THE UMBRAL CALCULUS
This is a volume in PURE AND APPLIED MATHEMATICS A Series of Monographs and Textbooks Editors: SAMUEL EILENBERC ANDHYMAN BASS
A list of recent titles in this series appears at the end of this volume.
THE UMBRAL CALCULUS
STEVEN ROMAN Drparfmp)tl of Mathematics Calforitto Statt Uniiierstty Fullrrtou, Cnl2fornia
1984
ACADEMIC PRESS, INC. (Harcourt Brace Jovanovich, Publishers)
Orlando San Diego San Francisco New York London Toronto Montreal Sidney Tokyo Sao Paulo
0
COPYRIGHT1984, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. N O PART OF THIS PUBLICATION MAY B E REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.
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United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD.
24/28 Oval Road, London NW1 IDX
Library 01' Congress Cataloging in Publication Data Roman. Steven The umbra1 calculus {Pure and applied mathcmatics ; ) lncludcs biblio:raphical references and index 1. Calculus. I . Title. 11. Series: Purc and applied rnatlicrnaticr (Academic Press) ; Q A 3 . P X I Q A 3 0 3 1 510s [.5151 X3-1 1940 ISBN 0-12-SY43XO-6
PRINTED IN THE. U N l T € D STATtS OF AMERICA 84858687
Y X 7 6 5 4 7 2 I
To Donna and to my parents
CONTENTS
ix
Preface
Chapter 1 INTRODUCTION I . A Definition of the Classical Umbral Calculus 2. Preliminaries
1
3
Chapter 2 SHEFFER SEQUENCES 1. The Umbral Algebra
6
2. Linear Operators 3. Sheffer Sequences 4. Associated Sequences 5. Appell Sequences 6. A Few Examples
12 17
25 26 28
Chapter 3 OPERATORS AND THEIR ADJOINTS I . Continuous Operators on P* 2. Automorphisms and Derivations on P* 3. Adjoints 4. Umbral Operators and Umbral Composition 5. Sheffer Operators and Umbral Composition 6. Umbral Shifts and the Recurrence Formula for Associated Sequences 7. Sheffer Shifts and the Recurrence Formula for Sheffer Sequences 8. The Transfer Formulas
32 33 34
37 42 45 49
50
Chapter 4 EXAMPLES 53 86 107 125
1. Associated Sequences 2. Appell Sequences 3. Sheffer Sequences 4. Other Shefi'er Sequences vii
C0N TEN TS
Chapter 5 TOPICS I . The Connection Constants Problem and Duplication Formulas
2. 3. 4. 5. 6. 7.
The Lagrange Inversion Formula Cross Sequences and Steffensen Sequences Operational Formulas Inverse Relations Sheffer Sequence Solutions to Recurrence Relations Binomial Convolution
131 138 140
144 147
156 160
Chapter 6 NONCLASSICAL UMBRAL CALCULI 1. 2. 3. 4.
Introduction The Principal Results A Particular Delta Series and the Gegenbauer Polynomials The +Umbra1 Calculus
REFERENCES
162 162 166 174
183
189
PREFACE
This monograph is intended to be an elementary introduction to the modern umbral calculus. Since we have in mind the largest possible audience, the only prerequisite is an acquaintance with the basic notions of algebra, and perhaps a dose of applied mathematics (such as differential equations) to help put the theory in some mathematical perspective. The title of this work really should have been The Modern Classical Umbra1 Calculus. Within the past few years many, indeed infinitely many, distinct umbral calculi have begun to be studied. Actually, the existence of distinct umbral calculi was recognized in a vague way as early as the 1930s but seems to have remained largely ignored until the past decade. In any case, we shall occupy the vast majority of our time in studying one particular umbral calculus-the one that dates back to the 1850s and that has received the attention (both good and bad) of mathematicians up to the present time. For this, we use the term classical umbral calculus. Only in the last chapter do we glimpse the newer, much less well established, nonclassical umbral calculi. The classical umbral calculus, as it was from 1850 to about 1970, consisted primarily of a symbolic technique for the manipulation of sequences, whose mathematical rigor left much to be desired. To drive this point home one need only look at Eric Temple Bell’s unsuccessful attempt (in 1940) to convince the mathematical community to accept the umbral calculus as a legitimate mathematical tool. (Even now some are still trying to achieve Eric Temple Bell’s original goal.) This old-style umbral calculus was, however, useful in deriving certain mathematical results; but unfortunately these results had to be verified by a different, more rigorous method. In the 1970s Gian-Carlo Rota, a mathematician with a superlative talent for handling just this sort of situation, began to construct a completely rigorous foundation for the theory-one that was based on the relatively ix
X
PREFACE
modern ideas of a linear functional, a linear operator, and an adjoint. In 1977, the author was fortunate enough to join in on this development. It is this modern classical umbral calculus that is the subject of the present monograph. Perusal of the table of contents will give the reader an idea of the organization of the book; but let us make a few remarks in this regard. A choice had to be made between the present organization of Chapters 2 -4 and the alternative of integrating these chapters by applying each new aspect of the theory to a running list of examples. We feel that the alternative approach has a tendency to minimize the effect of the theory, making it difficult to see just what the umbral calculus can do in a specific instance. On the other hand, we recognize that it can be difficult to remain motivated in the face of a large dose of theory, untempered by any examples. For this reason, we have included at the end of Chapter 2 a very brief discussion of some of the more accessible examples. Chapter 4 contains a more complete discussion of these and other examples. Let us emphasize, however, that we do not intend this book to be a treatise on any particular polynomial sequence, nor do we make any claims concerning the originality of the formulas contained herein. While Chapter 2 contains the definition and general properties of the principal object of study- the Sheffer sequenceit is Chapter 3 that really goes to the heart of the modern umbral method. In Chapter 6 we touch on some of the nonclassical umbral calculi, but only enough to whet the appetite for, it is hoped, a sequel to this volume. Before we begin, we should like to express our gratitude to Professor Gian-Carlo Rota. His help and encouragement have proved invaluable over the years.
CHAPTER 1
INTRODUCTION
1. A DEFINITION OF THE CLASSICAL UMBRAL CALCULUS
Sequences of polynomials play a fundamental role in applied mathematics. Such sequences can be described in various ways, for example, (1) by orthogonality conditions:
jab
Pn(x)Prn(x)w(x)dx = 6n,rn,
where w(x) is a weight function and dn,rn= 0 or 1 according as n # m or n = m; ( 2 ) as solutions to differential equations: for instance, the Hermite polynomials H,(x) satisfy the second-order linear differential equation
y" - 2xy' + 2ny = 0; (3) by generating functions: for instance, the Bernoulli polynomials Bf)(x) are characterized by
(4) by recurrence relations: as an example, the exponential polynomials 4n(x)satisfy 4n+
1(x) = x(4n(x) + 4r(x));
( 5 ) by operational formulas: for example, the Laguerre polynomials Lf)(x) satisfy Lf'(x) = X-aexDne-xX"+= (some put n! Lf)(x) on the left).
2
1.
INTRODUCTION
One of the simplest classes of polynomial sequences, yet still large enough to include many important instances, is the class of Sheffer sequences (also known, in a slightly different form, as sequences of Sheffer A-type zero or poweroids). This class may be defined in many ways, most commonly by a generating function and, as Sheffer himself did, by a differential recurrence relation. Although we shall not adopt either of these means of definition, let us point out now that a sequence s,(x) is a Sheffer sequence if and only if its generating function has the form
where A(t)= A,
+ A , t + A 2 t 2 + ...
( A , # 0)
and
The Sheffer class contains such important sequences as those formed by (1) the Hermite polynomials, which play an important role in applied mathematics and physics (such as Brownian motion and the Schrodinger wave equation); (2) the Laguerre polynomials, which also play a key role in applied mathematics and physics (they are involved in solutions to the wave equation of the hydrogen atom); (3) the Bernoulli polynomials, which find applications, for example, in number theory (evaluation of the Hurwitz zeta function, a generalization of the famous Riemann zeta function); (4) the Abel polynomials, which have a connection with geometric probability (the random placement of nonoverlapping arcs on a circle); (5) the central factorial polynomials, which play a role in the interpolation of functions.
Now to the point at hand. The modern classical umbra1 calculus can be described as a systematic study of the class of Sheffer sequences, made by employing the simplest techniques of modern algebra. More explicitly, if P i s the algebra of polynomials in a single variable, the set P* of all linear functionals on P is usually thought of as a vector space (under pointwise operations). However, it is well known that a linear functional on P can be identified with a formal power series. In fact, there is a oneto-one correspondence between linear functionals on P and formal power series in a single variable. For example, we may associate to each linear
2.
3
PRELIMINARIES
c,“=o
functional L the power series L ( x k ) t k / k !But . the set of formal power series is usually given the structure of an algebra (under formal addition and multiplication). This algebra structure, the additive part of which “agrees” with the vector space structure on P*, can be “transferred” to P*. The algebra so obtained is called the umbra1 algebra, and the umbral calculus is the study of this algebra. As a first step in this direction, since P* now has the structure of an algebra, we may consider, for two linear functionals L and M , the geometric sequence M , M L , M L 2 , M L 3,.... Then under mild conditions on L and M , the equations M L k ( s , ( x ) )= n! dnqk
for n,k 2 0 uniquely determine a sequence s,(x) of polynomials which turns out to be of Sheffer type, and, conversely, for any sequence s,(x) of Sheffer type there are linear functionals L and M for which the above equations hold. Thus we may characterize the class of Sheffer sequences by means of the umbral algebra. The resulting interplay between the umbral algebra and the algebra of polynomials allows for the natural development of some powerful adjointness properties wherein lies the real strength of the theory. The umbral calculus is, to be sure, formal mathematics. By this we mean that limiting processes, such as the convergence of infinite series, play no role. Formal mathematics, much of which comes under the headings of combinatorics, the calculus of finite differences, the theory of special functions, and formal solutions to differential equations, is, in the opinion of some, staging a comeback after many years of neglect. It is our hope that the present work will aid in this comeback.
2. PRELIMINARIES
Since formal power series play a predominant role in the umbral calculus, we should set down some basic facts concerning their use. The simple proofs either can be supplied by the reader or can be gleaned from other sources, such as the paper of Niven [11. Let C be a field of characteristic zero. Let B be the set of all formal power series in the variable t over C . Thus an element of B has the form (1.2.1)
for ak in C . Two formal power series are equal if and only if the coefficients of
1.
4
INTRODUCTION
like powers of t are equal. It is well known that if addition and multiplication are defined formally,
then 9 is an algebra (with no zero divisors). The order o(f(t)) of a power series f ( t ) is the smallest integer k for which the coefficient of rk does not vanish. We take o ( f ( t ) )= + cc if f ( t ) = 0. It is easy to see that o ( f ( M t ) ) = o(f(t)) + .(s(t))3 o(f(t) + g ( t ) ) 2 minb(f(t)), o ( g ( t ) ) ) .
The series f ( t ) has a multiplicative inverse, denoted by f ( t ) - or l/f(t), if and only if o(f(t)) = 0. We shall then say that f ( t ) is inuertible. Let f k ( t )be a sequence in 9and suppose o(fk(t)) + 00 as k -+ cc.Then for any series X
g(f) =
1
bkfk
k=O
we may form the new series
2
bkfk(f)
k=O
that is a well-defined formal power series in t . In particular, if&(t) o(f(t)) 2 1, then the composition of g ( t ) with f ( t ) is
= f ( ~ )with ~ ,
It is clear that o ( g ( f ( t ) ) ) = o(g(t))o(f(t)). Ifo(f(t)) = 1, then the formal power series f(t) has a compositional inverse J(t) satisfying f ( f ( t ) )= f(f(t)) = t . A series f ( t ) for which o(f(t)) = 1 will be called a delta series. The sequence f ( t ) k of powers of a delta series forms a pseudobasis for 9.That is, for any series g ( t ) in 9there exists a unique sequence of constants a, for which
If f(t) has the form (1.2.1), the formal derivative of f ( t )with respect to t is
2.
5
PRELIMINARIES
We shall also use the notation We use the notation
f"(t)
for c',.f'(t).
1 8n.k =
0
if n = k, if n # k
for the Kronecker delta function. There is a variety of notations for the lower (or falling) factorial x(x - l ) . . . ( . u - n + 1). We shall use the notations l)...(x- n
(x), = x ( x
-
,p)= x(.x
+ 1). . . (x + n
+ 1)
and -
I),
which are the usual ones in combinatorics and are also used in the calculus of finite differences. (Another common notation in the calculus of finite differences is x(") = x(x - l ) . . . ( . x - n + l), and in the theory of special functions one sees (x), = x ( x + 1 1.. .( x + n - I).) We abbreviate the expression Section k of Chapter n as Section n.k. References to the bibliography will be made simply by using the author's last name and the citation number, as in Riordan [l].
CHAPTER 2
SHEFFER SEQUENCES
1. THE UMBRAL ALGEBRA
Let P be the algebra of polynomials in the single variable x over the field C of characteristic zero. Let P* be the vector space of all linear functionals on P. We use the notation (Ll P W h borrowed from physics, to denote the action of a linear functional L on a polynomial p(x), and we recall that the vector space operations on P* are defined by ( L + MIP(4>
=
( L IP ( X ) >
+ (MI P ( 4 )
and (CLlP(X)) = C(LIP(X)> for any constant c in C. Since a linear functional is uniquely determined by its action on a basis, L is uniquely determined by the sequence of constants ( L I x">. As in Chapter 1, we let 9denote the algebra of formal power series in the variable t over the field C. The formal power series (2.1.1)
defines a linear functional on P by setting (2.1.2)
1.
7
T H E U M B R A L ALGEBRA
for all n 2 0. In particular, ( t k 1 x " ) = n! b , , k .
Actually, a n y linear functional L in P* has the form (2.1.1). For if (2.1.3) then from (2.1.2)we get (fL(f)
and so as linear functionals L Theorem 2.1.1 onto 9.
I x")
=
( LIX"),
= fL(r).
The map L + f L ( f ) is a vector space isomorphism from P*
Proof Since L = M if and only if ( L I x k ) = ( M I x k ) for all k 2 0, which holds if and only if f L ( t )= fM(f), the map is bijective. But we also have
=fL(f)
+ f&),
and, similarly, LL(f)
= CfL(f).
Thus our map is a vector space isomorphism. W e shall obscure the isomorphism of Theorem 2.1.1 by thinking of linear functionals on P asformal power series in t , using (2.1.2)and(2.1.3)as our guiding light. Henceforth, 9 will denote both the algebra of formal power series in t and the vector space of all linear functionals on P , and so an element f ( t ) of 9 will be thought of as both a formal power series and a linear functional. It is important to keep in mind that two elements f ( t ) and g ( t ) of 9are equal as formal power series if and only if they are equal as linear functionals. This follows directly from (2.1.2)and the corresponding definitions of equality. Thus we have automatically defined an algebra structure on the vector space of all linear functionals on P , namely, the algebra of formal power series. We shall call 9 the umbra1 algebra. Let us give an example. For y in C the evaluationfunctional is defined to be the power series e"'. From (2.1.2) we have (ey'Ix") = y",
8
2.
SHEFFER SEQUENCES
and so <eyrI P W )
(2.1.4)
= P(Y)
for all p(x) in P , which explains the name. If ez' is evaluation at z , then e~te=f=
e (+~z ) t
7
and so the product ofevaluation at y with evaluation at z is evaluation at y
+ z.
Notice that for all f ( t ) in 9 (2.1.5)
and for all polynomials p(x) (2.1.6) Let us consider some simple consequences of the results so far. Proposition 2.1.2 If f ( t ) and g(t) are in 9, then
Proof If we write both product is
J'(t) and g ( t ) in
the form of (2.1.5), then the formal
By applying both sides of this, as linear functionals, to X"and using (2.1.2),we get the desired result. It is easy to extend this result (by induction) to the product of several linear functionals. Proposition 2.1.3 If fl(t), . . .,f,(t) (fi(t)...fm(t)lx"> =
q.
11,.
are in 5, then n
..,1,. ) ( f 1 ( t ) I x 9 - . (.Lr(t)I xi,)
where the sum is over all i,, . . .,,i such that i,
+ .. . + i,
Proposition 2.1.4 If o ( f ( t ) )> degp(x), then (f(t) I p(x)) Proof
=
n.
= 0.
This follows easily from (2.1.2) since (f(t)Ix")
o ( f ( t ) ) > n.
7
=0
whenever
1. THE UMBRAL ALGEBRA
9
Proposition 2.1.5 If o ( f k ( t ) = ) k for all k 2 0, then
for all p ( x ) in P , the second sum being a finite one. Proof
Suppose that deg p ( x ) = d . Then
=
ak(fkffflP(X))~ k=O
Proposition 2.1.6 If o ( h ( t ) )= k for all k 2 0 and if
I p(-x)) = ( f k ( f ) 1 4(x)) for all k, then p ( x ) = q(x).
Proof' Since the sequence , f k ( t )forms a pseudobasis for 9, for all n 2 0 there exist constants an,kfor which f" = u,,,fk(t).Thus
c,"=,
=
f
un,k(fk(f)l
q(x))
k=O
=
(r" I Y(-4)
and so (2.1.6) shows that p ( x ) = q(x). Proposition 2.1.6 implies that if o ( j i ( t ) )= k and k 2 0, then p ( x ) = 0.
(fk(t)
Proposition 2.1.7 If degpk(x)= k for all k 2 0 and if
(f(f)I P k ( X ) ) = ( q ( l )1 P k ( X ) ) for all k, then f ( t ) = y ( t ) .
Proof
For each n 2 0 there exist constants a n , k for which
I p ( x ) ) = 0 for all
2.
10
SHEFFER SEQUENCES
Thus
= (&)I
and so (2.1.5) shows that f ( t ) = g(t). Proposition 2.1.7 implies that if degp,(x) = k and ( f ( t )I p k ( X ) ) k 2 0, then f ( t ) = 0. The reader may have noticed that for any polynomial p(x)
= 0 for all
( t k I P ( X ) ) = P'k'(0).
In words, the linear functional t k is the kth derivative evaluated at 0. The multiplicative identity to is simply evaluation at 0. When we are considering a delta series f ( t )in F as a linear functional we will refer to it as a delta functional. Similarly, when we are considering an invertible series as a linear functional, we use the term inuertible functional. Proposition 2.1.8
The series f ( t ) is a delta functional if and only if (f(t)ll> = 0
Proof
and
(f(t)lX)
z 0.
This follows directly from the definition of delta series and (2.1.5).
Proposition 2.1.9
The series f ( t ) is an invertible functional if and only if
( f ( t ) I 1 ) z 0.
Let us now give our first umbra1 result. Theorem 2.1.10 If f ( t ) is in 8, then ( f ( t ) I X P ( X ) > = (4f(t)I P ( X ) )
for all polynomials p(x). Proof
then
By linearity we need only verify this for p ( x ) = x". But if
1. THE UMBRAL
11
ALGEBRA
The following simple result will be used frequently. Proposition 2.1.11
For any f ( t ) in 9 and p ( x ) in P , (f(t)I P(a-u)) = ( f ( 4I P
W)
for all constants a in C . Proof Taking f ( t ) = t" and p ( x ) = x k , we have (t" I
= a k n !dn,k =
a"n! hn,k= ((at)" I x').
This may be extended by linearity to all f ( t ) and p ( x ) . We conclude this section with a few examples of linear functionals that will appear in the sequel. Example 1. The evaluation funcrional is the invertible functional e"' and we have
The forward diflirrnce functional is the delta functional ey' -
1 I P(.Y)) = P ( Y ) - P
O.
Example 3. The Abrlfunrtiunrd is the delta functional rey'. We have
and so
(ti"' I p(.u)) = p'(y).
'
Example 4. The invertible functional (1 - t ) - satisfies
It is associated with the Laguerre polynomials. Since ll!
