s

a.

been

the

and t h e w e i g h t

Given also

a,b,c,d,

I,

it

a se-

is required

to determine a function f(x) in such a way that I llf(x)m(x,c)Pn(x,a)dx for n = 0,i ..... N; an = ,

/if(x)~(x,C)Pn(X,d)dx for n

N+I, N+2 ..... M;

Iif(x)m(x,b)Pn(X,b)dx for n > M+I. Conditions on Pn(x,a), necessary for the solvability of this problem are determined and among the solutions Pn(x'a) are found also the BP. The author obtains as one of his results also the following formula (previously obtained by Ai-Salam [3]): (a+n-l)k (4k+c-n-a)n_ k Yn(X;a) = k=o~ (2) (c*n-l)k (2k*C)n-k

Yk(X;e)"

Other contributions of the author with Kelker to the theory and applications of BP have been mentioned in the Introduction, and in Chapters i, i0, and 14. 19.

H.L. Krall and O. Frink coined the name Bessel Polynomials in [68] and much of

their work has been discussed in preceding ch~ters.

Nevertheless [68], contains

also some topics, mainly related to the generalized BP yn(Z;a,b), a ~ 2, that were not discussed so far and they deserve mention at this place. Several recurrence relations occur in [68], not mentioned in Chapter 3. the differential equation for y'(z;a,b),

Also

159

dy~--= n (n+a+l) y~ x2 d 2 Y_,__~n + (ax+2x+b) d-~

(6)

dx 2 appears in [68].

From (6) it follows that y~(z;a,b)

the value a+2 for its principal par&meter.

is proportional

As y~(z;a,b)

n-l, it follows that y~(z;a,b) = ClYn_l(z;a+2,b),

to a BP with

is a polynomial of degree

or y~(z;a-2,b)

= CYn(Z;a,b ).

comparing the constant terms on both sides we obtain that c = n(n+a-l)/2. this result that was generalized by Ai-Salam 20.

P. J. McCarthy's contribution

mentioned in Chapter i0.

By

It is

[3] to (2).

in [74] to the location of the zeros of BP was

In [75] he shows how to generalize AI-Salam's formula (4)

(see [4]) to other systems of orthogonal polynomials. 21.

N. Obreshkov's paper

[82] has already been mentioned in connection with DeWey's

work on the zeros of BF in Chapter i0. expansions of holomorphic 22.

In the same paper the author also studies

functions in series of BP.

A. Pham-Ngoc Dinh [86] studies the function Y~(z) = e-Zz(l-a-2n)/nen(Z'a)" relations satisfied by Y~(z), o p e r a t o r s

The author obtains differential-difference

that raise, or lower the value of the parameter a, and expansions of Ya(z-utzl/2) n

in series of the form Cn(P,Z,tU),

X Cn(P,Z,tu)Y~+P(z), p=o

and convergent

for lutl ~ ~ ,

with rather complicated coefficients unless

a

is a negative integer or

a = +i. 23.

F.M. Rahab

~8] states and offers proofs for five summation formulae, of which

the following is the simplest: n

(7)

(n)

P(2-a)

2x, r

£(2-a-r)

,x .

(- --b-J Yr(2 ' a,b) = Y2n(X;a-2n,b).

r=o

It seems that certain restrictions values of

a

(not explicitly stated in [88]) on the

are needed, in order to insure the validity of the summation formulae.

So, e.g., if the arguments of the ga~na functions are zero, or negative integers, the ratio

F(2-a) £(2-a-r)

presumably has to be interpreted as its limiting value

(-l)r(a-l)a(a+l)...(a+r-2).

For a = 1 this product vanishes, except when r = 0, so

that (7) becomes yo( ~x ," l,b) = Y2n(X;l-2n,b), x yo( ~ ; l,b) = i, while Y2n(X;l-2n,b) Y2(x;-l,b)

= l+(4/b)x+(6/b2)x

2

.

which is obviously incorrect.

is a polynomial of degree 2n.

Indeed

For n = I, e.g.,

160

24.

A.K. Rajagopal

BP satisfies

[91] verifies that a certain function, closely related to the

a certain equation (known as Truesdall's equation) and from this fact

he draws some (essentially known) conclusions 25.

about the BP themselves.

H. van Rossum, in [93] and [94] studies the orthogonality of the BP on hand

of the Pad~ table.

He calls a sequence of real numbers c m "strictly totally posi-

tive" if all determinants m = 0,1,2,... ~,v)

ICm+i_jl

and n = 1,2,3 .....

of the power series

~

~,j = 0,1,2 ..... n-l) are positive for all If the Pa~e rational fraction of the square

Cm zm has the denominator V ,~(z), then the polynomials

m=o

B (k)(z)

= ZUVk,k+u(-z-l)

positive".

(k = fixed integer,~ = 0,i ...) are said to be "totally

They are orthogonal on the circle

with respect to the weight function

Izl = p+~

~(k)(z) =

~

(p = lira sup {m C~m}, c > 0),

(_l)m+k+icm+k+l z-m-l; they have

m=o

only positive coefficients; Izl < to c o

Cl/C o. =

cI

=

m d they have all their zeros inside the circle

The BP are an instance of totally positive polynomials, i.

The author's further contributions

corresponding

to the location of zeros of BP

[95] have already been mentioned in Chapter i0. 26.

P. Rusev [96] indicates sufficient conditions for thc convergence of series in

BP, similar to those

corresponding to series in Jacobi polynomials

inside an

ellipse with foci at +i and -I. 27.

H. Rutishauser

[97] uses continued fractions,

in order to give elegant proofs

of results previously obtained by Ai-Salam and Carlitz

[8] and by D. Dickinson

concerning the orthogonality on the real axis of certain functions

[47],

(denoted in [8]

by Un(X ) and Vn(X)), closely related to BP. 28.