=
1;
ll"P-"dU,
we get the integral representation ((1 - t ) - ' I p ( s ) ) =
Example 5. The functional
f(t)
s:
p(u)e-"du.
that satisfies
-
1
12
2 . SHEFFER SEQUENCES
for all polynomials p ( x ) can be determined from (2.1.5) to be
Its inverse -t/(e" - 1) is associated with the Bernoulli polynomials. In fact, the numbers En = ( t / ( e ' - 1 ) l x " )
are known as the Bernoulli numbers. We shall have more to say about this in Chapter 4. Example 6. The invertible functional (1 ((1
+ e y ' ) / 2 satisfies
+ e Y f ) / 2 1 p ( x ) )= (P(0)+ P(Y))/2.
Its inverse 2/(1 + ey')is associated with the Euler polynomials, also discussed in Chapter 4. 2. LINEAR OPERATORS
Let us begin this section by doing something that may seem, at first, to lead to some confusion. Namely, we use the notation t k for the kth derivative operator on P , that is,
where (n)k = n(n - I)...(n - k
+ 1). With this notation, any power series f(-t) =
k
(2.2.1)
(n)akx.-k.
(2.2.2)
'k
1 --t
k=Ok!
is a linear operator on P defined by f(t)x" = k=O
k
Notice that we use juxtaposition /(-t)p(x)to denote the action of the operator
f(r) on the polynomial p(x). Thus an element of 9 plays three roles in the umbra1 calculus-it is a formal power series, a linear functional, and a linear operator. A little familiarity should remove any discomfort that may be felt by the use of this trinity, and the notational difference between
("m)I P W >
and
will make the particular role of f ( t ) clear.
f(t)P(X)
2.
13
LINEAR OPERATORS
As with linear functionals, we note that two elements f(r)and g(t) in 9are equal as formal power series if and only if they are equal as linear operators. (To see the "if" part, take successively n = 0,1,2,. . . in (2.2.2).) Since (rkti)x"= t k + j . x " = t"(t'.u"), we conclude that
Lf(r)dt)lP L Y ) = f(OLY(t)P(-x)I
(2.2.3)
for all f ( r ) and y ( t ) in 3 and p ( x )in P, and so we may write f ( r ) g ( t ) p ( x )without ambiguity. Equation (2.2.3) shows that the product in 4 is also the composition of operators. We remark that .f(t)dt)P(.Y) = d t ) f ( f ) P ( X )
for all f ( t ) and y(t) in .F and p ( s ) in P . The operator t o is, of course. the identity operator. When we think of a delta (or invertible) series as an operator, we shall refer to it as a delta (or invertible) operutor. Let us consider the operator analogs of Propositions 2.1.4-2.1.7. Since the proofs are also analogous, we shall omit them.
) degp(.u), then f ( t ) p ( x ) = 0. Proposition 2.2.1 If o ( f ( t ) >
Proposition 2.2.2 If o(,fk(r))= k for all k 2 0, then
for all p ( ~in) P , the second sum being a finite one.
Proposition 2.2.3 If o ( f i ( r ) ) = k for all k 2 0 and if fk(t)p(x) = f k ( t ) q ( X ) for all k, then p(.x) = q(x). Proposition 2.2.4 1fdegpJ.u) all k, then f ( t ) = y ( t ) .
=
k for all k 2 0 and if f(r)pk(.u) = g ( f ) p k ( x for )
We now come to the key relationship between the functional f ( r ) and the operator f ( t ) . Theorem 2.2.5 If f ( f )and g ( r ) are in 3, then
( f ' ( r ) d r ) I P(.Y))
= ( d t )If ( r ) p ( . x ) )
for all polynomials p(.x). Proqf' Writing f ( t )and q ( t ) in the form
14
2.
SHEFFER SEQUENCES
and using (2.2.2),we get
which, according to Proposition 2.1.2, gives the desired result. Incidentally, consideration of Theorems 2.1.10 and 2.2.5 points out the advantages of the notation ( L I p ( x ) ) . From Theorem 2.2.5 we see that (f(t)I
P(X)> = (to
If(Mx)).
In words, applying the functional f ( t ) to p ( x ) is the same as applying the operator f ( t ) and then evaluating at x = 0. Let us consider the operator versions of the examples of Section 1. 1. The operator e" satisfies
and so e"p(x) = p(x
+ y).
For this reason ey' is called a translation operator. 2. The forward diference operator ey' - 1 satisfies (ef'
-
I)p(.u) = p ( x
+ y) - p ( x ) .
3. The Abel operator is the delta operator fey',and we have ref'p(x) = t p ( x 4. The operator (1
Since
-
1)-
+ y ) = p'(x + y).
' satisfies
2.
15
LINEAR OPERATORS
we have (1
-
t)-'p(x)=
so'
p(x
+ u)e-"du.
5. The operator (eyt- l)/r is easily seen to satisfy
and so eyr- 1 t
6. The operator (1
+ e Y r ) / 2is given by 1 + eyr P(X) + P(X + Y ) P(X) =
2 Unlike the case for linear functionals, not all linear operators on P take the form of a power series in 4.For example, since deg f ( t ) p ( x ) Ideg p ( x )for any f ( t ) in 4,the linear operator multiplication by x , XP(X), cannot have the form f ( t ) for any series in 4. We devote the remainder of As a point this section to various characterizations of the operators in 5. of semantics, we shall say that a linear operator T on P has the form g ( t ) if T p ( x ) = g ( t ) p ( x )for all polynomials p ( x ) . First we require a lemma. P(.Y)
+
Lemma 2.2.6 Let f ( t ) be a delta operator and let T be a linear operator on P that commutes with f(t), that is, for all polynomials p ( x ) . Then deg T p ( x ) I degp(x) for all p ( x ) .
Proof If the conclusion did not hold, there would exist an integer m 2 0 such that deg Tx" 2 rn 1. Now, since o ( f ( t ) )= 1, it is clear that
+
deg f ( t ) p ( x ) = deg P ( X ) whenever deg p ( x ) > 0. Thus we have
-
0 = Tf(f)"+lX"= f ( f ) " + ' T X "
1
# 0.
This contradiction establishes the lemma.
Theorem 2.2.7 A linear operator T o n P has the form g ( t ) if and only if it commutes with the derivative operator, that is, if and only if TCtP(41
for all polynomials p ( x ) .
=
tCTP(X)I
16
2.
SHEFFER SEQUENCES
Proof If T has the form g ( t ) , then it commutes with all operators in 9.For the converse, let
Then
But, using Theorem 2.2.5, we see that ( t o 1 Tx')
kl
= -(to
n! k!
= -n(!t o
I TTkxn) Itn-kT~")
and so
Thus T has the form g ( t ) and the proof is complete. Corollary 2.2.8 A linear operator T on P has the form g ( t ) if and only if it commutes with any delta operator.
Proof If T has the form g(t), then it commutes with any operator in 9. Conversely, suppose that T f ( t )= f ( r ) T for some delta operator f(t). Then there exist constants uk such that
Using Lemma 2.2.6, we have n
TCX"
=T
C akf(t)kxn
k=O
and so we may invoke Theorem 2.2.7 to conclude the proof.
3.
17
SHEFFER SEQUENCES
Corollary 2.2.9 A linear operator T on P is of the form g(t) if and only if it commutes with any translation operator eyt. Proof This follows from Corollary 2.2.8 and the fact that T commutes with e”‘ if and only if it commutes with the delta operator err - 1. 3. SHEFFER SEQUENCES
By a sequence s,(x) of polynomials we shall always imply that degs,(x) = n. Theorem 2.3.1 Let f ( t ) be a delta series and let g ( t ) be an invertible series. Then there exists a unique sequence s,(x) of polynomials satisfying the orthogonality conditions
(g(t)f(t)k1 sn(-x))
= n! 6n.k
(2.3.1)
for all n, k 2 0. Proof The uniqueness follows from Proposition 2.1.6. For the existence, if we set s,(x) = a,,,j ~and j Y(f).f(f)k = x?lk bk,ifiwith b k , k # 0, then (2.3.1) becomes
c;=,
Taking k
=
n, one obtains an.n
=
1lbn.n.
By successively taking k = n, n - 1,. . . ,0, we obtain a triangular system of equations that can be solved for an,k. We say that the sequence s,(x) in (2.3.1) is the ShefSer sequence for the pair (g(t),f(t)), or that s,(x) is Sheflkr for (g(t),f(t)). Notice that g(t) must be invertible and f ( t ) must be a delta series. There are two types of Sheffer sequences that deserve special consideration. The Sheffer sequence for (1, f ( t ) ) is the associated sequence for f ( t ) , and s,(x) is associated to f ( t ) . The Sheffer sequence for (g(t), t ) is the Appell sequence for g ( t ) , and s,(x) is Appell for g(t). Incidentally, the term Appell sequence appears in the literature, but usually with a slightly different
18
2.
SHEFFER SEQUENCES
meaning; s,(x) is Appell in our sense if and only if s,(x)/n! is Appell according to these other sources. For convenience, in the next two sections we reformulate the results of this section for these two special cases. Since ( t k I x " ) = n! b n , k , the sequence p,(x) = x n is associated to the delta functional f ( t ) = t . The next two results are among the most useful in the umbra1 calculus. The first one shows how to express any power series in terms of the geometric sequence g( t )f(t ) k .
Theorem 2.3.2 (The Expansion Theorem) Let s,(x) be Sheffer for (g(t),f(t)). Then for any h ( t ) in 3
Proof
The expansion theorem follows from Proposition 2.1.7 since
=
( W I)s,(x)>.
The polynomial analog of the expansion theorem shows how to express an arbitrary polynomial as a linear combination of polynomials from a Sheffer sequence.
Theorem 2.3.3 (The Polynomial Expansion Theorem) Let s,(x) be Sheffer for (g(r),f(t)). Then for any polynomial p ( x ) we have
Applying g(t)f(t) to either side of this equation gives the same result (g ( t ) f ( t ) kI p ( x ) ) . Hence Proposition 2.1.6 completes the proof.
Proof
It is our intention now to characterize Sheffer sequences in several ways. We begin with the generating function.
Theorem 2.3.4 The sequence s,(x) is Sheffer for (g(t),f(t))if and only if (2.3.2) for all y in C, where f c t ) is the compositional inverse of f(t). Proof
If s,(x) is Sheffer for (g(t),f(r)), then by the expansion theorem
3.
19
SHEFFER SEQUENCES
Thus
and finally p e ~ /1U )
=
y(f(t))
wtk, k=o
k!
For the converse, suppose that (2.3.2) holds. Then if r,(x) is Sheffer for ( d t ) ,f'(t)), we have 'k(y)
---f k=O
k!
k
=-
er/(f) =
q(f(t))
2
k=o
St(Y)rk,
k!
and so rk(y ) = s k ( y) for all y in C , which implies that r k ( x )= sk(x). If we allow the use of formal power series in the two variables x and t , we could write (2.3.2)as (2.3.3) While this is the usual form for a generating function, it has a small drawback in the present context, namely, when we think of x and t as formal variables they commute, but when we think of t as a linear operator they do not, for t acts on polynomials in x. This is not a serious problem and when we use (2.3.3) it is with the tacit understanding that t is nothing but a formal variable. Some caution must be exercised with the term Sheffer sequence when consulting the literature. For example, sequences of A-type zero are characterized by Sheffer as sequences u,(x) that have a generating function of the form
where o ( A ( t ) )= 0 and o ( B ( t ) )= I . Thus S,,(.Y) is a Sheffer sequence if and only if s,(.x)/n! is a sequence of Sheffer A-type zero. As we have already noted, the same remark applies to Appell sequences. Incidentally, sequences of Sheffer A-type zero are called poweroids by Steffensen [l] and sequences of generalized Appell type by the Bateman Manuscript Project (Erdelyi [l]). although in Boas and Buck [ l ] the latter term is used for a more general set of polynomial sequences. The generating function leads us to a representation for Sheffer sequences.
Theorem 2.3.5
The sequence s,(.Y) is Sheffer for ( g ( t ) ,f ( t ) ) if and only if (2.3.4)
2.
20
Proof
SHEFFER SEQUENCES
Applying the right side of (2.3.2)to .xn gives
and applying the left side to x n gives
Since this holds for all y in C , the result follows. Equation (2.3.4) is called the conjugate representation for s,(x).
Theorem 2.3.6 The sequence s,(x) is Sheffer for (g(r),f(r)) if and only if g(t)s,(.x) is the associated sequence for f(r). Proof
This follows directly from Theorem 2.2.5 and the definitions since ( f ( t ) k I g(f)sn(x)) = < g ( t ) f ( r ) k1 s n ( x ) )
= n ! 6n,k.
Theorem 2.3.6 says that each associated sequence gives rise to a class of Sheffer sequences, one sequence for each invertible operator g ( f )in 9. We next give an operator characterization of Sheffer sequences. Theorem 2.3.7 A sequence s,(x) is Sheffer for (g(t), f(t)), for some invertible y(t), if and only if (2.3.5) f ( t ) s n ( x ) = nsn - 1 (x) for all n 2 0. Proof
If s,(x) is Sheffer for (g(r),f(t)), then (g(t)f(r)kI f ( t ) s n ( x ) ) = ( g ( t ) f ( t ) k + l I sn(x)) = n! 6 n . k + 1 = n(n =
l)! s,-
l,k
( g ( t ) f ( r ) kI ns,-
I(X)),
and so f ( t ) s n ( x ) = ns,- l(x). Conversely, suppose that (2.3.5)holds and let p,(x) be associated to f ( t ) .We define a linear operator T on P by Tsn(x) = pn(x).
This operator is well defined and invertible since both s,(x) and pn(x) form a basis for P. Then, since the first part of this proof shows that
3.
21
SHEFFER SEQUENCES
f(t)Pn(-x) = npn - 1 (XI,we have
Tf(r)s,(.x) = nTsn- l(.x) = “Pn
- 1
(x)
= S(r)p,(x) = f(t)Ts,( x ),
and so Tf(t) = f’(t)T.From Corollary 2.2.8 we deduce the existence of an invertible series g ( t ) for which q(t)s,(x) = pn(x). The result follows from Theorem 2.3.6. From Theorems 2.3.6 and 2.3.7 we get a formula for the action of an operator in 9 on a Sheffer sequence.
Theorem 2.3.8 Let s,(x) be Sheffer for ( g (f ) , J(t)) and let p n ( x )be associated to f(t). Then for any h ( t ) in 9,
Proof
By the expansion theorem
Applying each side of this equation, as an operator, to s,(.K) gives
We turn now to an algebraic characterization of Sheffer sequences that generalizes the binomial formula.
Theorem 2.3.9 (The Sheffer Identity) A sequence s n ( x ) is Sheffer for ( g ( t ) , f ( t ) ) for , some invertible y(t), if and only if
for all y in C , where p,(x) is associated to . f ( t ) ,
2.
22
Proof
SHEFFER SEQUENCES
Suppose that s,(x) is Sheffer for ( g ( t ) , f ( t ) ) .By the expansion theorem
and applying both sides of this equation to s,(x) gives s,(x
+ y ) = eYrs,(x)
For the converse, suppose that the sequence s,(x) satisfies the Sheffer identity, where p , ( x ) is associated to f ( t ) . We define a linear operator T on P by W X )
= P,(X).
Then according to Theorem 2.3.6 it sufficesto prove that T has the form g ( t ) in
9. The first part of this proof shows that p , ( x ) satisfies the Sheffer identity and so
=
Ts,(x
+ y)
= Te”s,(x).
Therefore e y “ T= Te” and Corollary 2.2.9 shows that T has the form y(t) in 9. This concludes the proof. Incidentally, by interchanging x and y in the Sheffer identity and setting y = 0, we get
Thus, given a sequence p,,(x) associated to f ( t ) , each Sheffer sequence s,(x) that uses f ( t ) as its delta functional is uniquely determined by the sequence of constant terms s,(O). Moreover, any sequence of constants a, for which a, # 0
3.
23
SHEFFER SEQUENCES
gives rise in this way to a Sheffer sequence, using f ( t )as its delta functional. T o see this, suppose that a, is such a sequence of constants and define s,(x) by
Then since a, # 0, s,(x) is a sequence and
- ns,-
1 (.Y).
By Theorem 2.3.7 the sequence s,(.Y) is Sheffer for (g(t),f(r))for some g ( t ) . An important property of Sheffer sequences is their performance with One might wish t o compare the next result with respect t o multiplication in 9. Proposition 2.1.2. Theorem 2.3.10 Let s,(x) be Sheffer for ( g ( r ) , f ( t ) ) and let p , ( x ) be associated to f ( t ) .Then for any h ( t ) and I ( r ) in .Pwe have
We conclude this section with an illustration of how nicely umbra1 results can marry to produce useful formulas that are general enough to apply to all Sheffer sequences.
2.
24
We wish to determine the coefficients
SHEFFER SEQUENCES
in the expansion
n t I xsn(x)
=
1
Un,kSk(X),
k=O
where s,(x) is a Sheffer sequence. If s,(x) is Sheffer for ( g ( t ) , f ( t ) ) , then by the polynomial expansion theorem
where we take (7) = 0 if j < 0 or j > m.
It is also worth singling out a formula for sk(x) in terms of s k ( x ) . Theorem 2.3.12 If s,(.x) is Shefferfor (g(t),f ( t ) )and p n ( x )is associated to f ( r ) , then
Proof
By the polynomial expansion theorem
4.
25
ASSOCIATED SEQUENCES
4. ASSOCIATED SEQUENCES
We reformulate the results of the previous section for associated sequences. One should take special note of Theorem 2.4.5.
Theorem 2.4.1 (The Expansion Theorem) Let p,(x) be associated to Then for any h ( r ) in 9
Theorem 2.4.2 (The Polynomial Expansion Theorem) ciated to f ( t ) . Then for any polynomial p ( x ) we have
Theorem 2.4.3
f(t).
Let p,(x) be asso-
The sequence p,(.w) is associated to f ' ( t )if and only if
for all y in C.
Theorem 2.4.4
The sequence p , ( x ) is associated to f ( t ) if and only if
This is the conjugate representation for associated sequences. The operator characterization of associated sequences adds a new feature.
Theorem 2.4.5
The sequence p , ( x ) is associated to f ' ( r ) if and only if
(i) < t o I P , W > = 4I.0, (4 f ' ( t ) p , ( x ) = npn- Ib). Suppose that p,(x) is associated to f ( r ) . Then ( f ' ( t ) k I p , ( x ) ) = n! (Sn,kr and setting k = 0 gives (i). Part (ii) follows from Theorem 2.3.7. Conversely, if (i) and (ii) hold, then Proqf
( f ( t ) k Ip n ( x ) )
=
=
( to
I .f(l)kPn(X)> I(n)kpn-k(x))
= (n)k (5,-k.O
= n! (S,.k.
One should notice that (i) is equivalent to p O ( r )= 1 and p,(O)
=0
for n > 0.
2.
26
SHEFFER SEQUENCES
Theorem 2.4.6 If p,(x) is associated to f ( r ) ,then for any h ( t ) in 9
Theorem 2.4.7 (The Binomial Identity) The sequence p,(x) is an associated sequence if and only if
for all y in C. A sequence p,(x) that satisfies the binomial identity is known as a sequence of binomiul type. According to Theorem 2.4.7, a sequence is an associated sequence if and only if it is of binomial type. Theorem 2.4.8 If p , ( x ) is associated to f ( t ) , then
Theorem 2.4.9 If p,(.x) is an associated sequence, then
5. APPELL SEQUENCES
Let us recast the results of Section 3 for Appell sequences. Recall that our Appell sequences may differ from others in the literature by a factor of n!.The last result of this section has not appeared previously. Theorem 2.5.1 (The Expansion Theorem) Let s,(x) be the Appell sequence for g(t). Then for any h ( t ) in B
Theorem 2.5.2 (The Polynomial Expansion Theorem) Let s,(x) be Appell for y(t). Then for any polynomial p ( x ) we have
where p(II)(x)= t k p ( x )is the kth derivative of p(.x).
5.
27
APPELL SEQUENCES
Theorem 2.5.3 The sequence s,(.Y) is Appell for y(t) if and only if 1
--ey!
g(t)
=
u k=o
f
k
k!
for all y in C.
Theorem 2.5.4 The sequence s,(x) is Appell for g ( t ) if and only if
This is the conjugate representation for Appell sequences.
Theorem 2.5.5 The sequence s,(.Y) is Appell for g ( t ) if and only if S,(X)
= g ( t ) - IX".
Theorem 2.5.6 The sequence s,(x) is Appell for g ( t ) if and only if
that is, It is common in the literature to find that Appell sequences are defined by the condition ub(x) = u,- l(x). Then n! u,(x) is Appell in our sense.
Theorem 2.5.7 Let s,(x) be Appell for y(t). Then for any h ( t ) in .F
Theorem 2.5.8 (The Appell Identity) The sequence s,(x) is an Appell sequence if and only if
Theorem 2.5.9
If s,(x) is Appell for g ( t ) , then
Theorem 2.5.10 (The Multiplication Theorem) If s,(x) is the Appell sequence for g(r), then for any constant cx # 0
for all n 2 0.
2.
28
SHEFFER SEQUENCES
and so from which the result follows.
6. A FEW EXAMPLES
For reasons outlined in the Preface, we pause briefly to discuss a few examples of Sheffer sequences. No proofs or derivations will be given here and for convenience all results will be repeated in Chapter 4. Although it may seem from the coming discussion that in some cases the determination of the Sheffer sequence for a given pair of linear functionals is oblique, in the next chapter we shall obtain formulas for the explicit computation of these sequences. One may wish to refer to the examples of linear functionals at the end of Section 2.1 and the corresponding examples of operators in Section 2.2. Example I . The sequence p,(x) = x n is, of course, associated to the delta functional f ( r ) = f . The generating function
and the binomial identity (x
+ y)" = kf: =o
(")."y"-k
k
should come as no surprise.
Example 2. The lower factorial polynomials (x), = x ( x - l ) . . . ( x - n
+ l),
6.
29
A FEW EXAMPLES
also called the falling factorial or binomial polynomials, are associated to the forward difference functional f ( t ) = e' - 1.
This can easily be seen from Theorem 2.4.5. The generating function is
which is equivalent to the binomial expansion
The binomial identity is
which can be rewritten as
This is known as Vandermonde's convolution formula.
Example 3. The Abel polynomials A n ( x ; a )= x(x - an)"-'
are associated to the Abel functional ,f(r)
=
re"'.
This can readily be verified from Theorem 2.4.5. The generating function is
Incidentally, if we differentiate with respect to x, we obtain
Setting x = 0 gives
30
2.
SHEFFER SEQUENCES
The binomial identity in this setting is (x
+ y)(x + y
- un)"-' =
$ ( nk ) x y ( . ~- u k ) k - l ( y- a(n - k ) ) n - k - l ,
k=O
which is a formula due t o Abel Example 4.
The Hermite polynomials H,,(x)form the Appell sequence for g ( t ) = e12'2.
By Theorem 3.5.5 we have
The generating function is
The Appell identity (Theorem 2.5.8) is
Some care must be exercised here since the term Hermite polynomial is frequently used for the sequence s,(x) satisfying
This sequence is not exactly Appell, but it is Sheffer for (e1''4,t / 2 ) . In Chapter 4 we discuss the more general Hermite polynomials Hp'(x) of variance v. Exampfe 5.
The Bernoulli polynomials B,(.x) form the Appell sequence for g ( t ) = (e' -
l)/r.
Theorem 2.5.5 gives us B,(x) = ( [ / ( e l- 1 ) ) ~ " .
The generating function is
6.
31
A FEW EXAMPLES
Taking .Y
= 0, we
have
The numbers B,(O) are known as the Bernoulli numbers. In Chapter 4 we discuss the more general Bernoulli polynomials B?)(.Y)of order a.
Example 6. The Laguerre polynomials L1p’(x)of order M (sometimes known as the generalized Laguerre polynomials) form the Sheffer sequence for g(r)= (1
-
t ) - ” - l,
f(t) = t/(t-
I).
The sequence LL-” is associated to f ( t ) = t / ( t - 1). From the formulas of the next chapter we shall easily compute that
The generating function is
The Sheffer identity is
Many interesting properties of the Laguerre polynomials follow by umbra1 methods from the fact that f ( t ) = f’(t). Once again, some care mmt be exercised with the term Laguerre polynomials, for in the literature they are frequently defined as LF)(x)/n!.Also, for reasons related to orthogonality, the condition a > - 1 is sometimes invoked. Needless to say, we shall disregard this constraint.
CH,\PTER 3
OPERATORS AND THEIR ADJOINTS
1. CONTINUOUS OPERATORS ON P*
In this and the next two sections, we shall let P * denote the umbral algebra, This will conform to standard usage and which we have been denoting by 9. should not be misleading since, for the time being, we shall be concerned with the vector space structure of the umbral algebra only. Linear operators defined on the umbral algebra P* play an important role in the umbral calculus. An indispensible property of such an operator T is that it pass under infinite sums, that is, T
1
(3.1.1) whenever a,f,(t) exists, which, as we have remarked, is precisely when o ( f , ( t )+ ) z as k + K . But in order for (3.1.1) to be meaningful, the right side must also exist, that is, o( T f k ( f ) ) -+ ‘I; as k -+ CG. Thus a necessary condition for T to have the property expressed in (3.1.1) is that o(Tfk(f))
+
cc
whenever
o(fk(t))
-+
m.
This turns out to be sufficient as well. Theorem 3.1.1
If T is a linear operator on P* satisfying (3.1.2), then
c I
akfk(t)
=
k=O
for all sequences
fk(t)
1
nkTf,(t)
k-0
for which o ( f , ( r )-, ) cc as k 32
+ ;c.
(3.1.2)
2.
33
AUTOMORPHISMS AND DERIVATIONS ON P*
Proqf Suppose that o ( f k ( t ) )+ m. For any polynomial p ( x ) and any m 2 0 we have
(3.1.3) Now o(fk(r))+ co implies that o[c;=,+ a , f k ( t ) ]+ cc as m + m, and so by (3.1.2)wehaveo[TC,"=,+, a k f k ( f ) ] + CD asrn + cz.Thuswemaytakernlarge enough so that the second term on the right of (3.1.3) vanishes and so that o( Tfk(t)) > degp(x) for k > rn. Then
=
(
k$O
ak
Tfk(t)
I po>.
This proves the theorem. We shall say that a linear operator T on P* is continuous if it has the property expressed in (3.1.2). Actually, it is possible to define a topology on P* by specifying that the sequence fk(f) converges to f ( t ) if, for a n y n 2 0, there exists an integer k , such that if k > k,, then f k ( f ) and f ( t ) agree in the first II terms; in symbols, ( f k ( t ) I x') = (f(t) I x') for 0 I j I n - 1. I t can be shown that this makes P* into a topological vector space. Clearly, the sequence f k ( t ) converges to 0 if and only if o ( f k ( t ) ) + cc as k + m. Now a linear operator T is continuous if and converges to 0 whenever f k ( r ) converges to 0. In other words, T is only if Tfk(t) continuous if and only if it satisfies (3.1.2). Since we shall not require any topological notions other than continuity, we take (3.l.2) as the definition. 2. AUTOMORPHISMS AND DERIVATIONS ON P*
Let us recall that a linear operator T on P* is an algebra morphism if it preserves the product on P*,
7-[f(t)Y(r)l
=
Tf(t)Tg(t).
If in addition T is bijective, then it is said to be an autornorphisrn of P*. Theorem 3.2.1
If T is an automorphism of P*, then T preserves order, that is, o(Tf(t))= .(f(t,)
for all f ( t ) in P*.
3.
34
OPERATORS AND THEIR ADJOINTS
Proof First we observe that T i ( [ is ) invertible if and only if f ( t )is invertible. For if f ( t ) - ' exists, then Tf(t)T(f(t)-') = T ( f ( t ) f ( t ) - '= TI = 1. Hence Tf(t)is invertible, with inverse T ( f ( t ) - ' ) The . converse follows in a similar manner, using the automorphism T - In terms of order we have shown that o ( T f ( t ) )= 0 if and only if o(f(r)) = 0. In particular then, o ( T t )2 1. But if o( T t ) > 1, then there would be no g ( t )in P* for which Tg(t)= t. For any g(t)in P* can be written in the form g ( t ) = go + tgl(t), and then Tg(t)= go T ( t g , ( t )# ) t because o ( T t g , ( t ) )= o ( T t T g , ( t ) )2 o ( T t )> 1. Thus, since T is surjective, we must have u ( T t )= 1. This implies that o ( T t k )= k for all k 2 0. Finally, if o ( f ( t ) )= k, then f ( t ) = t k g ( r ) , where o ( g ( t ) ) = 0, and so O( Tf(t))= O ( TtkTg(t))= o( T t k )+ O( T g ( t ) )= k.
'.
+
Corollary 3.2.2 Any automorphism of P* is continuous.
As a consequence, we remark that any automorphism of P* is uniquely determined by its value on any delta functional, in particular by its value on t . A linear operator d on P* is a derivation on P* if
c"f(r)g(r)l = f(0& ( t ) + g ( t ) ?f(r) for all f ( t ) and g ( t ) in P*. Theorem 3.2.3 If d is a surjective derivation on P*, then dc = 0 for all constants c in C and o ( d f ( t ) )= k - 1 whenever o ( f ( t ) )= k 2 1. Proof Firstweobservethatdl = d I 2 = d l + a l , a n d s o d l =O.Thusdc=O for any constant c. Now any g ( t ) in P* has the form g ( t ) = go + tgl(r). Hence Sg(t) = t S g , ( t ) + gl(t)3t, and since o(rdg,(t)) 2 1, the fact that d is surjective implies o(&) = 0. Finally, if o ( f ( t ) )= k L 1, then f(t) = tkg(t),where o ( g ( t ) )= 0, and so o ( ? f ( t ) )= o ( t k ? g ( r ) + k t k - ' g ( t ) S t ) = k - 1.
Corollary 3.2.4 Any surjective derivation of P* is continuous. 3. ADJOINTS
If i,is a linear operator on P , its adjoint is the linear operator i.* on P* defined, for each f ( t ) in P*, by the condition
O.*f(t) I P(X))
= ( f ( t )I M x ) )
(3.3.1)
for all polynomials p ( x ) . For example, since (f(r)g(t)/p(x)) = (g(r)J f ( t ) p ( x ) ) , we see that the adjoint of the operator iL = f ( t ) is the operator multiplication by f ( t ) .That is, f(t)*g(t)=f(t)g(t)
for all g(t) in P*.
3.
35
ADJOINTS
It follows easily from (3.3.1) that (j.
+ p)*
=
i* + p*,
(c/.)*= cR*,
(i r. p)*
= p*
(3.3.2)
E.*,
(A-')* = ( I . * ) - ' ,
i. invertible.
Thus the adjoint map, which sends a linear operator i. on P to its adjoint /.* on P*, is a linear transformation from the vector space of all linear operators on P to the vector space of all linear operators on P*. Furthermore, if R* = 0, then ( f ( t )I i&)) = 0 for all f'(t) in P* and all p ( x ) in P . Thus EL = 0 and the adjoint map is injective. In the next theorem we determine its range.
Theorem 3.3.1 A linear operator T o n P* is the adjoint of a linear operator i. on P if and only if T is continuous. Proof
Suppose first that T
=
EL* for some operator jkon P, and let o(.f&))
+
x.
Then for all n 2 0 there exists an integer k, such that k > k, implies o ( f k ( t )> ) d e g i x j for all 0 I jI n . Thus if k > k,, we have ( T f ( f I)x j ) = (fk(t)I i x j ) = 0 for 0 I j I n, that is, o( Tfk(t))> n. Thus o(Tfk(t))4 00 and T is continuous. Conversely, suppose that T is continuous. We define a on P by linear operator i. i.x" =
( TtkI x") 1 x k>O k!
k,
the sum being a finite one since u( T t k )> n for k large. Then (3.*rmlxn)
= (tmlE..x")
=
(Tt" I x"),
and so i.*t" = Tt" for all m 2 0. Since both 3.* and T are continuous, i.*f(t) =
for all f ( t )in P*. Thus T
=
Tf(r)
i.and * the proof is complete.
In summary, we have established the following theorem. Theorem 3.3.2 The adjoint map (Fig. 3. l), which sends a linear operator on P to its adjoint /.* on P*, is a vector space isomorphism from the vector space of
3.
36
OPERATORS A N D THEIR ADJOINTS
Linear operators on P'
Linear operators on P
FIGURE 3.1 The adjoint map of Theorem 3.3.2.
all linear operators on P onto the vector space of all continuous linear operators on P*. Since both automorphisms and derivations on P a r e continuous, we may augment Fig. 3.1 as shown in Fig. 3.2. The linear operators on P that fill in the upper left-hand box in Fig. 3.2 are known as umbral operators and those that fill in the lower left-hand box are known as umbral shifts. We shall study these important operators in Sections 4 and 6, but first let us attend to one more result.
Linear operators on P
Linear operators on P'
FIGURE 3.2
4.
UMBRAL OPERATORS AND UMBRAL COMPOSITION
37
If T is a linear operator on P*, then it too has an adjoint T*, defined on P** by (T*4))f(O=
Wm)
for all $ in P** and f ( r ) in P*. Let us recall that while the vector spaces P and P** are not isomorphic, there is a vector space isomorphism 8, from P into P**, defined by (@(x)).f(t)= (J@fI P
W)
for all f ( t )in P* and p ( x ) in P. The isomorphism 8 is known as the canonical isomorphism. It is common to identify the image 8 ( P ) of the canonical isomorphism with the space P of polynomials, in other words, to think of a polynomial p ( x ) as a linear functional on P*, where p ( x ) f ( t )= (f(t) I p ( x ) ) . If T is a linear operator on P*, it is natural to wonder whether the adjoint T*:P** -+ P** maps "polynomials" in P** to "polynomials" in P**; that is, whether T*O(P) c O(P).The answer is given in the next theorem. Notice that it is in terms of the original operator T.
Theorem 3.3.3 Let T be a linear operator on P*. Then the adjoint T* satisfies T*8(P)c 8 ( P ) if and only if T is continuous. Proof Assume first that T*B(P)c O(P) and let o(f,(t))-+co. Then O - ' T * Q x j is in P for all j 2 0, and for all n 2 0 there exists an integer k, such that k > k, implies ( T * O x j ) , f k ( t=) ( f k ( t ) l O-~'T*H.x')= 0 for 0 I jI n. Thus k > k, implies ( Tfk(t)1 x j ) = (Hxj)(Tfk(t)) = ( T * O x j ) f k ( t )= 0 for 0 I j I n, that is, o(Tfk(t))> n. Hence T is continuous. For the converse, assume that T is continuous. Then T = i* for some operator 3, on P. We have
(T*OP(.d)f(t)) = (j.**Qp(.x))f(t) =
( H P ( 4 ) ( j L * f ( O )
= O - * f ( t )I P(.Y)) =
( f V ) IW x ) )
= (Oj.p(.x))f(r),
and so T*Bp(x)= Hj.p(.x) and T*O = 01.. Hence T*B(P) = OR(P) c B(P) and the proof is complete. 4. UMBRAL OPERATORS AND UMBRAL COMPOSITION
We return to the notation .% for the umbra1 algebra. Let p,(x) be the associated sequence for f ( t ) . Then the linear operator ion P, defined by i,x" = p , ( x ) ,
3.
38
OPERATORS A N D THEIR ADJOINTS
is called the umbra1 operator for p,(x) or for f(t). To indicate the dependence of 1 on f ( r ) we may use the notation i.,-. Incidentally, in the next section we shall study Shefer operators, which are linear operators %:x" + s,(x), where s,(x) is an arbitrary Sheffer sequence. Since deg p , ( x ) = n, it is clear that an umbral operator is a bijection. It is our intention in this section to characterize umbral operators by means of their adjoints. According to Theorem 3.3.1, the adjoint of an umbral operator is x" + p,(x) is an umbral continuous. Much more is true, however, since, if jL,-: operator, then (i./*f(r)k 1 x") = ( f ( r ) kI +x") =
(f(tIk IP n W )
= ? I ! bn.k
=
(rk
1 x"),
and so iff(r)l = tk
for all k 2 0. The continuity of 17 then implies that I f Y ( f ( t ) )= g(t)
for all g(t) in 9. In particular, if we take g ( t ) = T ( c )then ~,
(3.4.1)
for all g ( t ) in 9. In words, the operator lL? is substitution by f c t ) . From (3.4.1) it follows that 1.; Y(f ) h(r1 = g
(rc
t )) h (
f(t 1) = q g (t) j.7 h(t),
and so i f is an automorphism of P . It is a pleasant fact that this characterizes umbral operators. Theorem 3.4.1 A linear operator i. on P is an umbral operator if and only if its adjoint I * is an automorphism of 9. Moreover, if IJ is an umbral operator, then j.;s(t)
for all g ( t ) in 9. In particular, i;f(r)
= g(f(r,)
= t.
4.
UMBRAL OPERATORS AND UMRRAL COMPOSITION
39
Proof We have already seen that the adjoint of an umbral operator is an automorphism of 9satisfying (3.4.1).For the converse, suppose that A* is an automorphism of 9. It follows from Theorem 3.2.1 and the fact that i.* is a bijection that there exists a unique delta series j ( t )for which i.*f(r) = r. If p,(x) is associated to f ( r ) , then
( f ( t ) kI iL.urz>= (i.*f(t)k I x") = (tklx")
and so i.x" complete.
= p,(x).
=
n! dn,k
=
<.f(rIk I P " ( X ) ) ?
Therefore ,iis an umbral operator and the proof is
Corollary 3.2.2 and Theorems 3.3.1 and 3.4.1 imply that any automorphism of 9 is indeed the adjoint of an umbral operator. Thus we have the following result.
Theorem 3.4.2 The adjoint map, which sends a linear operator on P to its is a bijection between the set of all umbral operators on P and adjoint 3.* on 9, the set of all automorphisms of 9. We can now fill in one of the boxes in Fig. 3.2 of Section 3 (see Fig. 3.3). It is easy to see that the set of all automorphisms of 9 is a group under composition of operators. The adjoint map of Theorem 3.4.2 can be used to transfer this group structure to the set of umbral operators on P.
Linear operators on P
Linear operators on P'
FIGURE 3.3
3.
40
OPERATORS AND THEIR ADJOINTS
Theorem 3.4.3 The set of all umbral operators on P is a group under composition. In fact,
. /..,
/.,
=
i., j-,
i.;'
(3.4.2)
= ii.
Proof We give two proofs of the first part of this theorem. The first one uses Theorem 3.4.2. Let i. and p be umbral operators. Then (3. p)* = p* i.is* the composition of two automorphisms of 3 and so is an automorphism of 9. Thus by Theorem 3.4.2 the map i, p is an umbral operator. Similarly, since (I.-')*= (),*)-' is an automorphism of 6 , the map I.-' is also an umbral operator. Thus the umbral operators form a group under composition. The second proof is more direct and establishes (3.4.2)but is less elegant. We have from (3.4.1) (1
0
p
(i.j-
i.,)*h(r) = i.g* i f h ( t ) r
=
i.g*h( r(t))
= h(f(g(r0)) = h((g,)(r))
,
=i .g* j-h(t),
so (iLri4)* = i.g* and 2, (-j, = i., Similarly, 0
(j,; ' ) * h ( r ) = (i,*)' h ( r )= h ( f ( t ) )= i#z(t),
and so i,'i= 2,. Corollary 3.4.4 sequences.
An umbral operator maps associated sequences to associated
Proof Let p , ( x ) be an associated sequence and let 1. be an umbral operator. Then if p: X " + p,(x) is the umbral operator for p,(x), we have Ap,(x) = i.px". But j. p is an umbral operator and so i.p,(x) is an associated sequence. 0
Corollary 3.4.5 If p , ( x ) and q,,(x) are associated sequences and a: p,(x) + q,,(x) is a linear operator, then tl is an umbral operator. Proof Let j.:x n -+ p , ( x ) and p: x" + q,,(x) be umbral operators. Then pfi i.-'p,,(x) = q,,(x) = rp,(x), and so s1 = p i L -isl an umbral operator.