S. L. Soni [104] obtains operational

formulae for the generalized BP by using

operators different from D(= d/dx) and 6[= x d/dx). 29.

II. M. Srivastava

[105] establishes expansions of a large class of functions in

series of von Neumann type, involving 30.

L. Toscano,

(among other systems) also BP.

in addition to his work on gencrating functions

[109] quoted in

Chapter 6, also obtain (in [Ii0]) (a) Yn ( - 2 / p )

representations

of

= P f~ o Ln(l_2t)e-Ptdt

BP as Laplace transforms of a Legendre Polynomial: ' valid

notation, (b)

representations

of the type

f o r Re p > 0; a n d , i n a somewhat c h a n g e d

161

Yn(x-l;a) = x-nen(X;a) = F(a)-i i= o u a-i 2Fl(-n,n+a-l;~,-ux/2)e -u du. 31.

It is not possible to conclude this chapter, as well as this monograph, with-

out the sincerest apologies of the present author to the many contributors to the theory, or the applications of BP, whose work was either overlooked, or misinterpreted, or underevaluated.

Ideally, every one of them, who could be located, ought

to have received the intended text (at least in as far as it related to his/her work), with a request for approval and/or comments.

Due to the large number of

contributing mathematicians, their dispersal throughout the world, the many years that have elapsed since many of the papers were written - and, last, but not least, the fact that, unfortunately, dure was not practicable.

some of them are no

longer alive, this ideal proce-

Even the partial execution of the possible part of this

program would have entailed such an additional delay for the present publication, that the value of this monograph itself would have been put in question,

l:or this

reason the present author counts on the understanding of his colleagues, and on their forgiveness of his sins of omission and commission.

APPENDIX SOME OPEN PROBLEMS RELATED TO BP Quite frequently papers appear with improvements of kno~m results concerning BP.

So, e.g., new generating functions are obtained, sharper bounds for the loca-

tion of the zeros of BP are determined, new recursion formulae are established, etc. Such improvements of known results are of course always welcome; here, however, we want to call attention to a number of unsolved problems related to BP and of a somewhat different nature. I.

It seems that the polynomial solutions of equation (2.26), with a = I,

b = 2 have interesting properties.

This was observed already by Krall and Frink,

who suggested in [68] that such a study be undertaken.

It seems nevertheless that

these polynomials were never investigated thoroughly, although it may be worthwhile to do so.

As we have seen in Chapter 14, these polynomials occur in the inversion

of the Laplace transform.

Do they occur also in some different context?

Do they

have other practical, or theoretical applications? 2.

The BP satisfy a very specific condition, recently called Bessel orthogona-

lity (see [63]].

Are there other interesting sets of functions (perhaps even of

polynomials) that have this property? 3.

In Chapter 9 we studied expansion in series of BP.

While the work of Boas

and Buck (see [13]) is remarkable, our knowledge of these expansions in series of BP cannot be compared, as completeness, with the theory of expansions in Fourier Series or in series of classical, orthonormal polynomials.

Many open problems

remain, concerning, e.g., the speed of convergence, behavior of the series on the boundary of its domain of convergence, summability methods (see, however [i0] and [96]), Lebesgue constants and other similar questions.

The generalizations to the

case a # 2 have, apparently not even been touched. 4.

Are there reasonable generalizations of the BP to polynomials in several

variables? 5. yn(Z;a)

In Chapter i0, certain regions were determined, where all the zeros of (or of @n(Z;a) are located (see, e.g. the work of Parodi, DoPey, Saff and

Varga, etc).

In Section 12, on the other hand, following Olver, a curve is indica-

ted, close to which all these zeros lie, the approximation becoming better with increasing n.

There is a gap to be filled between these two types of results.

So,

e.g., one may determine a strip, enclosing Olver's curve, such that all the zeros of yn(Z;a) belong to that strip. 6.

Wimp [112] proved that the zeros ~!n)(a) j

strictly negative real parts. larger set A 1 of values of of

a.

(j = 1 2 ..... n) of yn(Z;a) have

One may conjecture that the property holds for a (Conjecture: A 1 =

~).

Also the set A 2 of values

a, such that yn(Z;a) = 0 has a real zero, if n is odd, may deserve study.

163

7.

Determine good upper and lower bounds for the real zero of yn(Z;a), n odd,

where a ~ A 2 (see Problem 6). 8.

Find explicit formulae for ~(n) and ~(n), the sums of the r-th powers of r -r

the zeros of yn(Z), and of 0n(Z), respectively; if possible, generalize to the case of arbitrary a ~ 2. 9.

Prove that yn(Z)

(hence, also 0n(Z)) is irreducible over the rational field

for all n; failing that, prove the irreducibility for a large class of values of n. i0.

If possible extend the results of 9 to yn(Z;a), for a ~ 2.

ii.

In Chapter 12 the proof that the Galois group of yn(Z) is the symmetric

group on n symbols required almost a case by case consideration for each n < 13. The problem is to find a unified approach that works for all these small values of n. 12. (2nz/e) -n

Define the function fn(Z;a) by (see (13.9)) fn(Z;a) = 23/2-ae-i/Zyn(Z;a); devclop fn(Z;a) into a complete asymptotic series.

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W.A. A 1 - S a l a m a n d L. C a r l i t z - Bernoulli N~mbers and Bessel Polynomials Math. J . , v o l . 26 ( 1 9 5 9 ) , p p . 4 3 7 - 4 4 5 ; MR 21 # 4 2 5 6 .

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I. Bai6ev - Convergence and summability of series of generalized Bessel Polynomials (Bulgarian, Russian and English Summaries) - B"igar. Akad. Nauk. Otdel. Mat. Fiz. Nauk. Izv. Mat. Inst., vol. I0 (1969), pp. 17-26; MR 44 #5703.