Let us see what effect Theorem 3.4.3 has on the corresponding associated sequences. According to the definition of umbral operator, the sequence A," /x" is associated to the delta functional g(f(r)). But if we write i r :X " -,p,(x), n
4.
UMBRAL OPERATORS AND UMBRAL COMPOSITION
41
then (3.4.3) In general, if p,(x) and q,(x) = Ci = q n . k ~are k sequences of polynomials, we define the umbral composition of' q , ( x ) with p,(x) to be the sequence n
qn(P(x)) =
qn.kPk(X). k=O
Thus (3.4.3)may be written as j-g j . x n
= qn(P(x)).
(3.4.4)
Theorem 3.4.6 The set of associated sequences is a group under umbral composition. In particular, if p , ( x ) is associated to f ( t ) and qn(x)is associated to g(t), then q , ( p ( x ) ) is associated to q ( f ( t ) ) . The identity under umbral composition is the sequence x n and the inverse of the sequence p,(x) is the associated sequence for fct).
Proof The second statement follows directly from (3.4.4).It is clear that the sequence x n is the identity under umbral composition. Setting g ( t ) = S ( t )in (3.4.4) gives x n = if j ~ =n q n ( p ( x ) ) , and so iJxn= qn(x)is the inverse of p,(x). But 2fx" is the associated sequence for f ( t ) . We close this section with a formula for the action of an automorphism of 9.
Theorem 3.4.7 Let ibe an umbral operator on P. Then as operators on P we have i,*q(t) = i - l q ( r ) i
for any operator y(t) in 9. Proof
For any f ( t ) in 9 and p(.x) in P, ( f ( f )I (j.*y(t))p(.x))
= ( f ( t ) ( A * g ( O )I P ( 4 ) =
(j.*Cs(t)O-*)- !m)l I P ( 4 )
tf@) IM x ) ) = (0. - ' ) * A t )I y ( t ) j d x ) ) = ( f ' ( r ) 1 j,- 'q(r)j.p(x)), = (d"j.*)F
3.
42
OPERATORS AND THEIR ADJOINTS
and so ik*g(t)p(x)= 2"- ' g ( t ) R p ( x ) .Thus as operators on P we have i.*g(t) = i.-'g(t)R.
In view of the fact that corollary.
Corollary 3.4.8
j.Tg(t) = g(f(t)),
we get the following useful
For any umbral operator i., we have j.,-"g(f(t))
= y(t)giwf
(3.4.5)
for all g ( t ) in F . 5. SHEFFER OPERATORS AND UMBRAL COMPOSITION
Let s,(x) be the Sheffer sequence for (g(t)f(t)). Then the linear operator E, on P defined by i x " = s,(x)
is called the She$%, operator for s,(x) or for ( g ( t ) , f ( t ) ) To . indicate the or ibg(r),f(r). Clearly, Sheffer dependence of i,on g ( t ) and f ( t ) we may write i,,,, operators are bijective. Since, if s n ( x ) is Sheffer for ( g ( t ) , f ( t ) ) ,then p , ( x ) = g(r)s,(x) is associated to f ( t ) ,we see that
i.s,lx"
= s,(x) = ~ ( t ')p n ( x ) = g(t)-'i.fx".
and so
ig,,= g(t)-'E,.
(3.5.1)
This makes the theory of Sheffer operators a corollary of the theory of umbral = i.;(g(t)- ')*, we have operators. In particular, since j.?.,-h(t)
= g(f(t))-'h(f(t))
(3.5.2)
for all h ( t ) in 9. Theorem 3.4.1 and Eq. (3.5.1) immediately give a characterization of Sheffer operators by their adjoints. Theorem 3.5.1 A linear operator E,, on P is a Sheffer operator if and only if its adjoint E,*. has the form of multiplication by an invertible series g ( t ) - followed by an automorphism 1.7 of 9. Moreover, if I.* has this form, then R = &. In fact, any operator on 4 that is of this form is the adjoint of a Sheffer operator.
5.
43
SHEFFER OPERATORS AND UMBRAL COMPOSITION
Let us consider the composition of Sheffer operators. If lLg(f),,(f) and jbh(f)q I(f) are Sheffer operators, then by (3.5.2) '
(lg(c).f(f) ih(f).
r(f))*u(t)
= %if).I ( r ) ' E L : ( t ) . f ( t ) u ( t ) =j&,l(1)9(m-
'.(.m)
= h ( W ' g ( m ) ) ) -' u ( f ( r ( N ) = (hr
/I((/.f)(r))-' g ( ( [ c , , f ) ( r ) ) - ' u ( ( / f)(r)) ~1
= j . h , , m .l
I m 4 0
for all u ( t ) in 9, and so 'Wf).f(f) -' A h ( f l . l ( O = ~ ~ ( f ) h ( f ( f ) ) ~ l ( ~ ( f ) ) .
We also have by (3.5.1) (j.,cff.f(f))*u(t)= Cj.n:, r' g(r)l*u(t) = &)*(Eq(:,)*u(t) = 9(t)>.f,,,u(t) = dt)u(f(t)) = ( 9 " .ff(f(O)u(.f(t)) = ~.,*((i.(,))'.i(t)u(')
for all u ( t ) in 9, and so - - I
' " g w ~ l - l f )= % f I f ) ) -
1.jll).
We have proved the following theorem.
Theorem 3.5.2 The set of all Sheffer operators is a group under composition. In fact, '*g(t).f ( f )
' 'h(r). I(r)
= 'l-glf)h(f(f)b. I(f(f))
).if;, f(,)= j.SIPlt,) -
I.
3
S(0'
(3.5.3)
Corollary 3.5.3 A Sheffer operator maps Sheffer sequences to Sheffer sequences. Let s,(x) be a Sheffer sequence and let i be a Sheffer operator. Then if p: xn + s,(x) is the Sheffer operator for s,,(.Y), we have j.s,(x) = 1.2 px". But 2. p
Proof
C-
is a Sheffer operator, and so j..s,,(x) is a Sheffer sequence.
Corollary 3.5.4 If s,(x) and r,,(x) are Sheffer sequences and 3: s,(x) linear operator, then a is a Sheffer operator.
+ r,(.u)
is a
3.
44
OPERATORS AND THEIR ADJOINTS
Proof Let i.: x n + s,(x) and p: x" + r,(x) be Sheffer operators. Then p E.-'s,(x) = r,(x) = as,(x) and so a = p 0 E.- is a Sheffer operator.
'
0
Now we come to the behavior of Sheffer sequences under umbral composition. Theorem 3.5.5 The set of Sheffer sequences is a group under umbral composition. In particular, if s,(x) is Sheffer for (g(t),f ( t ) )and r,(x) is Sheffer for (h(t),I([)), then r , ( s ( x ) ) is Sheffer for ( g ( t ) h ( f ( t ) I(f(t))). ), The identity under umbral composition is the sequence X" and the inverse of the sequence s,(x) is the Sheffer sequence for (g(f(r))-', f ( t ) ) . Proof
From (3.5.3) we have h(S(X)) =
&).
f(Or"(4
- j * g ( r ) . f ( t )'' j-hu). I ( r j X n - ' W ) h ( f U ) hl(f(r))xnt
and this proves the second statement of the theorem. The third statement is clear. To prove the last statement we observe that r,(s(x))= x" if and only if i . ~ ( ~ ) ~ ~ fis( the , ) ) identity, , , ( f ( r )which ) is true if and only if
s ( w o ( t ) =)
1 7
I(f(t)) = t .
Solving these equations shows that r,(x) is Sheffer for (hf%40) = (s(ffWl , fct,).
We summarize some of the previous results in a useful form in the next theorem.
6.
45
UMBRAL SHIFTS
6. UMBRAL SHIFTS AND THE RECURRENCE FORMULA FOR ASSOCIATED SEQUENCES
Let p , ( x ) be the associated sequence for f ( t ) .Then the linear operator 0 on P defined by 0P"(X) = P" + *(x) iscalled the umbrul shift for p , ( s ) or for f ( t ) . We use the notation 8, to indicate the dependence on f ( r ) . In the next section we shall study the more general Sheffer shfr @ sn(x)
-+
sn + 1
(XI
for s,(x) an arbitrary Sheffer sequence. Since degp,(.u) = n, we see that an umbral shift is injective, but it is not surjective. We proceed to characterize umbral shifts by means of their adjoints. If O,:p,(.x) .+ p n + l(x)is an umbral shift, we have
(Or*f(Ok IP , ( . d )
I 0JP"(X))
=
= <.f(t)kIPn+l(X)) =
(n
+ I)! 6 n +
L.k
= k n ! d,,, k + 1 =
=
k , f ( r ) k -I .
I pn(.X))
for all k 2 0, and so H,*f(r)'
By the expansion theorem any g ( t ) in 9 can be written in the form
and the continuity of H,* implies
Also, we have Off(t)y.(t)' = O ~ , f ( t ) " ' =(k
+ j)j(f)k+'-
(3.6.1)
3.
46 From this and the continuity of
OPERATORS AND THEIR ADJOINTS
of we get
+
H f g ( t ) h ( t ) = s ( t ) q h ( r ) N t )Ofs(t)
for all g ( t ) and h ( t ) in 9. Thus Of is a surjective derivation on 6. In view of (3.6.1), the map Of is actually the derivative with respect to f ( t ) ,which we denote by ZJ. That is, Of = i?f.Once again we have a characterization. Theorem 3.6.1 A linear operator 0 on P is an umbral shift if and only if its Moreover, if 0, is an umbral shift, adjoint 8* is a surjective derivation on 6. then Bf = ?f is the derivative with respect to f ( t ) , e;f(t)k
= kf(qk-1
for all k 2 0. In particular, O f f ( [ ) = 1. Proof We have already established one half of this theorem. Suppose now It follows from Theorem 3.2.3 that there that O* is a surjective derivation on 9. exists a delta functional f ( t )for which H*f(t)= 1. If p n ( x )is associated to f(t), then (f(tIk
I hJ.4) = ( f i * j ( t ) IkP , ( - 4 ) =
(kf(t,k-'fl*fcf)I P,(.4)
( k m k I P,(.) = (n + l)!h,+ =
-
1.k
= ( . f ' ( t I kI pn t l ( x ) ) v
and so U p , ( x )
= pn+ '(x) and
8 is the umbral shift for I(?).
Corollary 3.2.4, along with Theorems 3.3.1 and 3.6.1, implies that any surjective derivation on 9 is the adjoint of an umbral shift. Theorem 3.6.2 The adjoint map, which sends a linear operator 8 on P to its adjoint operator 8* on 9,is a bijection between the set of all umbral shifts on P and the set of all surjective derivations on 9.
We can now fill in the remaining box of our original diagram (see Fig. 3.4). Unlike the case for automorphisms, the set of surjective derivations on 9 does not form a group under composition. In casting about for some structure, we recall a well-known fact that almost every student of elementary calculus fails to learn, namely, any two derivations are related by a formula known as the chain rule! Theorem 3.6.3 (The Chain Rule) Let 9. Then
and
i, = ( ? , f ( t ) ) + .
?q
be surjective derivations on
6.
47
UMBRAL SHIFTS
Linear operators on P
Linear operators on P"
FIGURE 3.4
Proof
We have
zgf(f)k=
' i$f(t) = (dgf(l))dJf(t)k
for all k 2 0 and the result follows by continuity.
Corollary 3.6.4 If df and
?g
are surjective derivations on 8,then
The correspondence of Theorem 3.6.2 can be used to translate the chain rule into a relationship between umbral shifts.
48
3.
OPERATORS AND THEIR ADJOINTS
Taking g ( t ) = t in Theorem 3.6.5 and observing that dSx"= x"", that is, the operator 8,is multiplication by x, we get a very useful formula for 8,, 0,
= .x(?,f(r))- =
= .Y;;t
x ( f ' ( r ) ) -I .
Applying this to the associated sequence for recurrence formulas.
f(t)
leads to the following
Corollary 3.6.6 (The Recurrence Formulas) If p,(x) is associated to then (i) P n + 1 (ii) pn+ l(.x)
f(t),
= -xCf'(t)I - ' P , ( x )and =x~,[J(t)l'x~
for all n 2 0. Proof The first formula has been proved. For the second we use Corollary 3.4.8, pn+ l(x) = x [ f ' ( t ) ]
li,.x"
= xi.,[.f'(f(t))]-'x" = xi.,
[f(t ) ]'.x ".
Incidentally, the second form of the recurrence formula can be written as jLfxn+' = xi./[f(t)]'xn,
which is equivalent to the commutation rule -
2,x = xi.,[J(t)]'.
We can now derive a formula for surjective derivations on 8.
Theorem 3.6.7 Let 8 be an umbra1 shift. Then as operators on p we have
8*&)
= dt)O -
Wt)
for all operators g ( t ) in 3. Proof
For any f ( t )in 3 and p ( x ) in P we have (f'(t,
I ( f l * q ( r ) ) p ( . 4 )= ( f ( t ) ( o * d t ) ) I P W ) =
(O*f(t)dt) - dt)O*f(t)IP ( 4 )
= ( f l * f ( t ) . y ( t )I P ( 4 ) - ( g ( t ) O * f ( t ) I P ( 4 )
=
( f ( r ) lq(r)Qp(-4)- (f0)I @ q ( t ) p ( x ) )
=
( . f ' ( t )I Cg(t)O - W t ) l P ( 4 ) 3
from which the desired conclusion follows.
(3.6.2)
7.
49
SHEFFER SHIFTS
When 8: x n --r x"+ is the umbral shift for p , ( x ) (3.6.2) becomes
= x", the
map 8* is 8, and (3.6.3)
g'(t) = d t ) x - xg(t),
where both sides are considered as operators on P. The right side of this equation is known as the Pincherle derivative of the operator g ( t ) . 7. SHEFFER SHIFTS AND THE RECURRENCE FORMULA FOR SHEFFER SEQUENCES
Let s,(x) be the Sheffer sequence for ( g ( t) ,f(t)). The linear operator 8 on P defined by Hs,(.x) = sn+l(x) (3.7.1) is called the Sheffer shift for s,(x) or for ( g ( f ) f,' ( t ) ) . We also use the notation Of,,. Since deg s,(x) = n, a Sheffer shift is injective, but it is not surjective. If s,(.x) is Sheffer for (g(t),f ( r ) ) , then p,,(.x) = g(t)s,(x) is associated to f ( t ) . and so (3.7.1) becomes Q g . / g ( t ) - 'Pn(.u) = g([)-'pn+l(.x).
Thus
Og,f= d t ) - '@,ry(f)9 where OJ is the umbral shift for chain rule, we get
.f(t).
Using Theorems 3.6.6 and 3.6.7 and the
Qg,f = g(r)-l(!fdf) = g ( t )-
'C d f ) Q f - q q @ ) l
= Q,
dt)-
-
?,,,,df)
= 8, - g ( r ) - i,,,,t d,g(t) = xCf'(t)l-
-
d t ) - '")I
-
'g'(t)
We have proved the following result.
Theorem 3.7.1 If
Qg.f
is a Sheffer shift, then
This gives a recurrence formula for Sheffer sequences.
50
3.
OPERATORS AND THEIR ADJOINTS
Corollary 3.7.2 (The Recurrence Formula for Sheffer Sequences) If s,(x) is Sheffer for ( g ( t ) ,f ( t ) ) ,then
8. THE TRANSFER FORMULAS
We next derive two powerful formulas for the computation of an associated sequence from its delta functional. These formulas, together with Theorem 2.3.6, can be used to compute Sheffer sequences.
Theorem 3.8.1 (The Transfer Formulas) for f ( t ) ,then
If p,(x) is the associated sequence
for all n 2 0. Proof For convenience we set g ( t ) = f ( t ) / t .Then g ( t ) is invertible. Actually, the right sides of (i) and (ii) can be shown to be equal, using (3.6.3), . f ' ( t ) g ( t ) - " - lxn = [ t g ( t ) ] ' g ( t ) - " - I x n
+ r g ' ( t ) g ( t ) - " - lx" = g ( r ) - " s " + n g ' ( t ) g ( t ) - " - l.x"-l = g(t)-".x"
= g(r)-".u" -
[g(f)-"]'.u"-]
= g(r)-".u"
[y(t)-"x
-
-
.ug(t)-"]x"-
= .ug(f)-nXn- I .
To prove (i) we verify the two conditions of Theorem 2.4.5 for the sequence q,(.u) = j " ( f ) g ( f ) - " -'x". For the first condition, when n 2 1, the fourth equality above and Theorem 2.1.10 give (to1
q,(.u))
= ( t o I f"(t)y(r)-"- I X " ) =
(rolg(t)-"x" - [g(t)-"]'x"-')
= (g(t)-"l.u") - ([g(t)-"]'Ix"-l) = (g(t)-"s") = 0.
-
(g(r)-"lx")
8.
51
THE TRANSFER FORMULAS
If n = 0, then ( t o /4,(x)> condition
=
1, and so ( t o i qn(x)) = 8,,o. For the second
. f ( t ) ~ n ( - y )= . f ( f ) f ' ( t ) g ( t ) - " -'xn
=
f ~ g ( f ) f ' ( t ) y-,(t) IXn
=
nf'(t){](t)-"x"-
-
- nq, - I (x).
Thus q n ( x ) is associated to f ( t ) and the proof is complete. The next corollary gives a formula that directly relates any two associated sequences. Corollary 3.8.2
Let p,(x) be associated to f ( t )and let q n ( x )be associated to
y(t). Then for n 2 1,
[We remark that f(t)/g(t) is the invertible element of 9 defined by [f(t)/t] [g(t)/t] and that since p,(O) = 0 for n 2 1, the expression x-'p,(x) is indeed a polynomial.] Proof
By the first transfer formula we have (f(t)/t)" -s 'p,(.u) = x n- I = ( g ( t ) / t ) " x- '4,(x),
and the result follows by solving for q,,(x). We conclude this section with a rather useful result.
Theorem 3.8.3 If s,(x) is Sheffer for ( y ( t ) ,f ( t ) ) ,then r,(x) = S,(X
+ sl + bn)
is Sheffer for
Proof
The sequence r , ( x ) is Sheffer for the delta functional e - P ' j ( t ) since e @ff(t)r,(x) ~
= c fiff(t)e(a+pn)r Sn(.y) - l , e ~+iO n -
=
~
@)I
Sn - 1
(-XI
nr, . I (x).
Now suppose that r,(x) is Sheffer for ( h ( t )e, -O'f(t)).Corollary 3.8.2 implies that
52
3.
OPERATORS AND THEIR ADJOINTS
the associated sequence for e - p t f ( t ) is q n ( x )= xepnrx- 'p,(x)
and so
This gives the desired result.
CHAPTER 4
EXAMPLES
In this chapter we discuss some important examples of Sheffer sequences. It is not our intention here to give a treatise on any polynomial sequence, but rather to give enough examples of umbral techniques to allow the interested reader to continue on his own. In the next chapter we shall develop additional umbral techniques and apply them to the examples of this chapter. Our primary source for the definitions of special polynomial sequences is the Bateman Manuscript Project (Erdelyi [ l]), and we remind the reader of the remarks made in this regard following Theorem 2.3.4. We shall keep the reader informed of any discrepancies, albeit they are only minor, between our definitions and those of our sources.. In the hope of achieving greater clarity each example is divided into five parts, whose defining lines are by no means precise: definitions, the generating function and the Sheffer identity, recurrence relations and operational formulas, expansions, and miscellaneous results.
1. ASSOCIATED SEQUENCES
For ease of reference we list the key results. Let the sequence p , ( x ) be associated to f(r). Generating function
53
54
4. EXAMPLES Binomial identity
Expansion theorem
Polynomial expansion theorem
or
1,
55
ASSOCIATED SEQUENCES
Theorem 2.4.9
or
Conjugate representation
1.1. The Sequence x'
The sequence p,(x) = X " is, of course, associated to f ( t ) = t. The results in this case present no surprise and we list them quickly. The generating function is n
.*k
and the binomial identity is (x
+ y)" =
(n)xkg"-k. k=O
The expansion theorem is
which for g ( t ) = ey' is Taylor's expansion
Applying this to p ( x ) gives
k
4. EXAMPLES
56
1.2. The Lower Factorial Polynomials
DEFINITIONS Let us consider the forward difference functional f ( t ) = ear - 1
for a # 0. We note that f ' ( t ) = ae"', f c t ) = a-'log(l
+ t).
The associated sequence for f ( t )is easily determined by using Theorem 3.5.6, Y n ( Y ) = (eB'I P n ( X ) ) - (eY7v)
I -u">
- ( @' log(1 =
=
((1
(5)
+ t)lX">
+ t ) " o 1 xn>
n
Thus p,(x)
= (;)n
=
;( +.(; -
-
n
+ I).
These are the lower factorial polynomials, also known as the falling factorial or binomial polynomials. The first form of the recurrence formula would have worked as well to determine the sequence p,(x), Pn
+ 1 (x)= xCf'(t)I ' P ~ ( x ) = a-'xe-"'p,(x) X
= -pn(x - a),
a
and since p,(x) = 1, we are lead to p , ( x ) = (x/a),,. For ease of expression we couch some of the following results for a = 1, although it is really no more difficult to obtain them for arbitrary values of u # 0.
1.
57
ASSOCIATED SEQUENCES
We express (x), in powers of x by
The coefficients in this expression are known as the Stirling numbers of the first kind,
Thus
We shall discuss these important numbers later in this example and in the next one.
GENERATING FUNCTION AND SHEFFER IDENTITY The generating function for ( x ) , is
which can be written in the familiar form
f(")+ k
k=O
= (1
+ t)".
The binomial identity is
which can also be written as
This is known as the Vandermonde convolution formula.
RECURRENCE RELATIONS AND OPERATIONAL FORMULAS Theorem 2.4.5 gives the formula
(e' - I ) ( x ) , = n(x)n- 1,
4. EXAMPLES
58
which can be written as (x
+ 1 In - (x), = 4 x 1 , -
1.
The second form of the recurrence formula is (,x),,+
=.~i.~[f(t)l'x~
+ t ) - '.Y"
I
=
n
= xi.,-
C( -
t)ktkxn
k=O
EXPANSIONS The expansion theorem is
Taking g ( t ) = ey' we get the beautiful formula
This is actually Newton's forward difference interpolation formula. If we use the notation E': p ( x ) + p ( x
+ y),
Aa: P(X) + P ( X
+
U)
- P(.Y)
for the translation and forward difference operators, it takes the more familiar form
Setting f ( t ) = t"' in the expansion theorem gives
59
1. ASSOCIATED SEQUENCES
By applying this to a polynomial p ( x ) and observing that
(4.1.1)
we obtain a classical formula for numerical differentiation,
Expanding the functional (eyr- l ) / t in terms of the forward difference gives ey'
- 1
--
t
-2
k=O
((es' - I)/' I ( x ) k ) ( e , k!
-
~ ) k ,
If we apply this to a polynomial p ( x ) ,we get
This is known as Gregory's formula (Jordan [ 1, p. 2843). It was discovered by Gregory in 1670 and is reported to be the earliest formula of numerical integration. We shall discuss the coefficients of this expansion when we come to the Bernoulli polynomials of the second kind in Section 3. The polynomial expansion theorem is (4.1.2)
and, using (4.1.l), this becomes
If we take a
=
1 and p ( x ) = x" in (4.1.2), we obtain xn=
1" ((e'
- 1)klX")
k=O
k!
Xk.
These coefficients are known as the Stirling numbers of the second kind, 1
S(n, k ) = - ( ( d k!
-
l)k Ix").
4. EXAMPLES
60
Thus
c S(n,k)(X),. fl
X" =
(4.1.3)
k=O
We shall say more about these numbers later, but for now we notice that (4.1.1) gives
Again, by the polynomial expansion theorem
From Theorem 3.5.6 we get
and, using Proposition 2.1.3, we have ((pa' -
1)'I ( d b ) " )
Either of these forms can be substituted into the above expansion. According to Theorem 2.4.8,
and since (l),
= 8,.
+ d,,,,
we get
.x(:)"=
a(;)"+l
f
q). n
1.
61
ASSOCIATED SEQUENCES
Theorem 2.4.9 provides a formula for the derivative of the lower factorial polynomials,
RESULTS MISCELLANEOUS Let us explore some of the properties of the Stirling numbers, using umbra1 methods. A great deal has been written about these numbers, and they have important combinatorial significance which we cannot discuss here. The interested reader might wish to consult Comtet 113 or Riordan [2,3]. As to the Stirling numbers of the first kind
let us apply t k I k ! to both sides of the recurrence x(x), = (x), + 1
+ n(x),*
Using Theorem 2.1.10, we get 1 k!
-(PI
x(x),) =
1 ~
(k - l)!
(tk-l[(x),)
=
s ( n , k - 1)
and
and so s(n + 1, k ) = ~ ( nk , - 1) - H S ( H , k ) . This recurrence, together with the initial conditions
s(n,O) = 0,
n > 0,
~ ( 0k ,) = 0,
k > 0,
s(0,O) = 0, enables one to compute the numbers s(n, k ) . Applying t k / k ! to the formula obtained from the second form of the recurrence formula gives s ( n + I , k ) = x ( - l ) ' - j ( n ) n - j s ( j , k - 1). j=O
The conjugate representation for (x), is (x), =
x
" 1
-([log(l k=ok!
+ t)lklX">Xk,
62
4. EXAMPLES
and so 1 s(n, k ) = -([log( k!
1
+ t)Ik1 x").
From Proposition 2.1.3 we then get
-(-
1)n-k-
n! 1 7. k! il+...+ik=nll ' " ~ k
1
i, > 0
When the binomial identity for (x), is written out and the coefficients of like powers of x and y are compared, one obtains
Turning to the Stirling numbers of the second kind, we have already remarked that
Proposition 2.1.3 gives, as we have seen,
Finally, for k 2 1 and n 2 1, we have from Theorem 2.1.10 1 S(n,k ) = -((e' k!
--
-
1
(k - l)!
-
(e'(e' - l ) k - l ~ x n ~ ' )
1
(k - l)! =
l ) k1 x")
((PI
kS(n - 1 , k )
-
1
1)klxn-l) + ___ ((e'
+ S(n - I,k
(k - l)!
-
1).
-
t ) k - l Ixn- ')
63
1. ASSOCIATED SEQUENCES
From the initial conditions S(n,O) = 0,
n > 0,
S(0,k )
k > 0,
= 0,
S(0,O) = I , one can compute the numbers S(n, k ) . We shall discuss the connection between the Stirling numbers of the first and second kind in the next example. The multinomial version of Theorem 2.3.10 and Eq. (4.1.1) lead to
for any associated sequence. This specializes to give some interesting combinatorial identities. For example, if we take p , ( x ) = (bx/a),,this becomes
Incidentally, the backward difference functional f ( t ) = 1 - e-a'
satisfying
(f@ I P ()X ) ) P ( O ) - P(4
and
f(t)P(X) = P(X) - P(X
-
4
is associated to the sequence of rising factorial polynomials
(y)(:)(;
(;
+ 1). . . + n
=
-
1).
All of the above results have analogs for the rising factorial sequence, and we shall not give them here except to mention that the role of the Stirling numbers of the first kind s ( n , k ) is taken by their absolute values Is(n,k)l, which are known as the signless Stirling numbers of the first kind, that is, x(n)
=
f I s(n,k f I xk.
k=O
1.3. The Exponential Polynomials DEFINITIONS Let us consider the delta functional f ( t ) = log(1
+ t).
64
4.
EXAMPLES
Since f ( t ) = e' - 1 is the forward difference, Theorem 3.4.6 implies that the associated sequence for f ( t ) is inverse to the lower factorial sequence under umbra1 composition. Equation (4.1.3) gives us the sequence +"(X) =
2 S(n,k)xk.
k=O
These are called the exponential polynomials.
GENERATING FUNCTION AND SHEFFER IDENTITY The generating function for the exponential polynomials is
and the binomial identity is
RECURRENCERELATIONS AND OPERATIONAL FORMULAS The recurrence formula is +n+
1(x)
+ [)+.(XI
= xCf'(t)I-'+n(x) = ~ ( 1
+ 6b(x)).
= -x(+n(x)
From this we also get the operational formula
b",(X) = [ x ( l + t)]"l. But e - x ( x t ) e x x "= e - x x ( e x x "+ n e X x " - l )= x(1 + t ) ~ " ,
and so, as operators, x( 1
+ t ) = e -x(xr)ex.
This gives the operational formula &(x)
= e -x(xt)nex.
The second form of the recurrence formula is
6"+1(x) = x ~ f c f ( t ) l ' x " = xAf e'x" = xLf(x
+ 1)" (4.1.5)
1.
65
ASSOCIATED SEQUENCES
EXPANSIONS The expansion theorem is
For g ( t ) = ey' we get a formula for the exponential function in terms of the logarithm,
Since f'(f(t)) = 1/(1
+ (e' - 1))= e -',Theorem 2.4.8 gives the expression
Theorem 2.4.9 is
MISCELLANEOUS RESULTS If we set pn(x)= ( x ) ~then , &(p(x)) = x n is just (4.1.3).But we also have Pn(4(x)) = xn. Writing
(we are extending ourselves here a bit in writing this, but it can easily be justified), we have
4.
66
EXAMPLES
Thus for any polynomial p ( x ) ,
If we take p ( x ) = x", then $"(X) = e - *
c kn
-xk, k=ok!
which is known as Dobinski's formula. Let us turn again to the Stirling numbers. When the binomial identity for 4 J x ) is written out and the coefficients of like powers of x and yare compared, one obtains
applying t j to the recurrence (4.1.5) gives
=
ik
(">j!S(k,j - 1 )
k=O
or S(n -t 1 , j ) = k=O
( n ) S ( k , j - 1). k
Applying t' to the expansion of x ~ " ( xgives, ) after dividing by j ! ,
The umbra1 composition $"(p(x)) = x", where p , ( x ) = (x),,, is equivalent to n
=
f S(n,k)
k=O
k
j=O
s(k,j)xj
1.
67
ASSOCIATED SEQUENCES
and comparing coefficients of x i gives
If we let
j.:.yn --t
(x), be the umbral operator for (x),, then ( x ) , = 1.-
0
jL(x),
s(n, k)E,-
=
9
Axk
k=O
=
5 s(n,k)R-'(x),
k=O
k
=
s(n,k) C s(k,j)
k
j
j=O
i=O
k=O
S(j,i)xi
and so
Many more properties of the Stirling numbers can be obtained by similar umbral methods. 1.4. The Gould Polynomials and the Central Factorial Polynomials
DEFINITIONS We generalize the forward difference functional by setting f ( t ) = eor(ehr - I),
where b # 0. Using Corollary 3.8.2 to relate the associated sequence for f ( t )to the lower factorial sequence, we have q n ( x )= . x e ~ " " ' x -
These are the Gould polynomials, which we denote by G,(x; a, b).
68
4. EXAMPLES In case a
=
-
b/2 we get the central difference series d b ( t ) = ebri2 - e - b r i 2
which satisfies (sb(t)
I p(x))
= p(b/2) - p(-b/2)
and h b ( t ) p ( ? c )=
p(x
+ )b)
-
p(x
-
th).
This series plays an important role in interpolation theory (Steffensen[a]). The associated sequence for b , ( t ) is formed by the central factorial polynomials .Y"'] = G,,(x: - 1/2,1) = x(x + i n - I),, 1 . GENERATING FUNCTION AND SHEFFER IDENTITY
The generating function for the Gould polynomials is
By differentiating with respect to x and setting x
where f ( t )
= errr(ebr -
= 0 we get
1). Let us compute G;(O; a, b) by umbra1 techniques,
G;(O; a, b) = ( t I Gk(x; a, b ) )
(equution continues)
1.
69
ASSOCIATED SEQUENCES
Thus we obtain (4.1.6) The binomial identity is a generalized Vandermonde convolution,
=i-
x
~ = o . Y-
1’
ak y
-
(
(.Y -
a(n - k )
uk)/h k
)(
(y
-
~ ( -n k ) ) / ~ n-k
Such convolutions play an important role in combinatorics (Riordan [2]). RECURRENCE RELATIONS AND OPERATIONAL FORMULAS Equation (4.1.6) can be used in connection with the recurrence formula to give the recurrence -
G,,
(x; u, b) = xibff(t )’.Y‘‘
For the central factorial polynomials, the first form of the recurrence formula is ++ll
= y[.f”(r)]
- 2.,..&/Z
I.k+nl
+
= .Yp(t)- l . d n 1 ,
-i/Z]
-
l.ylnl
(4.1.7)
70
4.
EXAMPLES
where p ( t ) is the averaging operator
Theorem 2.4.5 is (x
+ 1/2)["'- (x
-
1/2)'"' = nx["- I].
EXPANSIONS The expansion theorem can be used to derive various classical interpolation formulas, such as those of Gauss, Stirling, Bessel, and Everett. We shall give only a brief sampling, referring the reader to Steffensen [2, Article 181 for more details. The expansion theorem gives
and so
We wish to differentiate this with respect to d , ( t ) . Using the chain rule (Theorem 3.6.3), we have
+
Sdl(l)(eJ-J + e - 1'1 ) = (761,r,tSl(eyl e-yl) =
[?,d,(t)]-'
+
L7,(eJf
= p ( t )- ly(e"' - e - y f ) ,
and so
Thus we have
Adding this to (4.1.8)gives Stirling's formula
e-y')
1.
71
ASSOCIATED SEQUENCES
As another example, the expansion theorem gives
for any series f(t), and so from (4.1.7) we deduce that
Setting f ( t )
= e', we
obtain a formula of Steffensen [l, p. 3621
MISCELLANEOUS RESULTS Riordan [2, p. 2121 gives a discussion of the numbers t(n,k ) and T(n,k ) defined by
and
Note the analogy with the Stirling numbers. We leave it to the interested reader to obtain results analogous to those obtained earlier for the Stirling numbers. We conclude this example with an amusing application of Proposition 2.1.3. If p,(x) is an associated sequence, we have
4.
12
EXAMPLES
and so
i(!)
j=O
J
( - 1Ik -'p,(bj
Taking p n ( x ) = xn, a
+ ak)
= 1
and a
+ b = - 1, we get
i, odd
Except for the factor ( - 2 ) k , the right side counts the number of ways of distributing n balls into k boxes, subject to the condition that each box receive an odd number of balls! 1.5. The Abel Polynomials
DEFINITIONS The associated sequence for the Abel functional
f ( r ) = tea'
(a # 0 )
can be determined from the transfer formula to be
A,(x;a)= x
~
(f -nXn- xe -an1 X n 1 -
= x(x - an)"-
These are the Abel polynomials.
GENERATING FUNCTION AND SHEFFER IDENTITY The generating function for the Abel polynomials is
1.
13
ASSOCIATED SEQUENCES
where f ( t ) = te". Differentiating with respect to x and setting x
=
0 give
The binomial identity in this case is
There are a host of identities of this form, and Riordan [2, p. 181 gives a thorough account. For example, one of Abel's generalizations of the binomial formula is [Riordan 2, p. 18, Eq. (1 3)]
x-'(x
+ y - nu)" = kc= o
(x
k
-
~ k ) ~ - ' (-y a(n - k))"-'.
(4.1.9)
Since te"'(x - an)" = ne"'(x - an)"- = n(x - a(n - 1))"-
l,
we see that the sequence s,(x) = (x - an)" is nothing but a Sheffer sequence for the Abel functional f ( t ) = tea' and that (4.1.9) is nothing but the Sheffer identity for s,(x).
RECURRENCERELATIONS AND OPERATIONAL FORMULAS The recurrence formula for Abel polynomials is A , + 1 (x; 4
lA,(x; 4 a t ) - 'e-"'A,,(x; a) at)-'A,(x - a;a)
= xCf'(t)l-
+ + 1 akA$k'(x - u;u),
= x( 1
= x(1 =
x
k=O
and the second form gives A,+,(x;u) = xAf[f(t)]'x"
Theorem 2.4.5 is the simple equation
14
4.
EXAMPLES
EXPANSIONS The expansion theorem is
Taking g ( t ) = eyr,we get
and for any polynomial y(y - ak)k-'
k!
k 2 0
p'yx
+ ak).
Taking y ( t ) = r" in the expansion theorem and observing that for k 2 n 2 1
( t " J x ( x- a k ) ' - ' ) = ( n t " - ' l ( x - a k ) k - l )
= n(k - 1)"- '( -ak)k-",
we have t"= k=n
n(k - 1)" (-~k)~-"r~e"~'. k!
Applying this to a polynomial p ( x ) gives a formula for ~ ' " ' ( 0in) terms of Pfk'(4,
The polynomial expansion theorem is
or P(X)
=
c p'k'(ak) k! ~
kto
Taking, for instance, p ( x ) = .Y" gives
X(X
- ak)k-'
1.
75
ASSOCIATED SEQUENCES
This shows that the inverse, under umbra1 composition, of the Abel sequence A , ( x ; a) is the sequence
This sequence is associated to f ( t ) = Theorem 2.4.8 is
2;;
-ak)k-'tk/k!.
1.6. The Mittag-Leffler Polynomials DEFINITIONS We next consider the delta series
and note that f'(t) = -'
(e'
2e'
+ I)2
Let us denote the associated sequence for f ( t ) by M,(x). We call the M,(x) Mittag-Leffler polynomials(Bateman [I], Erdelyi [ 1, Vol. 3, p. 2483). It should be noted that both the aforementioned sources define the Mittag-Leffler polynomials as M,(x)/n!.
4. EXAMPLES
76
We obtain a nice expression for the Mittag-Leffler polynomials by using Theorem 3.5.6, Mn(Y)= (e"
I W(X)>
1 --t
=((I
+")'I.> 1-t
and so
(")("
M,(.W)= k - 0 k
n -- k1)2*(x)k
GENERATING FUNCTION AND SHEFFER IDENTITY
The generating function for the Mittag-Leffler polynomials is
These polynomials were used by Mittag-Leffler in the study of the integrals and invariants of linear homogeneous differential equations, wherein he used the conformal mapping w = ((1
+ z)/(l - z))"
for x constant. They were also used by Pidduck (see Bateman [I]) to study the propagation of a disturbance in a fluid acted on by gravity. The binomial identity is
1.
ASSOCIATED SEQUENCES
77
RECURRENCE RELATIONS AND OPERATIONAL FORMULAS Theorem 2.4.5 is e' - 1 M,(.X) e'+ 1
= I7h'f,.
~(x),
which is equivalent to (e' - 1)M,(x) = n(e'
+ l)Mn-t(x)
or Mn(x
+ 1) - Mn(x)= n [ M n - l(x + 1) + M n - l ( x ) ] .
The recurrence formula is M n + l ( x ) = xCf'(t)I-'Mn(x)
+ I)'M,,(x)
= +xe-'(e' = +x(e'
+ 2 + e-')M,(x) + I ) + 2 M n ( x )+ M,(x - I)],
= +XCM,(X
and the second form gives the recurrence M,+l(X) = . d J [ f ( t ) ] Y
= x%,(2/(1
-
t2))x"
c tZkxn x
=2
.4
k=O
EXPANSIONS The expansion theorem for the Mittag-Leffler polynomials is
For g ( t ) = ey' we get e'
+1
78
4.
EXAMPLES
Since ,f'(J(f)) = (1 - t 2 ) / 2 ,Theorem 2.4.8 yields
1
= - [ M , , + l(x) - n(n - l)Mn-l(x)].