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103.

fc+i~ c-l~ eXP F(p)dp - Izdat. "Nauka i Tehnika", Minsk, 1964, MR 29 # 7 2 5 .

- The distribution o f zeros of polynomials connected with the numerical integration of the Laplace transform (Russian) - Dokl. Akad.

N. S k o b l j a

Nauk BSSR 9 (1965), pp. 288-291; MR 32 #4841. 104.

S.L. Soni - A note on Bessel Polynomials - Proc. Indian Acad. Sci. Sect. A, vol. 71 (1970), pp. 93-99; MR 41 #7166.

105.

H.M. Srivastava - On Bessel, Jacobi and Laffuerre Polynomials - Rend.Sem. Mat. Univ. Padova, vol. 35 (1965), pp. 424-432; MR 33 #4346.

106.

L.

- (a) Synthesis of constant-time-delay ladder nevwork using Bessel Polynomials - Proc. IRE, vol. 42, pp. 1666-1676, Nov. 1954; (b) An application of modern network synthesis to the design of constant-time-delay networks with low-q elements 1954 IRE Convention

Storch

Record, Part 2, Circuit Theory, pp. 105-117; ~ 107.

W.E.

Thomson

-

16-1182.

Delay network having maximally flat frequency characteristics

Proc. Instituto Electr. Engineers, vol. 96 (1949), Part Ill, p. 487.

-

170

108.

W.E. Thomson - Networks with maximally flat delay - Wireless Engineer, (1952), pp. 256-263.

109.

L. Toscano -Funzioni generatrici di partico!ari polinomi di Laguerre e di altri da essi dipendenti - Bull. Un. Mat. Ital., vol.(3) 7 (1952), pp. 160167; MR 14, p. 269.

ii0.

L. Toscano - Osservazioni, confronti e c o ~ l e m e n t i su particolari polinomi ipergeometrici - Le Matematiche, vol. i0 (1955), pp. 121-133; MR 17, p. 733.

iii.

H.S. Wall - Polynomials whose zeros have negative real parts - Amer. Math. Monthly, vol. 52 (1945), pp. 308-325; MR 7-62.

112.

J. Wimp - On the zeros of a confluent hypergeometric function - Proc. Amer. Math. Soc., vol. 16 (1965), pp. 281-283; MR 30 #4001.

113.

A. Wragg and C. Underhill - Remarks on the zeros of Bessel Polynomials - Amer. Math. Monthly, vol. 83 (1976), pp. 122-126; ~ 52 #1146.

vol. 29

SUPPLEt6ENTA2Y BIBLIOGRAPHY OF TITLL~ OBTAINED AFTER COAIPLETION OF THE PRESENT MONOGRAPH. 114.

W. Miller, Jr. - Encyclopedia of mathematics and its applications, vol. 4, Sy~netry and separation of uariables - Addison-Wesley Publ. Co., Reading, Mass., 1977.

115.

L. Weinberg - Network analysis and synthesis - McGraw-Hill Book Co., Inc., New York, 1955; Krieger Publishing Co., Huntington, N.Y., 1975.

116.

A.H. Marshak, D.E. Johnson and J.R. Johnson - A Bessel ratior~! filter IEEB Trans. Circ. Supt. vol. CAS - 21 (1974), 797-799.

GENERAL LITERATURE,

NOT DIRECTLY

RELATED TO BESSEL POLYNOMIALS

and I.E. Segun - Handbook of Mathematical Functions - Dover PubliInc. New York, 1968.

i.

M. Abramovitz cations,

2.

R. Askey - Dual equations and classical orthogonal polynomials - J. Math. Anal. Appl., vol. 24 (1968), pp. 677-685.

3.

I. Bendixson - Sur les racines d'une ~quation fondamentale - Acta Math., (1902), pp. 359-365.

4.

R.P. Boas

5.

R.P. Boas and R.C. Buck - Polynomials d e ~ n e d by generating relations - Amer. Math Monthly, vol. 63 (1956), pp. 626-632.

6.

W.N.

7.

L. Bondesson

8.

R. Breusch - Zur Verallgemeinerung des Bertrandschen Postulates - Math. vol. 34 (1932), pp. 505-524.

9.

W. Burnside - The theory of groups o f ~ n i t e University Press, Cambridge (England),

I0.

- Entire functions - Academic Press, New York,

vol. 25

1954.

Bailey - Generalized hypergeometric series - Combridge Tracts in ~ t h e m a tics and Mathematical Physics, No. 32 - Cambridge University Press, Cambridge (England), 1935.

- A general result on in]~nite divisibility

(to appear). Z.,

orders, 2nd edition - Cambridge 1911.

- Integral Functions - Combridge tracts in Mathematics and Mathematical Physics, No. 44 - Cambridge University Press, Cambridge

M.L. Cartwright (England),

1962.

II.

H. Davenport - The higher arithmetic - Harper Torchbook New York, 1960.

12.

M.G. Dumas - Sur quelques cas d zrreducvzbzl~te des polynomes a coefficients rationels - J. Math. Pures Appl. (6), vol. 2 (1906), pp. 191-258.

13.

G. Doetsch - Theorie und Anwendung der Laplace Transfor~ation - Springer Verlag, Berlin (1937).

14.

G. Enestr~m - Remarques sur un tneoreme ~ ~ relatif xax racines de 1 '~ equatzon... Tohoku Math. J., vol. 18 (1920), pp. 34-36.

15.

L. Euler - Introductio analysin in~nitori°~n, Commentarii Acad. Scientiarum Imperialis

t"

Operae S e r .

I, vol.

I

, •

•

•

- Harper

J

& Brothers,

~

-

L~asanne 1 (1748), Chapter 18 Petropolitanae,

vol. 9 (1737)

-

18.