2
Theorem 2.4.9 gives
and since
we get
n-k-j 1.7. The Bessel Polynomials
DEFINITIONS
(.y
Krall and Frink [1) made a study of the polynomials
1 (n-+k)!k)!k ! n
Yn(.u)=
k=O(n
2 '
which are called Bessel polynomials in view of their connection with Bessel functions. The y n ( x )satisfy the differential equation x2y"
+ ( 2 x + 2)y' + n(n + 1 ) y = 0,
and their derivatives form an orthogonal sequence of polynomials. Krall and Frink also discuss applications of these polynomials to spherical waves. Carlitz [ 2 ] defined a related set of polynomials Pnfx) = X " Y ~- 1 ( 1 / ~ )
and gave many of their properties. Now, by the transfer formula, the associated sequence for f ( t )= r - t2/2
1.
79
ASSOCIATED SEQUENCES
is
Thus
(2n - k - I)! (L)n-k Xk k = l ( k - I)!(' - k)! 2
=t is associated to f ( t ) = t - t 2 / 2 .
GENERATING FUNCTION AND SHEFFER IDENTITY
The generating function for p,(x) is
and the binomial identity is
which translates into (provided we set y - *(x) = 1 )
RECURRENCERELATIONS AND OPERATIONAL FORMULAS
According to Theorem 2.4.5, we have ( t - ( t ' / 2 ) ) P n ( X ) = upn
-
1 (.y)
or P;(.U)
- 2pL(.u)
+ 2npn- ,(?I) = 0.
80
4. EXAMPLES
c pkk'(x) n
=x
k=O
as well as
or, by performing the indicated derivative and simplifying, (X
+ l)pn+ 1(x)
-
XP:+
,(-XI - x2pn(x) = 0.
From the second form of the recurrence formula we obtain pn+ ,(x) = x j y c m l ' ~ " = x i f (1 - 2 t ) -
ll2xn
- 1/2 k=O
=x
2 (')>[Ik
. 3 . . . ( 2 k - 3)(2k - 1 ) ] ~ " - ~
k=o
EXPANSIONS The polynomial expansion theorem is
and so
1 . ASSOCIATED SEQUENCES
81
Taking p(x) = X" gives
c ( )c( k
x" =
k?n/2
n
-
-;>.-kpk(x).
k k!
Theorem 2.4.8 gives
Theorem 2.4.9 yields
n (2n - 2 k - 2 ) ! = k?o(k) (n- k - I)! n-l
(i> 1
n-k-l
Pk(x).
Let us also expand x2p,(x) as n+ 2
x2Pn(x) =
C
k-0
( ( t - t 2 / 2 ) kI x2p,(x)) k! Pk(x).
82
4. EXAMPLES 1.8. The Bell Polynomials
DEFINITIONS Let us consider a somewhat different type of example. Let K be a field of characteristic zero and let x l , x2,. . . be a sequence of independent variables. We take the field C to be K with the variables xi adjoined,
C
=
K ( x , , x 2 , . . .).
The generic delta series is the series
and < g ( t )I x ” > == x n
for all n 2 1. Let b,(x; x l , x 2 , . . .) denote the associated sequence for @), the compositional inverse of the generic delta functional, and set k=O
The sequence b,(x; xl, x2,.. .) is the generic associated sequence, so named since any associated sequence is but a special case of b,(x; x l , x 2 , . . .) for an appropriatechoice of the variables xk. We shall see some examples in the subsection on miscellaneous results. Results obtained for the sequence b,(x; x x 2 , . . .) are therefore generic results that apply to all associated sequences. The same is true for the generic coefficients B,,k(Xl,x 2 . . . .). The conjugate representation provides an explicit formula for the coefficients B , , k ( X , , x2,. . .),
1.
83
ASSOCIATED SEQUENCES
The B n , k ( x 1 , x 2 , . . .) are known as the Bell polynomials (Comtet [ I , p. 133]), and they have important combinatorial significance.
GENERATJNG FUNCTION AND SHEFFER IDENTITY The generating function for hn(.x;.xlrx2,.. .) is
is an associated sequence if and only if the coefficients a n , k satisfy
(
i+j
)%.i+j
=
(i)uk,ian-k,j.
k=O
Taking i
=
1 and j
=
1 - 1 in (4.1.10),we get the recurrence formula
RECURRENCERELATIONS AND OPERATIONAL FORMULAS The second form of the recurrence formula is b,(x; X I , x2,. . .) = x n , [ g ( t ) ] ' x " -
4.
84
EXAMPLES
Applying t k to both sides gives the recurrence
MISCELLANEOUS RESULTS Other identities involving the Bell polynomials follow easily by umbra1 methods. For example, setting ( g 1 ( t ) I x " ) = xn+ 1/(n + 1)
for n 2 1 and ( g , ( t ) l x o )
= 0, we
(4.1.11)
have
= t(X1
+ 9l(t)).
Hence g(t)k = j=o
(
-jgl(t)jtk
J
From (4.1.1 1) we see that ( g l ( t ) kI x " ) ing (4.1.12) to X " gives the identity
=
(4.1.12 )
k ! B n . k ( x 2 / 2xr , / 3 , . . .). and so apply-
If we adjoin additional variables y l , y 2 , ... to C and set
( W )I x n > = x n + Yn for n 2 1 and ( h ( t ) l x o ) = 0, then the associated sequence for h(t) is ~,,(.Y;x, Y , , x ~ y2, ...). NOW
+
+
( h ( t ) kI x n > =
([so) + ( 4 0 - s(t))lkI x n >
whence we obtain
Let us consider some special cases of the Bell polynomials. If we set xk = 1
for all k 2 1, then the generic functional becomes
1.
85
ASSOCIATED SEQUENCES
and so bn(x; 1,1,
'. .) = 4 n ( X ) r
where 4,,(x) are the exponential polynomials. Thus Bn.k(l, I,...) = S(n,k) are the Stirling numbers of the second kind. If we take xk =
(k - l)!
Z ( - l ) k - l -
(k - I)! t k = log(1 k!
for all k 2 1, then m
g(t)=
k= 1
+ t),
and so
df)= ( 4 n . Thus Bn,k(O!,- 1 !, 2 ! , - 3!, . . .) = S(n,k) are the Stirling numbers of the first kind. If we take xk
= kak-'
for all k 2 1, then
which is the Abel functional. As we saw in the subsection on expansions in Section 1 S, the associated sequence for i j ( t ) is
b,(x; 1,2a, 3 a 2 , .. . ) = Thus B n , k ( 1 , 2 a , 3 a 2 ,.). .=
(3
(ak)n-k
For a = 1 these numbers are known as the idempotent numbers (Comtet [I, P. 911). Finally, let us take x k = k!
4. EXAMPLES
86
for all k 2 1. Then
The transfer formula gives us the associated sequence for g(t), b,(.x;1!,2!,3! ,...)
= x(l
=
+c)~x~-'
f:( n-1
k=1
n!
)-xk
k - 1 k!
Thus
These are the Lah numbers (Comtet 11, p. 1563). We shall encounter them again in connection with the Laguerre polynomials.
2. APPELL SEQUENCES
For ease of reference we list the key results. Let s,(x) be Appell for g(c). Generating function
pk 1 f
k=o
- -ex'.
k!
dt)
Appell identity s,(x
+ y)=
f:( nk) s k ( y ) x " - k .
k=O
Theorem 2.5.5 s,(x) = g ( t ) - l x n .
Theorem 2.5.6 ts,(x) = ns,
Recurrence formula
-
1(x).
2.
87
APPELL SEQUENCES
Expansion theorem
Polynomial expansion theorem
or
Conjugate representation SAX)
=
.f (") (g(t)k
IXn-k)Xk.
k=O
Multiplication theorem
2.1. The Hermite Polynomials DEFINITIONS The Hermite polynomials H r ' ( x ) of variance v form the Appell sequence for y ( t ) = e"z'Z.
The case v
=
1 is the most familiar, and we write HL"(x) = H,(x).
88
4.
EXAMPLES
Theorem 2.5.5 easily gives an explicit expression for the Hermite polynomials.
From this we deduce that H ~ ' ( x=) v " ~ ~ H , ( 'I2). x/v Incidentally, the operator evtz/2satisfies, for v > 0,
and has been called the Weierstrass operator.
GENERATING FUNCTION AND
SHEFFER IDENTITY
The generating function for the Hermit polynomials is
It should be noted that many sources, including Erdelyi [I, Vol. 2, p. 1921, define the Hermite polynomials by means of the generating function
According to this definition, the sequence k! uk(x)is Sheffer, but it is not Appell, for the delta functional is f ( t ) = t / 2 . In any case, since u,(x) = 2 " ' 2 H r ' ( 2 ' ' 2 ~ ) ,
the discrepancy in the definitions is only minor (even though it is a major nuisance). The Appell identity in this case is
and applying e
rt2i2
gives the appealing identity (4.2.1)
2.
89
APPELL SEQUENCES
Taking p = - v gives (x
+ y)" = k = o (") Hf'(x)Hbl",'(y). k
We shall say more about identities of the form (4.2.1)when we discuss cross sequences in Chapter 5.
RECURRENCERELATIONSAND OPERATIONAL FORMULAS Since H:'(x) is Appell, we have tH!,"'(x)= nH!,"!l(x). The recurrence formula is
H:! l(x)
= (x
-
vt)H?'(x)
= xH:'(x) -
vnH!,? l(x).
(4.2.2)
Applying the operator r to both sides and recalling that tx - xt is the identity [Eq (3.6.3)],we get ( n + 1)H:'(x)
= tH!,"! l(x) =
r.xH:'(x) - Vt2H:'(.W)
+ H?'(x)- vt2H?'(x),
=~tH?'(x)
which simplifies to vt2H!,")(x)- xrH!,"'(x)+ n H r ' ( x ) = 0.
This is the familiar second-order differential equation for the Hermite polynomials. One easily sees, by taking p(x) = x", that (x - v t ) p ( x ) = -eXZi2v(vt)e-x2'2vp(x),
and so by iterating the first equation in (4.2.2)we get the classical Rodrigues formula H?yx)= ( - l)nexZ/2v(vt)"e-"2'2'
EXPANSIONS The polynomial expansion theorem is
4.
90
EXAMPLES
If we take p ( x ) = x", then
For n
= 2m,
l o
for k
odd,
and so
F o r n = 2 m + I,
for k
even,
and so
One of the nicest properties of the Hermite polynomials is expressed in the next lemma. Lemma 4.2.1
For any polynomial p ( x ) , ( e r z i 21 H , ( x ) p ( x ) ) = (t"e'zi2I p ( x ) )
for all n 2 0. Proof
We verify this for p ( x ) = x m by induction on m. If m = 0, then (er2I2I H,,(x)) = i5n,o= ( t n e f z i I2X O >
for all n 2 0. Suppose that for all 1s m (e12'2I H,,(x)x') = ( tner2'2I x ' )
for all n 2 0. Then we show that this also holds for 1 = m
+ 1. Now
2.
91
APPELL SEQUENCES
( e ' 2 ~ 2 1 H , , ( x ) x m += 1)
IH,,(X)X~)
= ( t e t 2 i 2I ~ , , ( x ) x ~ )
I tH,,(x)xm) = (etZ1' I nH,- l ( ~ ) ~+m mH,,(x)x"-') = (et2I2
-
(nrn-1er2/21Xm)
-
(nt"-
1 e12/2 +
+
(mtnet2/21xm-1)
t" + l e 1 2 / 2
Ixm>
= (ci,tne'2/2I x m ) -
(tne12/2
Ix m + l )
for all n 2 0. This completes the proof.
This lemma is most effective when used with the polynomial expansion theorem. If p ( x ) is any polynomial, then P(x)Hm(x)
=
11
-(etzi2 k t o k !
I fkp(x)Hm(x))Hk(x).
From the Leibniz formula and the lemma we have
and so, for any polynomial p ( x ) ,
In particular, if s,(x) is an Appell sequence and we take p ( x ) = s,,(x), then tm-k+2j sn(x)=
(n)m - k + 2 j s n - m + k -
- (n)j(n - j ) m
-
k
2j(x)
+j s n - m + k
-
2j(X),
92
4.
EXAMPLES
and so
In case s n ( x ) = Hn(x), (er”z l H n - m + k - Z j ( x ) )
= bn-m+k-Zj.O - bn
- m + k . Zj 3
and so for n 2 m,
+
This can be put into a more familiar form by observing that n - m k is even if and only if n + m - k is even, and so we may set n m - k = 21. Then 2 j = n - m + k = 2n - 21,andso j = n - 1. Also, k - j = m - land we have
+
This well-known formula goes back to 1918. MISCELLANEOUS
RESULTS
The umbral composition of Hermite polynomials is relatively simple, Hr)(H(fl)(.y)) = e-pr2/’Hjlv)(x) e-pr2/2 e -vtz/2 xn -
=
-(v+p)t2/ZXn
H;+”(.Y).
In particular, H n ( H ( x ) ) = H‘,’)(X)
For p
= -v
= 2ni2Hn(~/21’z).
we get
H!,“)(H‘-’’’(.~)) = HIp’(x)
=
x”,
and so H r ’ ( x ) and Hk- ”’(x)are inverses under umbral composition. This nice behavior with respect to umbral composition is certainly not unique to the Hermite polynomials, and we shall speak briefly about this when we discuss cross sequences in Chapter 5.
2.
93
APPELL SEQUENCES
We conclude with the multiplication theorem, Hr)(x)
H r ) ( a x )= an- ’(’)
’@/a) -
-a-*)vr2/2
H !,“)( x)
= ~“Hrla2)(x),
which, of course, follows directly from the explicit expression of H % ) ( x ) .
2.2. The Bernoulli Polynomials DEFINITIONS The Bernoulli polynomials B ~ ) ( . uof) order a form the Appell sequence for
$70)=
(i)
e*-1
a
( a # 0).
We recall from Chapter 2 that
and
The polynomials Bb”(x) = B J x ) are by far the most common. The interested reader may consult Milne-Thomson [l] for a discussion of the higher order Bernoulli polynomials from a classical point of view. It follows from Theorem 2.5.5 that
and so
A simple expression for the Bernoulli polynomials in terms of the lower factorials can be obtained from the polynomial expansion theorem,
94
4.
EXAMPLES
Since for k 2 1 ( ( e ' - 1)'I B,,(x)) = =
I ax"-')
((e' -
= n(k
-
~ ) ! S (-H l , k - I),
we have
GENERATING FUNCTION AND SHEFFER IDENTITY The generating function of the Bernoulli polynomials is
Setting x
= 0 gives
the expansion
where the numbers Bf)(O)are the well-known Bernoulli numbers. The Appell identity is
B t ' ( x + y ) = kf=(O " k) B ~ ' ( y ) ~ " - ~ . Applying [[/(el -
] ) I b , we have 51P+b'(X
and for u
+h =0
Setting J
=0
+ y) =
in the Appell identity, we get
B!,@(x)= k=O
(")Bt! k
k(0)~k.
RECURRENCERELATIONS AND OPERATIONAL FORMULAS Since BjP'(x) is Appell, t B ~ ' ( -=~ nBf! )
,(.Y).
(4.2.4)
2.
95
APPELL SEQUENCES
Next we observe that (e' - l)B!,@(.x)= (el - 1)
=(A) a- I
(4.2.5)
tx"
=
nBy--,"(x),
which may be written as B:'(X
+ 1) = BF'(.x) + FIB~I,"(x).
In particular, (e' - l)B,,(x)= nxY"-l
or B,(x
+ 1 ) = B,(x) + nx"-'.
Also,
The recurrence formula is
=[.x-il(l+l+f-p')] t(e' - 1)
B:'(X).
(4.2.6)
4.
96 Applying t to both sides and recalling that
tx
-
l+t-e' e' - 1 = (X
-
xt
EXAMPLES
is the identity, we get
1
B',"'(X)
t a ) t B f ' ( x )+ B',"'(x)+ uB',"'(x)- u-B~'(x) el-
= n(x - a)B!,"!1
1
+ B!'(X) + u B ~ ' ( x ) u B ~ + " ( x ) ,
( ~ )
-
and so (4.2.7)
+
which expresses the Bernoulli polynomials of order a 1 in terms of those of order a (for a # 0). By the transfer formula the associated sequence for the delta series
is p,(x) = x ( q ) - " x n - l =x
~
:
Jnnxn-
'
= xB!,?'~(x).
But if a = 1, then
f(t) =
e'
-
1 is the forward difference, and so we have (x), = x B ~ '(x), !
which gives
BPI 1 (x) = x - ' (x),
= (X - 1),
-1
.
Applying c " - ~ -for ' ,k In - 1, we obtain the operational formula (4.2.8) EXPANSIONS The expansion theorem is
k=O
2.
97
APPELL SEQUENCES
Taking h ( t ) = er', we obtain the famous Euler-Maclaurin expansion,
Applying this to a polynomial p ( x ) gives, for a p(x
+y)=
1 hz0
y
+
=
1,
p'k'(u) du.
Recalling the expansion of r m in powers of the forward difference, we have
Applying this to Bf'(x) gives
Since for any m 2 0
we have
Applying this to x n gives
For m = 1 this is the well-known formula
MISCELLANEOUS RESULTS The multiplication theorem is
4. EXAMPLES
98 Now if u = m is a positive integer. then
m- 1
- ,,,n-l
1 ekrimBn(x)
k=O
which is the classical multiplication theorem for BernoulIi polynomials. From Proposition 2.1.1 1 we see that
and so the sequence s,(x) = BIp'( -.Y)
is Sheffer for
Therefore, since p , ( x ) = (-x)" is associated to f ( t ) = - t ,
= ( - 1)"e O f BIp'(x) =(-
I)"BIp'(x+ a).
This is known as the complementary argument theorem. Let us discuss briefly the Bernoulli numbers. Taking x = 0 in (4.2.6) gives
By'( 1) = By'(0)+ nBt--,"(O). Taking x = 0 in (4.2.7), with a replaced by a - 1, gives
Combining these two formulas. we obtain
2.
99
APPELL SEQUENCES
The operational formula (4.2.8)is equivalent to t"(x), =
n! B?!+,"(X ( n - m)! ~
+ 1)
for m 2 I . This leads to a connection between the Bernoulli numbers and the Stirling numbers of the first kind,
1
I t"(x),)
= -(to
m!
Thus
We may connect the Bernoulli numbers to the Stirling numbers of the second kind as follows: 1 S ( n , k) = -((e'
k!
1 x")
-
and so
One of the most appealing aspects of the Bernoulli numbers is their connection with the partial sums of the harmonic series. If we let 1
1
1
h , = 1 +-+-+.-.+-, 2 3 n
4. EXAMPLES
100
then since dn+
= x(x
+ l)(x + 2 ) - - - ( x+ n)
we have
But p J x ) = x'") is associated to the backward difference functional f ( t ) = 1 - e-',
as can easily be seen by the recurrence formula. Thus the transfer formula gives
Using the complementary argument theorem, we finally get I + - +1 . . . + - = - 1 2 n
(-1y-l
(n - l)!
Er-+,')(O).
2.3. The Euler Polynomials DEFINITIONS
The Euler polynomials E',"'(x) of order a form the Appell sequence for
2.
101
APPELL SEQUENCES
The properties of the Euler polynomials are similar in spirit to those of the Bernoulli polynomials. We write Eb”(x) = En(x). It follows from Theorem 2.5.5 that
and so
Since
(&y
=
(I/(
1
+
=
g;a)(y)J,
we have the expression
(4.2.9) From Theorem 2.3.8 we get
and so
E?’(X) =
1[ C
k=O
.
j=O
]
-TS(k,j) x”-~.
Equation (4.2.9) gives another formula for E y ’ ( x ) .Since
we have n
(e’ - 1 ) ’ ~ ” =
1 S(n,k)(k)j(x)k-j, k=j
and so
This is known as the Newton expansion of the Euler polynomials.
(4.2.10)
102
4. EXAMPLES
Since p,(.w) = (x - a/2)" is Appell for the series e"'", the expansion of E',"'(x)in terms of p , ( x ) is
The numbers 2"Ejp)(a/2)are known as the Euler numbers, in contradistinction to the Bernoulli case. (This terminology is due to Norlund.)