16.

E. Feldheim - Relations entre les polynSmes de Jacobi, Laguerre et Hermite Acta Math., vol. 74 (1941), pp. 117-138.

17.

W. Feller - An introduction to probability theor~ and its applications 3rd edition - John W i l e y & Sons, New York, 1967.

18.

L.R. Ford - Differential equations, New York, 19S5.

19.

B.V. Gnedenko

20.

H.W. Gould - Combinatorial identies - Morgantown,

(2 vol.),

2nd edition - McGraw Hill Book Co., Inc.,

and A.N. Kolmogorov - Limit distributions for s ~ s of independent random variables - Addison-Wesley Publ. Co., Inc., Cambridge (Mass.), 1954. W. Va.,

1972.

172

21.

H.W.

Gould - Generalization of binomial identies of Carlitz, Grosswald and Riordan ~to appear).

22.

H.W.

Gould - Private communication (letter of 9 August

23.

E.

24.

E. Grosswald 1966.

25.

E. Grosswald and H.L. Krall - Evaluation of ~wo determinants - Amer. Math. Monthly (Advanced Problems; to appear).

26.

H. Hamburger - B e i t r ~ e zur Konvergenz~heorie der Stieltjesschen Kettenbr~che Math. Z., vol. 4 (1919),pp. 186-222.

27.

H. Hamburger - Ueber eine E r ~ e i t e ~ n g des Stieltjesschen Momentproblems - Math. Ann., vol. 81 (1920), pp. 235-319; vol. 82 (1920), pp. 120-164 and pp. 168-187.

28.

K. Hensel and O. Landsberg Leipzig, 1902.

29.

C. Hemnite - Sur la fonction exponentielle - C. R. Acad. Sci. Paris, vol. (1873), pp. 18-24; 74-79; 226-233; 285-293; also Oeuvres, vol. 3, pp. IS0-181.

30.

I.N. Herstein

- Topics in algebra - Blaisdell

31.

I.N. Herstein

- Private co~unication

32.

E. Hille - Ar~lytic function theor~q - Ginn & Co., Boston,

1977).

Grosswald - On a simple property of the derivatives of Legendre Polynomials Proc. Amer. Math. Soc., vol. 1 (1950), pp. 553-554.

- Topics from the theory

of numbers - The Macmillan Co., New York,

- Theorie der algebraischen Funktionen - Teubner,

Publ. Co., Waltham

(letter of 21 November

77

(Mass.),

1964.

1951).

1962.

f

33.

M.A. Hirsch - Sur les racines d'une equation fonde~nentale - Acta Math., (1902), pp. 367-370.

34.

A. Hurwitz

(1913),

-

vol.

25

- Ueber einen Satz des Herrn Kakeya - Tohoku Math• J., vol. 4 pp.

89-93.

Ismail - Bessel functions and the inIKnite divisibility of the Student t-distribution - Ann. Probability, vol. S (1977), pp. 582-585.

35.

M.E.H.

36.

M.E.H.

Ismail - Integral representations and complete monotonicity of various quotients of Bessel functions - Canad. J. Math., vol. 29 (1977), pp. 1198-

1207. 37.

K.E.

Iverson Aids to

- The zeros Computation

of

the

(~AC),

partial

vol.

s~ms

7 (1953),

of

e z

pp.

Math Tables 165-168.

and other

-

38.

D. Jackson - Fourier Series and orthogonal polynomials - Carus Monograph No. 6, The Math Assoc. of America , Oberlin (Ohio), 1941.

39.

C. J o r d a n - Sur la limite de transztzvzte des groupes non altern~s Math. France, vol. 1 (1872/3), pp. 40-71.

40.

S.

K a k e y a - On the limits of the roots of an algebraic equation ~., vol. 2 (1912), pp. 140-142.

41.

E.

Laguerre

•

•

.

-

-

Bull.

Tohoku

Soc.

Math.

- Sur la distribution dans le plan des racines d'une equation alg~brique, dont le premier membre satisfait ~ une Equation lineaire du second ordre - C.R. Acad. Sci. Paris, vol. 94 (1882), pp. 412-416; 508-510;

173

Oeuvres,

vol.

i, pp.

161-166.

Lambert - Memoire sur ~aelques proprietes remarquables des q u a n t ~ e s transaendentes, circulaires et l o g a r i t ~ i q u e s - Histoire de l'Acad, ro£ale des sciences et belles-lettres - Berlin, 1761 and 1768.

42.

J.H.

43.

E. Landau - Hand~uch der Lehre yon der Verteilung der Primzahlen Leipzig, 1909.

44.

P. L~vy - T~eor~e " ~ " de l'addition des variables aleato~res ~ " Paris, 1937.

45.

M. Marcus and H. M i n c - A survey of matrix theory and matri~r inequalities Prindle, Weber and Schmidt, Boston, 1964.

46.

F.W.J. Olver - The asymptotic solution of linear diJ~'erential equations of the second order Ibr large values of a par~neter - Philos. Trans. Roy. Soc. London, Set. A, vol. 247 (1954/5),pp. 307-327.

47.

O. Perron

48.

N. du Plessis - A note abou~ the dewLuatives of Legendre's Polynomials Amer. Math. Soc., vol. 2 (1951), p. 950.

49.

G. P~lya aad G. Szeg~ - Aufgaben und Lehrs~tze ~as der Analyse - Julius ger, Berlin, 1925.

80.

E. Rainville - On a simple property mials - Unpublished manuscript

51.

I. Schur - Einige Sa~ze u~er Pz~mzahlen - Sitzun$sberichte der Preussischen Akad. der Wissenschaften (1929), pp. 125-136; Gesammelte Abhandlungen, vol. III, No. 64, pp.140-151.

52.

I. Schur - Gleichung~en ohne A~fekt - Sitzungsberichte der Preussischen Akad. der Wissenschaften (1930), pp. 443-449; Gesammelte Abhandlungen, vol. III, No. 67, pp. 191-197.

53.

C.L. Siegel - Transcendental Numbers - Annals of Mat~em. ' ~ StuZies No. 16 Princeton University Press, Princeton, 1949.

54.

T.J. Stieltjes - Recherches sur les ~ a c t i o n s convinues - Annales de la Fac. des Sci. Toulouse, vol. 8 (1894), 122 pages and vol. 9 (1895), 47 pages; Oeuvres Completes, vol. 2, pp. 402-566.

55.

J.A.

56.

G. Szeg~ - Orthogonal polynomials Math. Soc. , New York, 1939.

57.

P.L. Tchebycheff - Sur l'Interpolation - Zapiski Akad. ment 8 (1864); Oeuvres, vol. ii, pp. 539-590.

58.

O. Thorin - On the infinite divisibility of the log normal distribution Scand. Actuar. J. (1977), pp. 121-]48.

59.

B.L. van der Waerden - Modez~ Algebra Unsar Publ. Co., New York, 1949.

60.

J.H. Wahab - New cases of irreducibility vol. 19 (1952),pp. 165-176.

.

,

H

- Die Lehre yon den Ketvenbruchen

Stratton

- Electromagnetic

- Teubner,

- Teubner,

- Gauthier-Villars,

Leipzig,

-

1929.

- Proc.

Sprin-

of the derivatives of Legendre's Polyno(letter of 21 October 19SO).

Theory - McGraw Hill, New York, - Colloqui~,~ Publication,

(Revised English

vol.

Nauk.,

1947. 23 - Amer.

vol. 4, Supple-

edition)

-

- Frederick

for Legendre Polynomials-Duke

Math.J.,

174

61.

G.N. Watson - A treatise on the theory of Bessel Functions, 2nd edition Cambridge University Press, Cambridge (England), 1946.

62.

L. Weisner - Group theoretic origin of certain generating functions J. Math., vol. 5 (1955), pp. 1033-1039.

63.

L. Welsher - Generating functions for Hermite functions ii (1959), pp. 141-147.

64.

L. W e i s n e r - Generatin~ ~ n c t i o n s II (1959), pp. 148-155.

65.

F.J.W. Whipple - Some transfo~nations of generalized hypergeometric Proc. London Math. Soc. (2), vol. 26 (1927), pp. 257-272.

66.

E.J. Whittaker - Sur les sertes de base de polynomes quelcon~ues Villars, Paris, 1949.

67.

E.J. Whittaker and G.N. Watson - Modern Analysis, 4th edition - Cambridge University Press, Cambridge (England), 1927.

68.

D.V. Widder - The Laplace Transform. - Princeton University Press, Princeton, 1941.

69.

H. Wilf - Mathematics for Physical Sciences and London, 1962.

for Bessel functions

-

- Pacific

- Canad. J. Math., vol.

- Canad. J. Math., vol.

series -

- Gauthier-

- J. Wiley & Sons, Inc., New York

SUBJECT

The

numbezs I

following

each

entry

for

page

in

snands

Arithmetic

properties

Asymptotic Asymptotic

properties, series -

Bendixson-Hirsch Bernoulli numbers Bernstein's

Cauchy Center

theorem

-

BP

-

-

-

-

-

to

page

numbers

Introduction.

2,4,82,93,124-130,]63.

3,90,91.

I, ! 3 7 ,

138.

- 102. (orthogonality) see

refer

the

155.

or relations 124,163.

Theorem - 154.

transform

Cardioid

any

of

Bertrand's postulate Bessel alternative "Black-Box" - 140. Borel

INDEX

Laplace

26,162. transform.

88.

product of mass

Characteristic Characteristic Christoffel

-

45. 83,84.

functions - !36. (eigen-)values constants - 147,149.

Christoffel-Darboux Commutation relations Continued fractions Convergence Cramer's rule Cycle - !17.

88-91.

(type) identities - 47,48. - 3,34, 59-63,81,

-

154,

Ig6.

160.

64,65,66,68,69,72,74,155,157,160,162. - 20.

Delay, flat

-

!45

maximally phase time

-

flat

-

2, 1 4 5 .

144. 141,142,143.

Differential

equations

of

BP

D1 f f e r e n t i a l Differential Differential

equations equations equations

of of of

~ -form - 9-12,14-15,35. Sturn-Liouville form y~(x) ]58.

Differential Discriminant Distortion

-

I, 1 , 4 - 1 7 , 3 6 ,

- difference relations - I16, IJ8, I19. factor - 143.

-

in series in series formulae -

of BP of other 151-153,

Filter - 141,143,144. Fourier analysis - 133. Fourier transforms - 26,136. Frequency - I, 1 3 3 , 1 3 6 , 1 4 0 , 1 4 1 , Frobenius' method - 5,12. Functions Associate

Legendre

-

1,46,64-74, functions 160.

49.

159.

Eigenvalues - see Characteristic values Eisenstein's criterion (irreducibility) Elements (of Newton polygons) - 10"7,109. Expansions Expansions Expansion

38,85.

-

102.

157, 159, 160, - 11,47,91.

144.

134,135.

Bessel - 2,3,4,!9,35,38,47,91,152,157,158. Bessel,modified - 4,5,18,19,34,76,91,92,137,144,158.

162.

176

Beta - 36,38,!32. Exponential - 1,3,67,68,7~,74,125, Exponential Greatest Hankel Hanke],

type

-

integer (see a!sc modified

Hypergeometric Hypergeometric, Lebesgue MacDonald Meijer's

- 1,97,()8, Bessel) (see a] s c -

J 3J.