GENERATING FUNCTION AND SHEFFER IDENTITY The generating function for the Euler polynomials is
Incidentally, if we set a = 1, t = 2iu, and x = 0, we get
and if we set a
=
1, t = 2iu, and x = f, then
For this reason the numbers 2 k E f ' ( 0 )and 2'&(4) are known as the tangent and secant coefficients, respectively. The Appell identity is E y ( x + y ) = k = O (n)E:"'(y)x"-k, k
and as before we have
and
103
2. APPELL SEQUENCES Setting y
=0
in the Appell identity gives Ef’(x)
(”)k
E r ! k(0)xk.
= k=O
RECURRENCE RELATIONS AND OPERATIONAL FORMULAS Since Er’(x)is Appell, tEf’(.u) = HE;! I(X).
Further, we have
X” = 2Ef-”(x),
and so
Ef’(x+ 1)
=
2Er-”(x)
-
Er’(x).
(4.2.11)
Next we have
The recurrence formula is E?! (x)= ( x -
2)
E?’(x)=
a
= xEf’(x) - - E ? + ” ( x
2
( )“’ + -
e‘
+ 1)
or a -
2
Effl’(x + 1)= xEf’(x)- Ef’(x)- E?l1(x).
1
Ef’(x)
4.
104
Combining this with (4.2.1I), we get the recurrence
EXPANSIONS The expansion theorem is tk.
Taking h ( t ) = ey', we get Book's summation formula
Applying this to p ( . ~for ) a
=
I gives
EXAMPLES
2.
105
APPELL SEQUENCES
MISCELLANEOUS RESULTS The multiplication theorem is
If 3
=
2m
+ 1 is a positive odd integer, then I
2m
c
1 I'
k = O [-p'"2m+
1
= -
- ~ - ~ r / ( 2 m + l )2 m + l
1
- [-pr/(2m+l)
1
1 +e' 1
+
er/(Zm+l)'
and so
2 in
k-0
For x
=
2rn a positive integer we must be a bit more enterprising and write
E,(2mx) = (2rn)"
1
1
+ e'
+ erI2" E , ( x )
= 2(2m)" 1
1
,
+ p ~ / ~ m x
2(2rn)"( ____ )B,,+l(x) n + 1 1 + e"'"
-
c
2(2m)" 2 m 1 [ - e f ' 2 m l k B" + 1 (x) n + 1 k=O
-
-~
-
____ 2(2m)" 2 m1
~
~
n+ 1
+ ").
( - l)kBn+l(x
2rn
k=O
Setting m = 1 gives a nice formula connecting the Euler polynomials with the Bernoulli polynomials, E,(2x)
=
2"+1
-~
n+I
[B,+ ,(.u) - B,+
,
4. EXAMPLES
106
By the multiplication theorem for Bernoulli polynomials
and so
Since E f ' ( - x) is Sheffer for -t),
we have
= (-
+
l)"Ejp'(x a),
which is the complementary argument theorem for the Euler polynomials. By the transfer formula the associated sequence for the delta series
s( t ) =
e-s'/2(er+ 1)" t 2"
= r[coshi].
is
This fact can be used to derive additional properties of the Euler polynomials.
3.
107
SHEFFER SEQUENCES
3. SHEFFER SEQUENCES
For ease of reference we list the key results. Let s,(x) be Sheffer for
(m, f @I).
Generating function
f s,(x.)tk k=O
k!
-1
-
exJ(t),
g(f(t))
Sheffer identity
Theorem 2.3.6 SAX) =
~
'PAX).
Theorem 2.3.7
f ( t ) s n ( x ) = ns, Recurrence formula
Expansion theorem
Polynomial expansion theorem
-l(4.
4.
108
EXAMPLES
Theorem 2.3.12
or
Conjugate represen tat ion
3.1. The Laguerre Polynomials
DEFINITIONS The Laguerre polynomials LIp'(,x)of order LY form the Sheffer sequence for the pair y ( t ) = (1 -
r)-@-',
f ( t ) = t / ( t - 1).
Evidentally, the sequence LL- "(x) = L,(.u) is associated to the delta series f ( t ) . However, the polynomials LLo'(x)are generally known as the (simple) Laguerre polynomials. Once again there is some variation in the definition of Laguerre polynomials. Many sources, including Erdelyi [11, take the Laguerre polynomials to be L f ' ( x ) / n ! . We note that f ' ( t ) = - 1/(1 - t)'
and
f ( t ) = t / ( t - 1) = f(t).
From Theorem 2.3.6 we see that LIp'(x) = (1 - t)"+ l L " ( X ) ,
and so ( 1 - t ) P L f ' ( X ) = Lf+P'(X).
(4.3.1)
3.
109
SHEFFER SEQUENCES
The transfer formula gives an explicit expression for the sequence L,(x),
=
-(I
- t)2(f - l)"+lX"
=(-I)"(l
-
f)n-l.xn
Incidentally, the coefficients of ( - x ) ~are the Lah numbers BnJ 1 !, 2 ! ,3 ! ,. . .) that we encountered in Section 1.8. From this and (4.3.1) we get LIp)(x) = ( 1 =
-
(-
t)"
I)"( 1
+
L,(x)
+ f)=+"Xn
It is worth noting separately that L',"'(X)= ( - 1 1 7 1
- tfe+nX".
To derive the classical Rodrigues formula we observe that, as operators, 1
-
r
= -exre-",
and so (1
-
t ) " = ( - 1)"e"t"e -".
Furthermore, it is easy to see that x-"(l
- t)nX*+"
= (1 -
l)U+"X",
and so LIp'(X) = ( - 1)"(1 - t ) " + " X n = (-
I ) " x - ~-( r)"xatn I
= x - e x~t n . C x e a t n .
110
4.
EXAMPLES
GENERATING FUNCTION A N D SHEFFER IDENTITY The generating function for the Laguerre polynomials is
The Sheffer identity is
Applying ( 1 - t ) B + gives
RECURRENCERELATIONS AND
OPERATIONAL
FORMULAS
Theorem 2.3.7 is t
-Ljp’(x)
= nL?!
1 - 1
(4.3.2)
l(x),
which is equivalent to t L f ’ ( x )= n t L;! ( x ) - n Lf!
,(x)
and to tL?’(x) = - nL;-+,”(x).
The recurrence formula is
+ 1)(1 = ( x t - x + a + 1)(1 = (xt - x + a + = -[x -(a
-
t)-’](I -
- t)’LIp’(x)
t)LF)(x) ”(x).
(4.3.3)
From this we see that LIpl,(x) = ( x t - x
and if m
= 0, we
+ + I)(n)L;+m)(X), c1
have a formula of Carlitz, L1p)(x)= ( x t
-
x
+ a + 1)(’)1.
From (4.3.2) and the recurrence formula we easily obtain the familiar second-order differential equation for the Laguerre polynomials. Writing
3.
111
SHEFFER SEQUENCES
(4.3.2)as (1 - t)Lr! I(x)= - ( l / n ) t L ? ) ( x )
+ 1 has been replaced by n, we get L?)(x) = -(l/n)(xr - x + + l)tL?’(x),
and substituting into (4.3.3),where n
c(
which simplifies to
[xt2
+ (a + 1
-
x)t
+ n ] ~ ? ’ ( x=) 0.
EXPANSIONS Theorem 2.3.1 1 is
But (g’(l)lL:’(.Y))
=(a
+ 1)((1
-
r)-“-2IL:)(x)) = ( a
+ l)(tlLi-2’(x))
Thus
=
-Lfll(x)
-(n(a
+ ( a + 1 + 2n)L?’(x)
+ 1) + n(n
-
I))L?! l(x),
which is the recurrence
LIpl l(x)
- (2n
+a + 1
-
X)Lf’(X)
+ n(n + a)L?! l(x) = 0.
I12
4.
EXAMPLES
Theorem 2.3.12 gives
The polynomial expansion theorem is
Let
US
set p(xJ= srLF’(..t).Then, since
= kn!bn,k =
- (3 + 1
+
k)tl!dn,k+i
nn! Ls,,h - ( 3 + n)n!Ci,_
I.k,
we have stL:’(.u)
=
,
n L y ( s ) - n(a + n)L:! (x).
This is just a small sample of the many formulas involving the Laguerre polynomials which can easily be obtained by umbral methods.
MISCELLANEOUS RESULTS Let us discuss briefly the umbral composition of Laguerre polynomials. To compute LF’(L‘”’(.u))we employ Theorem 3.5.5, where ( q ( r ) . f ( r ) ) = ((1
-
t)-8-1,t/(t
-
f)-’
-
I))
and (h(f)./(fJ)
=((I
’ , t / ( f - 1)).
3.
113
SHEFFER SEQUENCES
Then, according to this theorem, L;)(L(")(.y)) is Sheffer for ( y ( r ) h ( f ( r ) ) , / ( f ' ( r ) ) )= ((1
-
r Y p , r),
and so L;'(L'qx))
If we set ct
=
( 1 - t)P -*xn= ( - I)"L!#D-"-"'(x).
= /j, then
LF'(L ' y x)) = x", which shows that the Laguerre polynomials are self-inverse under umbral composition. If we let sF'(.y) = ( 1 -
r)-j.y" I
,
then s?)(.y)
= ( - 1)"Lb-i-"'(.y)
and (4.3.4)
LIp'(L"J'(.u)) = s:-p'(.x).
Furthermore, since s!#')(x)is Sheffer for ((1 - r)i,r), another application of is Sheffer for ((1 - t)-i-v-',r/(t - 1)). Theorem 3.5.5 shows that s!#')(L'I')(x)) Thus
.
.s?)(L(qX))= La + P y x ) .
(4.3.5)
Combining (4.3.4) and (4.3.5), we obtain Rota's umbral composition law for Laguerre polynomials as
..
. .)))
L ~ I ) ( L ( Z Z ) ( L ( ~ ~ ()L( ". L J ( ~ ~ ) ) . (31
~
z2+2,-
'"+Zk)
(x), (x),
k odd, k even.
3.2. The Bernoulli Polynomials of the Second Kind DEFINITIONS The Bernoulli polynomials b,(u) of the second kind form the Sheffer sequence for r =
~
el-
I'
, f ( r ) = e r - 1.
114
4.
EXAMPLES
We recall that
and
Jordan [ 11makes a study of these polynomials, but he defines them as b,(x)/n!. Theorem 2.3.6 is e' - 1 b,(x) = _ _ (XI, t
Also,
rb,(x) = (e'
-
l)(x), = n(x),-
and from this we get
1:
b,(x) - b,(O) = n
(u),,- du.
Theorem 2.3.7 is (e' - l)b,(x) = nb,- l(x)
or h,(x
Setting x
+ 1) = b,(x) + nb,- l(x).
= 0 gives
If we combine this with (4.3.6), we get b,(O) = Let us call the numbers
s:
the Bernoulli numhers of the second kind.
(u),du.
(4.3.6)
3.
115
SHEFFER SEQUENCES
In order to expand b,(x) in powers o f x", we observe that for k 2 1
n k
= -s(n -
1 , k - l),
(4.3.7)
where s(n,k ) are the Stirling numbers of the first kind. Thus b,(x)
=
1 " n 1" r(tkI b n ( x ) ) x k= b,(O) + 1 -s(n - 1,k - l ) x k k.
k=O
k = l k
(cf. the corresponding formula for Bernoulli polynomials in the subsection on definitions of Section 2.2). Next we compute Jh b,(x) dx by using the formula for integration by parts, which is easily seen to be e'
-
t
1
=
el
10. e'-1
-
'
Then
=
b,(l) - nb,(O) - n(n - l ) b n - l ( 0 )
= (1 -
n)b,(O)
+ n(2
-
n)b,-,(O).
(4.3.8)
4.
116
GENERATING FUNCTION AND SHEFFER IDENTITY The generating function for b,(x) is f * k k=o
k!
t -
t
log(1
+ r ) (1 + t)".
If .x = 0, we have the expansion
The Sheffer identity is
Taking y
=0
gives
Applying t to this yields
From this one can compute the numbers b,(O).
EXPANSIONS The expansion theorem is
Taking h ( r ) = 1 and multiplying by (e' - I)/t, we get
EXAMPLES
3.
117
SHEFFER SEQUENCES
This is Gregory's formula, encountered in the subsection on expansions in Section 1.2, where we pointed out that Gregory discovered this formula in 1670 and that it was reportedly the first formula for numerical integration. Applying Gregory's formula to h,,(x) gives another expression for Jh M.u)dx,
lo1
h,(x) dx
=
(F1
bn(x))
The polynomial expansion theorem is
If we replace p ( x ) by [(e' - l)/t]p(.x), we obtain a formula for the integral of a polynomial in terms of its differences at 0,
Let us next expand b,(x) in terms of the Bernoulli polynomials (of the first kind) B,,(x).Using (4.3.7) and (4.3.8),we have
= (1 - n)b,(O)
+42
-
n)b,- ,(O)
+ nb,-
,(O)B,(x)
118
4. EXAMPLES
We may expand B,,(x)in terms of b,(x) by
In order to evaluate ( t / ( e ' - 1 ) I B,,(x))we require the analog of integration by parts for the operator t/(e' - l), which is
Then
= B,,(O) - nBn- ,(O) - nB,,(O) =(I
- n)B,,(O)- nBn-l(0).
Hence
= (1 - n)B,,(O)- n B n - l ( 0 )
" n(n - 1 ) +
kz2
+ nBn-1(O)bl(x)
S ( n - 2, k
-
2)b,(x).
3.
119
SHEFFER SEQUENCES
MISCELLANEOUS RESULTS Symmetry in the Bernoulli polynomials of the second kind is easily detected by observing that, according to Proposition 2.1.1 1, the sequence b,( -x) is Sheffer for
(-t/(e'
-
l),e-'
-
I).
Now ((e-'-
I ) k l ( - l ) n x ( n ~= ) (-l)n-k((l - e-')klx(n))
=(-1)"-'n!6,,k
= n!6,,k,
and so ( - l)"x(")is associated to e -' - 1. Thus
Replacing x by 1 - i n
= (-
e' I)"e-'-(x
= (-
1)"e-'b,(x
= (-
l)"b,(x
1
-
t
+ n - l),
+ n - 1)
+n
-
2).
+ x, we get
b,(tn - 1 - x) = ( - l)"b,,($n - 1 + x).
3.3. The Poisson-Charlier Polynomials
DEFINITIONS The Poisson-Charlier polynomials c,,(x;a) form the Sheffer sequence for g(r) = e"'"-
I),
f ( t ) = a(& - 1)
for a # 0. These polynomials are discussed by Jordan [l, p. 4731, Erdelyi [ 1, Vol. 2, p. 2261, and Szego [ 1, p. 341. They derive their importance from the fact that, for a > 0, they are orthogonal with respect to the Poisson distribution; that is, 'x
j(k)c,(k;a)c,,,(k;a) = a-"n! d,,,, k=O
where j(k) is the Poisson density j ( k ) = (ak/k!)Ca
f o r k = 0 , 1 , 2 ,....
4.
I20
EXAMPLES
Since ( a k ( & - l ) k l a - n ( x ) n= ) a k - " ( ( e l- l ) k l ( x ) n ) =
n! 6,.k,
we see that the associated sequence for f ( r ) is p,(x) = a - " ( x ) , . Thus from Theorem 2.3.6 we obtain cn(.u;a)= y(t)- 'p,(.u) - a -n,
I)
-n(e'-
(-4n
Recalling that (?'I ( x k ) = j ! s ( k , j ) , we get the explicit expression
= j-0
[f k=o
(")(-
1
l ) n - k a - k s ( kj, ) x'.
k
It is interesting to note that
= en(.-€;u),
where L1p'(x)are the Laguerre polynomials.
GENERATING FUNCTION AND SHEFFER IDENTITY The generating function for the Poisson-Charlier polynomials is
3.
121
SHEFFER SEQUENCES
The Sheffer identity is
RECURRENCE RELATIONS AND OPERATIONAL FORMULAS Theorem 2.3.7 is a ( e ' - I)c,(.x;a) = n c , - , ( x ; a ) ,
which is the recurrence ac,(x
+ 1;a)
-
uc,(x;a) - n c , - , ( x ; u ) =
0.
(4.3.9)
The recurrence formula is
1 ae'
= (x - ae')-c = u Ixc,(.x -
-
(x;a)
1; a) - c,(x; a)
or ac,+,(x;a)
+ uc,(x;a)
-
xc,(x - 1;a) = 0.
(4.3.10)
If we solve for c,- , ( x ; u ) in (4.3.9) and substitute into (4.3.10), where n is replaced by n - 1, we get ac,(x
+ 1;a) + (n - a
-
.x)c,(.x;u)
+ xc,(x
-
1;a) = 0.
122
4.
EXAMPLES
Thus
- uc,+,(x;a) -
+ (a + n ) c n ( x ; a )+ n c , - , ( x ; u )
or ac,+,(x;a) + ( a
+ n - x)c,(x;a) + nc,-,(x;a)
= 0.
According to Theorem 2.3.12,
MISCELLANEOUS RESULTS An interesting feature of the Poisson-Charlier sequence is its inverse under umbra1 composition. Theorem 3.5.5 implies that this inverse s,(x) is Sheffer for
Proposition 2.1.1 1 tells us that the associated sequence for log(l
+ t / a ) is
~ " ( u x )where , & ( x ) are the exponential polynomials. Thus sn(x) = e'd,(ax) = dn(a(X
+ I)),
and so Cn(+(x))
= ((x - ~YQ)".
Recalling a formula in the subsection on miscellaneous results of Section 1.3,
which is valid for any polynomial ~ ( x )and , letting p ( x ) = c,(x; a), we have
3.
123
SHEFFER SEQUENCES
3.4. The Actuarial Polynomials DEFINITIONS The polynomials aiD’(x),Sheffer for g ( t ) = (1
-
t)-P,
f ( t ) = log(1 - f ) ,
are known as the actuarial polynomials, since they were introduced in connection with problems of actuarial mathematics. They are discussed briefly in Erdelyi [ 1, Vol. 3, p. 2541 and Boas and Buck [ 1, p. 421. Since the associated sequence for f ( t ) = log(1 - t ) is p,(x) = 4,J -x), where 4,J.x) are the exponential polynomials, we have aifl’(x) = (1 - f)flc$,(-X) =
f (”)(k
- l)ktk4,( -x) =
k=O
4kk)(-x).
(4.3.11)
Also, (1
Expanding pression
-
t)Aup(.x) = akfl+”‘(x).
(4.3.12)
4Lk)(-.x) and substituting into (4.3.11), we get the explicit ex-
GENERATING FUNCTION A N D SHEFFER IDENTITY The generating function for U ! , ~ ’ ( . Xis)
The Sheffer identity is
and applying ( 1
-
gives
124
4.
EXAMPLES
RECURRENCERELATIONSAND OPERATIONAL FORMULAS
Letting 2. = 1 in (4.3.12), we obtain the formula ulp+ ”(x) = u:o’(x) - tu:o’(x).
(4.3.13)
The recurrence formula is
= -(x - p(1 = (xt
-
x
r)-l)(l
- t)u:/yx)
+ fl)af’(x),
and so u : q . Y ) = (.ut
- ?(
+ p)”1
(cf. the corresponding formula for Laguerre polynomials in the subsection on recurrence relations and operational formulas of Section 3.1). Using (4.3.12), we have
u:q
I(?()
= - ( x - p( 1 - f)-
- -xu:o”’(X)
‘)(1
-
r)ap(x)
+ pulp’(x).
Theorem 2.3.12 is
Combining the last result with (4.3.13) gives the recurrence
4.
OTHER SHEFFER SEQUENCES
125
MISCELLANEOUS RESULTS Let us conclude with an umbral composition. The sequence n!,o'( -.Y) is Shefferfor (( 1 t ) O, log( 1 t ) )according to Proposition 2.1.1 1. By Theorem 3.5.5 the inverse, under umbral composition, of a~p'(-.x) is the Sheffer sequence s,(.Y) for
+
~
+
(eS',e'
-
I),
and so S,(.Y)
='L
LjyY),.