6"7,69,72. I02, 132. 91,92. Besse!, modified)

2,34,38,[50. generalized

integrable - see Besse] G - 157.

136. , or

-

-

I.

34.

Hanke]

,

modified.

Positive real (p.r.) 141,147. Step - 154. Whittaker - 34, 37, 38, ]53. Gauss-Lucas Generating

theorem functions

-

Generating Gershgorin's

functions, theorem

-

-

-

Symmet£ic

-

Hermitian

matrices

Impedance Indiciai

(driving equation

-

formula transform

-

i~$,

120-122,

point, 5. -

66. see

!.aplace

of

Lagrange Laguerre's Laplace

-

-

algebra -

Moments Monotonicity, Monotonity, Multiplication Networks, Networks, Networks,

12!,

123,

-

Operators,

arid

inverse

Laplacian.

70,71. 41,46,47. -

154,

31 . !63.

[46.

- 82,83,84. transofrm

constants - 162. equation - I, 1 3 4 . rule 16,29,53,55,132.

Lemniscate Lie group

Measure

see

i, 1 2 3 - ]

1,26,32,146,154.

theorem (or Borel)

Lap!acian

2,141,142.

transform.

razional~

interpolation

Lebesgue Legendre's Leibniz's

Lie

e r (r

-

I, 3, ! ] 6 - 1 3 9 .

of ~ - I, 1 3 1 - 1 3 3 . - 2,9g-i 17, 120, -

162.

l(i3.

complex)

Irrationality Irreducibility delta

160,

163. ~ 20-!22.

Irrationality

Kronecker

153,

90.

Infinite divisibility Inner product - 25,26. Inversion Inverse

131,

116-!18,120,121. i16,

-

150,

44.

25. 116-118,120,122.

1,2, i16-123, - I[A-I13,

Transitive

36,4]-50,z7,58,67,70,

bilJnear - 9,88,90,01.

Gram-Schmidt method Group, Alternating Galois Primitive

84. 1,2,

48.

137. 25-32,51,53. Monotonic complete theorems electrical "Bessel"case passive -

-

- 1,68,75,78,80. 1,3, 136-139. - !53.

2,3, 139-14b. - 142. 141, 142.

transform

-

1,66,67,69, 137, !40-142,144-149.

138,

177

Networks, R-L-C Newton ~dentities Newton

polygon

-

Ohm-Kirchhoff

-

l aplacian Shift

-

Other Operational

-

-

equations 152,156. - 1,9-12,

93,94,97,98.

-

13g-1.q7 .

14, 15, 35,46,47,51,

133, 155-157,

160.

I, 1 3 3 .

159,!60. formulae (see

native) Orthonormality table

-

48,49.

Orthogonality

-

155,157,160.

also

Polynomials,

1,25-32,64, - 26. and

Permutation Po]ar axis

zeros)

100-115.

laws,

Operators Difference Differen%iai

Pad6

!43. (coefficients,

-

orzhogonal

135, 154,

approximanzs

-

and

the

Bessel

a] t e r -

155, !58, !60.

3, 1 6 0 .

117,121. I, ] 3 3 .

Pole (of center Polynomials

of

mass)

-

83,84.

Appe!i (classical, or generalized) 65-68,70. Associate BP - 153. Basic BP -151. Genera]ized BP - 12-~7, 22-24,27,29-~2, 35-38,42-45,49-51,54,5[~,64, 65,70,75,80-82,85-88,93,94, 125,128-130, 148, ~49, Hermite Hurwitz

15~-!63. 26,45, 154. 141-~43.

-

Jacobi Laguerre Legendre

26,34, 35,38,43, 15] , 154, ~60. - 26,34,36,38,44,152,154,155,!60. - 1,3g, 134, 135, ~45, 160.

Lommel

34,38,46,81.

-

Monic - 99,103,1!8,119,123. Orthogonal, c]assica] Orthogonal, Reverse BP Totally Probability

other - 7,8,12.

positive distribution,

Pseudogenerating Quadrature, Quadrature,

-

Random variable - 136. Recurrence relations Re] ations of Representation

160. densizy

functions

formulae Gaussian

BP

to other formulae

3,26,34,47,64,154,162.

153, 157-160.

of

BP

function -

.

136.

1,42.

146,148. 145,147.

1,2,9,]8-24,36,5(],60,86,87,94,96,143,i5!-153, !55, [58, 162. functions -

2,54-40,154,i55,[57,158.

of BP by determinants - 3,20-22,88, 155. of BP by other functions - 35-38,155. of BP by integrals - ] 5 ! , 1 5 6 , 15"I, i 6 0 , 1 6 1 . of other functicns by BP - 46. others - 45,46,151,153,155. Resultant - 118,119. Rodrigues (type) formulae - 16,36,51-58. Scheme Scheme Schmidt

of of

factorization factorization,

method

-

see

-

± O 9 - ] I I, 1 1 3 , lJ 5. compatibility or incompatibility

Gram-Schmidt

method.

-

~09,110, 112,115.

178

Separation of variables - I, 1 3 3 , ~ 34. Singularities, singular points - 4,69,!48. Special functions - 3. Spots - IOO, 104-iO9, I~ I. Stifling formula - I, 1 2 4 , 1 2 5 , 1 2 7 , 1 2 9 , 1 3 1 , 1 3 2 . Stirling numbers - 9. Student t-distribution - I, i 3 6 - I 3 9 . Summability - 64-66,69,72,74,155,[62. Synthesis (of a network) - 3,140. Transfer function - 2, 1 4 1 - 3 4 3 , Transposition - 122. Truesdell equation - 160. Turin inequality - 155,[56. Vector

space

-

25.

Wave(s) - I, 1 3 3 , 1 3 6 . Wave equation - I, 1 3 3 . Wave, monochromatic - I, [ 3 3 . Wave, propagating - I. Wave, stationary - I. Weight functions, w(x) , p(x) , Whittaker Zeros Zeros Zeros Zeros

of of of of

basis

]45.

series

-

etc.

-

25-27, 30-'~2,51-53,64, 155,15~, 160.

146, 148, 153,

69.

BP - 1,2,3, 137, 145, 148, ] 56, 157. BP, location - 2, 3 , 7 5 - 9 3 , 125, 138, 154, 155, 157, 159, 160, 162, 163. BP, properties - 1,2, 3,75-93, 129, 145, ~54, 163. BP, tabulations - 2,143,179.

NAME The F

numbers and

I

following

each

name

for

page

of

stand

any

INDEX

stand the

for

the

Foreword,

page

or

numbers;

of

the

the

letters

Introduction,

re-

spectively. W.H. M.

Abdi

-

F, 150,

Abramovitz

-

R.P.

Agarwal

W.A.

Ai-Salam

F.M.G.

164.

F,171.

-

F,2, -

149,

151,

164.

F,2,43,44,73,

149,

Askey

-

Erd@lyi

L.

Euler

-

102.

77,171.

-

166.

171.

Eweida

155,164.

E.

Feldheim

-

171.

W.

Feller

-

C.C.

-

N.

Bailey

P.

Banerjee

W.

Barnes

-

3,90,171.

Berlekamp

R.P.

Bernstein Boas,

S.

Bochner

L.

Bondesson

E.

Borel

155,164.

3,164.

Bendixson

-

-

157,166.

-

43,171.

-

171.

Feng

-

49,166.

L.R. Ford O. Frink

-

8. B.V.

-

H.W.

136,171.

E.

66.

Boudra,

Jr.

-

Gershgorin Gnedenko Gould

-

-

3,88,90.

-

I, 171.

F,57,

Grosswald

171,172.

-

1,2,3,39,

Brafman

-

2,45,46,149,153,164.

V.

Grosswald

-

F.

R.

Breusch

-

102,171.

E.

Guillemin

-

3,167.

Brillhart

-

J.W. R.C.

Brown Buck

J.L.

171. Burchnall

-

164. 2,8,64,65,67,!62,164, -

W.Hahn - 1,167. H. Hamburger -

1,2,6,10,15,42,43,

A.M.

Hamza Hazony

-

3,167.

K.

Hensel

-

172.

Carlitz 160,

-

117.171.

C. Hcrmite I.N. Herstein E.

- 2,45,57,73,74,154-156, 164,165.

B.C.

Carlson

M.L.

Cartwright

A.L.

Cauchy

S.K.

Chatterjea

C.K.

Chatterjee

-

T.W.

Chaundy

M.P.

Chen

T.S.

Chihara

J.A.

C±ma

M.E.H.

-

46,49,154,155,]57,

-

156,166.

Das

-

Davenport

R.

Dedeking

1,2,6,

M.

Dutta

3, 1 3 7 ,

158,

167,

158,

167.

172.

D.Jackson

-

172.

155,157,166.

D.E.

Johnson

-

167.

J.R.

Johnson

-

167.

-

26,167.

C.

Jordan

-

1i7,172.

Kakeya

171.

-

117.

S.

-

77,172.

-

166.

S. Karlin D.N. Kelker

155. - 1,3,

-

2,38,46,73,81,

J 57,

K.

2,!3,81,86,124,125,128,

-

-

F,I,

Jayne

130,157,159,!62,166. G. Doetsch M.G. Dumas

-

J.W.

-

Dickinson

-

Ismail 172. Iversen

46,157,166.

Dhawan

Do~ev

3,90,172. 77,78,141,142,143,172.

154,!64.

-

160,166. K.

-

!O, J64.

49,!66.

-

H.

D.

-

172.

171.

117.

K.E.

-

-

1,1,26,172. - 39,172.

165. -

-

Hille

M.A. Hirsch A. Hurwitz

165,166.

G.K.

25,172.

158,167.

D.

L.

M.K.

-

156,164. Burnside

166,

121.

49,51,65,70,76,93,94,143,149, W.

121,

167,172.

167.

F. J.D.

12,24,44,64,

145,149,152,158,162,167.

1,164.

-

171. F,I, 1,2,7,

-

1,137. 2,8,64,65,67,162,

Jr.

-

-

121.

164,171.

P.W.

A.

-

158,171.

V Baieev

R.

Enestr6m

M.T.

151,154,158-160,164. R.

Eisenstein

G.

171. 1OO,171.

157,166.

Kloss

-

136,

137,

121.

A.N.

Kolmogorov

A.M. H.L.

Krall Krall

V.I. V.N.

Krylov - 149,167. Kublanowskaya -

-

-

1,171.

F,26, 167, 168. F,I, 1,2,7, 12,24,44,64,

145,149,152,158,162,167,172. 149,167.

180

Laguerre H.

-

26,82,83,84,172.

Lambert

Landau

-

S.

Laplace

M.

Legendre -

Stieltjes

P.

Stokes

-

172.

J

-

I.

G

Szeg5

-

82,173.

P

Szeg5

-

J55.

L

-

I,

iO3.

Storch A.

-

Luke

-

P.L.

Tchebycheff Thomson

O.

Thorin

L.

Toscano

Marcus

-

173.

E.G.

McBride

-

McCarthy

Mittal

L.

Moore

D.

Morton

-

170.

C.

-

-

121.

Oberhettinger Obreshkov Olds

-

-

166.

!3,124,157,159,168.

1,168.

Olver

-

R.S.

Varga

B.L.

van

J

H.

Wahab

A

S.

Wall

G

N.

Watson

Paliwal

-

166.

M.

Parodi

-

3,88,89,90,162,168.

O.

Perron

-

173.

A.

Pham-Ngoc

E.

Phragm@n

-

R.

Piessens

-

N.

du

Dinh

-

159,168.

66. 149,168.

Plessis -

-

39,

173.

66,82,173.

F.M.

Ragab

E.D.

Rainville

-

2,149,159,168. -

2,39,43,151,168,

-

160,168.

173. A.K.

Rajagopal

V.

Romanovsky

-

1,168.

H.

van

-

160,169.

Rossum

P.

Rusev

H.

Rutishauser

-

E.B.

Saff

H.E.

Salzer

I.

Schur

160,169. -

-

I.E.

Segun

R.R.

Shepard

C.L.

Siegel

145,149,169.

39,102,116,117,173. -

F, 171. -

3,169.

I, 131,173.

G.

Sizemore-Ballard

N.

Skoblja

-

T.N.

Smirnova

S.L.

Soni

H.M. R.

-

-

F.

149,167,169. -

149,167.

160,169.

Srivastava Stauduhar

160,169.

2,86,162,169. -

-

-

166.

-

F,3,90,

170.

2,86,162,169.

-

121.

160,169

der

F

J.W.

E

J.

Waerden -

-

Weisner

D.V.

2,91,92,93,162,168,

173.

G.P61ya

-

26,168.

2,65,81,168.

F.

D.D.

2,3,160,170.

Tricomi

168. 157,166.

-

N. C.D.

136,173. -

Underhill

L

F.W.J.

-

25,102,173.

2,3,143,169,170.

173.

B.

Newman

-

2,81,159,168.

Jr. -

M.

173.

46,167.

-

Miller,

Nasif

-

W.E. 166.

M.

3,144,169.

66.

149,167.

-

Minc

-

25,173.

167.

Stratton

Magnus

B.

-

1,173.

Lindel~f L.

J.

V

173.

Landsberg

L@vy

T

59,173.

-

i07, 174.

46,174.

Whipple Whittaker Widder

Will

-

J.

Wimp

-

A.

Wragg

-

-

45,174.

-

66,69,174.

174.

174. 2,81, -

173.

170. -

H.

-

!73.

~62,

F, 3,90,

!70. 170.

PARTIAL LIST OF SYMBOLS

£ Z

field of complex numbers the rational integers

-

Z+

Jv(z),I

(z) ,K (z)

- the positive integers standard notations for Bessel and MacDonald functions -

H (I) (z) H (2) (z)

standard notations

Yn (z) , Yn (z ; a, b)

Resse] Polynomial, simple and generalized, standardization of Krall and Frink. Bessel Polynomial, simple and generalized, standardization of Burchnall.

On(Z),en(Z;a,b) Pn(a'b) (z)

for Hankel functions

- Jacobi Polynomial

Ln (x)

Legendre Polynomial

Ln(a) (z)

Laguerre

Polynomial

- Hermite Polynomial

Hn(Z) Rn,~(z)

Lommel Polynomial

pq(z)

associate

Wk,m(Z)

- Whittaker

pFq(a 1 . . . . . ap;b 1 . . . . . bq;Z)

r(z)

Legendre function function

generalized hypergeometric

function

An(S)

gamma function - beta function greatest integer function complementary error function - notations for weight functions - transfer function - driving point impedance distortion factor

An = [An(i~to) [

- size of the distortion

B(u,v)

Ix]

erfe(z) w(x) ,p (x;a,b) T(s)

Z(s)

A

factor for s = i~t

o

Laplacian operator d differential operator x ~-~

R

- right shift operator

L

- left shift operator

L(x, d

El' ~)

a

L(f) (t) L- 1 (g) (x) Mk(f,P)

- linear, ordinary differential operator that depends also linearly on a parameter - difference operator with respect to the parameter a. -

-

-

the Laplace transform of f(x) at t the inverse Laplace transform of g(t) at x k-th moment of the function f with weight function p

- binomial coefficient [ a o , a I, •.., a n , •.. ]

- continued fraction

B

- n-th Bernoulli number

n

A!n)

- Christoffel constants

1

s(m)

- Stirling numbers of the second kind

n

(~..

- takes only the values O, or i; w h e n used specifically as a Kronecker delta (6ij = i if and only if i = j), this is

±3

stated explicitly. ~k(n) a(n)(a,b) ' k

[n) B~n) (a,b) ~k '

- k-th zero of yn(Z), or of yn(z;a,b), respectively. - k-th zero o f @n(Z), or of @n(Z;a,b),

respectively.

- sum o f r-th powers of the zeros o f yn(Z)

r

(a,b) e m (P)

-

greatest common divisor of a and b spot of coordinates a,b a divides b the exact power of p that divides m

a-b f~ g f = O(g) P n

-

a is congruent b with respect to some modulus asymptotic, or formal e q u a l i t y means If(z)/g(z)' < C, some constant p e r m u t a t i o n on n symbols

[a,b]

alb

(a,b,c)

(d,e)

N.p. A*

cycles of a permutation; also used for a scheme of factorization. - Newton polygon - complex conjugate of the m a t r i x A - notations for c h a r a c t e r i s t i c (= eigen) values of a m a t r i x - resultant o f the polynomials f and g - denotes u s u a l l y a discriminant, D also a d i f f e r e n t i a t i o n operator - sum of the p-adic digits o f n

R(f,g) D, Dn (n), ap (n)

~z

- center of mass with respect to the pole z

a (n)

a m'

- m-th coefficient of the n - t h pol}momial, o f yn(Z), or @n(Z)

111

a m,(a)m,or

a(m )

- abbreviation for a(a+l)...(a+m-l)

a (m) - abbreviation for a(a-l)...(a-m+l)

often