Thus
and so for any polynomial p(.u) we have
Taking p ( x ) = .y", we have shown (Whittaker and Watson [ I , p. 3361) that
4. OTHER SHEFFER SEQUENCES
We conclude this chapter by mentioning some additional examples of Sheffer sequences. 4.1. The Meixner Polynomials of the First Kind
These polynomials form the Sheffer sequence for
126
4.
EXAMPLES
for c # 1. The generating function is
Meixner has shown that this sequence of polynomials satisfies the familiar three-term recurrence relation sn
+ 1 (x) = (x
-
bn)sn(x) - d n s n
~
1 (x),
so(x) = 1, which characterizes orthogonal polynomial sequences (Chihara [1,
p. 1751, Erdelyi [ 1, Vol. 2, p. 2253). We shall discuss this recurrence relation in the next chapter. Incidentally, the Krawtchouk polynomials, which are orthogonal with respect to the binomial distribution, are a special case of the Meixner polynomials of the first kind (Erdelyi [ l , Vol. 2, p. 2241, Szego [l, P.351).
4.2. The Meixner Polynomials of the Second Kind
These polynomials are Sheffer for
s(r) = C( 1 + b f ( r ) ) 2+ S(t)21”2, f ( t ) = tan-
t
1 +&’
and so the generating function is
Like the Meixner polynomials of the first kind, these polynomials form an orthogonal sequence (Chihara [ I , p. 1791). We mention them again in this connection in the next chapter. 4.3. The Pidduck Polynomials
These polynomials are ShefTer for
4.
127
OTHER SHEFFER SEQUENCES
and the generating function is
We recall that the associated sequence for f ( t )is formed by the Mittag-Leffler polynomials M,(x), and so = $(e'
P,(x)
+ l)kfn(x).
Bateman [ 11, Erdelyi [ 1 , Vol. 3, p. 2481, and Boas and Buck [ 1, p. 381 discuss the Pidduck polynomials. 4.4. The Narumi Polynomials
The Narumi polynomials form the Sheffer sequence for
The generating function is
The Narumi polynomials are mentioned in Boas and Buck [ l , p. 371 and Erdelyi [l, Vol. 3, p. 258, Eq. (3)]. 4.5. The Boole Polynomials
These form the Sheffer sequence for g(t) = 1
f ( t ) = e'
+ e", -
1.
Thus the generating function is
1 k.
sk(x)
=
-tk
[l
k=o
+ (1 +
t)A]-1(1
+ t)".
Jordan [l] and Boas and Buck [ 1, p. 371 discuss these polynomials. Actually, Jordan gives a detailed discussion of the polynomials n ! r,(x), where r,(x) is Sheffer for
s(t)=
1 ~
+el
2
'
f ( t ) = e'
-
1.
4. EXAMPLES
128
4.6. The Peters Polynomials
These are a generalization of the Boole polynomials, being Sheffer for q ( t )= ( 1
-
e“)”,
f ( t ) = e’ - 1.
Thus
They are mentioned in Boas and Buck [ 1, p. 371.
4.7. The Squared Hermite Polynomials According to Erdelyi [ l , Vol. 3, p. 2501, if H,(x) are the Hermite polynomials (by our definition, not that of Erdelyi), then
’
Thus the squared Hermite polynomials [H,,(X”~)]form the Sheffer sequence for g ( t ) = (1 - 2 t ) ” 2 / ( 1
-
t),
f(t) = t/(l - t).
See also Boas and Buck [ 1, p. 411. 4.8. The Stirling Polynomials
The Stirling polynomials S,(x) form the Sheffer sequence for (g(t),f(t)), where y(t) = e-‘,
The generating function is therefore
(cf. Erdelyi [ 1, Vol. 3, p. 2573).
4. OTHER SHEFFER SEQUENCES
129
The Stirling polynomials may be connected to the Stirling numbers as follows. By Theorem 3.5.6 and Proposition 2.1.1 1 we have S,(m)= ( e m fI S,,(.U))= (g(f(t))-'emf'f)l x n > 1 - e-' m+ 1
e'
-
1
Now, if m is an integer and rn 2 n, then
where BLmf"(x) are the Bernoulli polynomials, s(m + 1, rn - n + 1) are the Stirling numbers of the first kind, and the last equality follows from a result obtained in the subsection on miscellaneous results of Section 2.2. If m is a negative integer, then
-
-
( - l)"n! (n - m - l)! ( t - " - ' I $ n - r r - ~ ( x ) ) ( - l)"n!
( n - r n - l)!
where S ( n - rn
-
1,
-
m
-
S(n - m
-
1,
-
rn
-
l),
1) are the Stirling numbers of the second kind.
4.9. The Mahler Polynomials Mahler introduced the polynomials s,(x), associated to f ( t ) ,where f ( t )= 1
+r
-
e',
4.
130
EXAMPLES
for the investigation of the zeros of the incomplete gamma function (Erdelyi [1, Vol. 3, p. 2543). The generating function is
4.10. Polynomials Related to the Hermite Polynomials
The associated sequence for f ( t ) ,where -
1 2
f ( t ) = t - it
-
lOg(1 + t ) = - f t 3
+ at4 -
is related to the Hermite polynomials and was introduced for the study of the asymptotic properties of the Hermite polynomials (Erdelyi [ 1, Vol. 3, p. 2561). The generating function is
4.1 1. Polynomials Related to Hyperbolic Differential Equations The Sheffer sequence for g ( t ) = (1 f(t) =
+t)
1 - (1
-c,
+t)-Z
has been applied to the theory of hyperbolic differential equations (Erdelyi [l, Vol. 3, pp. 257-2581). The generating function is
4.12. The Mott Polynomials
Mott has considered the associated sequence for f(t) =
-2t/(l - t 2 )
in connection with problems in the theory of electrons (Erdelyi [l, Vol. 3, p. 2511). The generating function is
CHAPTER 5
TOPICS
In this chapter we discuss some topics to which umbral methods can be fruitfully applied. We shall consider only enough examples in each case to give an indication of the success of the method. The sections of this chapter may be read independently of each other.
1. THE CONNECTION CONSTANTS PROBLEM AND
DUPLICATION FORMULAS
The connection constants problem consists in determining the connection constants c , , ~in the expression
where s,(x) and r,(x) are sequences of polynomials. Umbra1 methods give an easy solution to this problem when the sequences involved are Sheffer. Let s,(x) be Sheffer for (g(r),{ ( t ) ) and let r n ( x )be Sheffer for (h(t),l(t)).In (5.1.1) we recognize an umbral composition
131
5.
132
TOPICS
and so from Theorem 3.5.2 we deduce that n
tn(.x)
=
cn.k-Y
k
k=O
Thus the sequence
is Sheffer for the pair
This is one form of our solution. To get an explicit expression for the connection constants we use (3.5.2).
A duplication formula is a formula of the form n
where u is a nonzero constant. When rn(.x)is a Sheffer sequence, this is a special case of the connection constants problem, for if r , ( x ) is Sheffer for ( h ( t ) , l ( t ) ) ,then, according to Proposition 2.1. I 1, (h(a
I
t ) l ( u - t I k I r n ( u x ) ) = (h ( t ) l ( t ) kI r,,(x)) = n! bn.k
'
and so r,,(u.Y) is Sheffer for ( h ( u t ) , l(u I t ) ) . Thus we see that ~
1. CONNECTION CONSTANTS PROBLEM AND DUPLICATION FORMULAS
I33
is Sheffer for
and
Let us give some examples. Example I
Consider the formula
where x"'' = .u(x + l ) . . . ( . u + n 1 - e-'. We have
-
I ) is associated to the backward difference j ( t ) = e' - 1,
q ( 1 ) = I,
h ( t )= I ,
I(t) = I
-
e-t,
and so t , ( s ) is Sheffer for
Thus
L,,( -I), where L , ( s ) are the Laguerre polynomials. In particular, f,l(.Y) =
and
Example 2
Consider the formula
1
.u"i' =
",l,J.Y)k.
h=O
The central factorials are Sheffer for q ( r ) = 1,
j ( r ) = rf
-F " 2 ,
and the lower factorials are Sheffer for h ( t ) = 1,
I ( [ ) = f'f
-
1.
134
5.
TOPICS
Thus the sequence rn(.x) is Sheffer for
By the transfer formula r n ( x ) = X( 1
+ r)nizXn-l
and so
Example 3 Consider next
c n
=
Cn.k4k(-X),
k=O
where Lf'(x) are the Laguerre polynomials and &(x) are the exponential polynomials. Here we have g(r) = (1 - r ) - = - ' ,
f(r)
= r/(r - 1)
and
I ( r ) = log(1 - t ) .
h(r)= 1 ,
Thus r n ( x )is Sheffer for (e-'"+l)l
, l -e-I)
and so rn(x) = d a +')'(x)(") = (x
+ a + I)(")
c Is(n,j)l(x + n
=
j=O
a
+ 1)'
(Comtet [ l , p. 2133)
where s ( n , j ) are the Stirling numbers of the first kind. Thus we have
1.
CONNECTION CONSTANTS PROBLEM AND DUPLICATION FORMULAS
For
c(
= -1
135
we get
c Is(n, n
Ln(x)
=
k=O
Example 4
k)l
4k(-x).
Consider k=O
where H r ' ( x ) are the Hermite polynomials of variance v. Here we have g ( t ) = e"'",
f ( t )= t,
h(t) = ei'2'2,
[ ( t )= t.
Thus t,(x) is Sheffer for (e'"-""'/z 3
r)
and tn(x) = e ( A - - v ) r 2 / 2 . y n
and so
Example 5
Let us consider
where E,(x) are the Euler polynomials and B,(x) are the Bernoulli polynomials. Here y(t) = (e'
+ 1)/2,
h ( t )= (e' - l)/t,
and so r,(x) is Sheffer for
f ' ( t )= t ,
l(t)= t,
136
5.
TOPICS
Hence
-
e'-1
n+l
f
--
rE,+ l(.x)
=
1 ~
n+l
(e' -
1(x)
2
1
(el + 1 - 2)En+,(.u)= -(.xnfl n+l
n+l = k=O
1
(
n+l k
-
En+l(x))
-2 );;;-i""k(o)"k.
and so
Example 6
Consider n
cn(.x;b)=
1
cn.k(.k(X;a),
k=O
where c,(x; a) are the Poisson-Charlier polynomials. In this case
and so r , ( s ) is Sheffer for
Therefore
and
1.
137
CONNECTION CONSTANTS PROBLEM AND DUPLICATION FORMULAS
Example 7 Consider the duplication formula for Hermite polynomials,
n
-
k
odd.
Thus
Example 8 Consider the duplication formula for Laguerre polynomials,
We have h ( t ) = (1 - t ) - = - ' ,
/ ( t )= r/(t - I),
and so
-a-l-k (a -
j -a-
n
l)'(t'lx"-k
1-k) =(n -
k
+
)
a)n! -ak(l H - k k!
-
5. TOPICS
I38 Thus we obtain
Example 9 Let us obtain the coefficients in
where MJx) are the Mittag-Leffler polynomials. Recall that Mn(x)are associated to I ( t ) = (e' - I)/(& I) and that I ( t ) = log((1 t)/(l - 0).
+
+
Thus the sequence t n ( x )is associated to
+ t)/(l + r)/(l - (1 + -(1
1(2fir)) =
((1
- f))2 -
((1
f))2
r)2
(I
+ t)2+(1
1
+1
- t)2 - t)2
- 2t - 1 +f2'
By the transfer formula,
and so
2. THE LAGRANGE INVERSION FORMULA
The Lagrange inversion formula is a formula for the compositional inverse of a delta series. Actually, the most common version of the Lagrange formula
2.
THE LAGRANGE INVERSION FORMULA
139
is more general and there are variants. A concise treatment from the classical point of view, with references, can be found in Comtet [I]. Let g ( t ) be any formal power series and let f ( t ) be a delta series. The Lagrange Inversion Formula The coefficient oft" in g ( f ( t ) ) , multiplied by n, equals the coefficient oft"-' in g'(t)(f(t)/r)-". In the symbols of the umbral calculus,
Hermite's Version of the Lagrange Inversion Formula The coefficient oft" in t g ( f ( t ) ) / f ( t ) f ' ( f ( tequals )) the coefficient oft" in g(t)(f(t)/t)-". In symbols,
When g ( t ) = t , the Lagrange formula reduces to
a formula for the compositional inverse of f(t). Notice that the key feature of these formulas, indeed of any variant of the Lagrange formula, is the presence of f ( t ) on one side and its absence on the other. There have been many different proofs of these formulas, many of which assume that the power series involved are convergent and represent functions analytic in a disk (for an example see Whittaker and Watson [l]). Cauchy's theorem is frequently invoked. But, as Henrici [l] points out, the essential character of the Lagrange formula is purely formal. Using umbral methods, the proof of the Lagrange inversion formula and its variants reduces to a simple computation. From the second form of the transfer formula we have ( s ( f ( t ) I)x"> = ( i / * g ( t )I x">
(do1if."> = ( d t ) I x ( f ( r ) / t ) - " x " -l > = (y'(t) I (fW) -"Xn> =
=
(g'(t)(f(t)/t)-"l x " - l ) .
140
5.
TOPICS
It is hard to imagine a simpler proof. Moreover, from this proof we see how to obtain variants of the formula. Using the first form of the transfer formula, we get
<s(f(t,) I x")
= < d t )1i-J-Y") = (get) If'(t)(f(t)/t)-".x") = ( g ( t ) f ' ( t ) ( f ( t ) / t ) - " - I x").
Thus we have another version of Lagrange's formula. A Third Variant of the Lagrange Inversion Formula The coefficient oft" in g ( f ( t ) )equals the coefficient oft" in g ( t ) f ' ( r ) ( f ( t ) / t ) - " - '. In symbols,
If we replace g ( t ) by g ( t ) f ( t ) / r f ' ( twe ) obtain Hermite's variant.
3. CROSS SEQUENCES AND STEFFENSEN SEQUENCES
The reader may have observed that many of the examples of the previous chapter share a common property. Namely, they have the form S?)(X)
= h(t)'S,,(x),
where s,(x) is a Sheffer sequence, h(t) is invertible, and i.ranges over the real numbers. The sequence sl;")(x)is known as the Steflensen sequence for h(t) and S,(.Y) of index 1.. In case s,(x) is an associated sequence, then s!,"(x) is known as the cross sequence for h ( f )and s,(x) of index i.If.s,(x) = x", then ska)(x)= h(t)'x" is the Appell cross sequence for h ( t ) of index 2.. For example. the Hermite, Bernoulli (of the first kind), and Euler polynomials form Appell cross sequences. The Poisson-Charlier and actuarial polynomials form cross sequences, and the Laguerre polynomials form a Steffensen sequence. Of course, any ShefTer sequence s,(x) generates, in a natural way, a cross sequence. For if s,(x) is Sheffer for ( g ( r ) ,f ( t ) ) and p,(x) is associated to f(t), then sn(x) = g ( t ) pn(x),
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CROSS SEQUENCES AND STEFFENSEN SEQUENCES
and so SY)(X)
= g(f)”n(x)
is a cross sequence for which s l - ”(x) = S ” ( X ) .
The theory of Steffensen sequences per se has not been well developed (but see Rota et al. [l], and J. W. Brown [l]). We shall present here only a few results.
Theorem 5.3.1 A sequence p v ) ( . ~is) a cross sequence if and only if (5.3.1)
for all n 2 0, all y in C , and all real numbers 1. and p.
Proof Let p y ’ ( x ) = h(r)’p,,(x)be a cross sequence. If p,(x) is associated to f ( t ) , then f(t)p!,”(x)= np!,’l1(x), and so for each 1, the sequence p!,’)(x) is Sheffer and
Applying h(t)$ to this equation gives (5.3.1). Conversely, if (5.3.1) holds, then taking p = 0 shows that p a ’ ( x ) is Sheffer, say for ( h , ( t ) , f ( t ) ) . Thus pY’(x) = h,(t)- ‘p,,(x).NOW
< h , i ( ~ ) - l h ~ ( t ) F ’ I P n (= x))
and so h,(t)h,(t)
(hi(f)F1l~k(x))(kp(t)F~ lPn-k(X))
-
p~+”’(o)
=
(h,+,(t)-’
= h’+,,(t), which
I pn(x)),
implies that h,(t) = A([)’. Thus
p Y ’ ( x ) = h(t)’p,,(x) is a cross sequence.
Theorem 5.3.2 A sequence sY’(.u)is a Steffensen sequence if and only if
(5.3.2) for some cross sequence p r ) ( x ) .
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Proof If s?)(x) is a Steffensen sequence, then s F ) ( x )= g(r)-'p;*)(x), where p!,')(x)is a cross sequence. Applying g ( t ) to (5.3.1) with x and y interchanged, we get (5.3.2). For the converse, taking p = 0 in (5.3.2),we deduce the existence, . for each i., of an operator yA(t)for which g'(t)sl;")(x)= p i A ) ( x )Then
'
+
g A + p ( t ) - ' p ~ + " ( X y ) = S?+P'(X
+ y)
= g'(t)-'pl;l+''(x
+ y),
and so g i + , ( t ) = g A ( t ) ,which implies that yi(t) = g O ( t ) .Thus SF)(X) = g & -
1p?'(x)
is a Steffensen sequence. The umbral composition of Appell cross sequences is very simple.
Theorem 5.3.3
If p!,')(x) is an Appell cross sequence, then p?'(p'"(x)) = P?+"'(X).
Proof
This follows from the fact that the umbral operator for p;')(x) is h(t)'.
We conclude this section with some more exotic results.
Theorem 5.3.4 If s?)(x) = h(t)'sn(x) is a Steffensen sequence, where s,(x) is Sheffer for (g(r),f(t)), then the sequence slp"'(x)
is Sheffer for
Proof
First we observe that h(t)-"f(t)slP"'(x)= h(t)-"f((t)h(t)""s,(x) = nh(t)"("-'Isn= nsf!
"'(x).
l(x)
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CROSS SEQUENCES AND STEFFENSEN SEQUENCES
Also, if p , ( x ) = g(t)s,,(x) is associated to
=
, f ( t ) , then
( h v f ( f ) ) (h ( r ) - " f t ) t
)
-n-l
.xn,
which, by the transfer formula, is the associated sequence for h(t)-y(t). This concludes the proof. Theorem 5.3.5 If sr)(x) = h(r)'[C.f''(t)]- ' p , ( x ) is a Steffensen sequence, where pn(.u)is associated to J'(t), then the sequence
.
yg(an) -
I
(x)
for n 2 1, is associated to h ( t ) ? f ( r ) . froof By the transfer formula and the recurrence formula we see that the associated sequence for h(t)-:f(r)is
=
.ull(t)un.u-'pn(x)
=
uh( t )""x lx [ f ' (r )] -
-
= .uh(t)""f'(t)]-'p,~
- .us(un) n
pn
-
1 (x)
I(.x)
l(-4.
Appell cross sequences fit the requirements of this theorem with f ( r ) = r. Corollary 5.3.6
If s?)(.Y)= h(r)'r" is a n Appell cross sequence, then s . s ~ " , (.x)
is associated to h ( t ) " t . ~
Corollary 5.3.7 If sa'(x) = k ( r ) " [ f ' ( t ) ] 'p,,(.u) is a Steffensen sequence, where p,(.u) is associated to f ( r ) , then ~
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4. OPERATIONAL FORMULAS
In this section we present a simple umbral technique for deriving operational formulas. A well-known example of an operational formula is n
(.xO" =
1S ( n ,k ) x k t k ,
(5.4.1)
k=O
where t is the derivative operator and S(n, k ) are the Stirling numbers of the second kind. Many formulas of this type are obtained by catch-as-catch-can methods and then proved by induction. The umbral approach gives a systematic method for generating such formulas. Let s,(.Y) be Sheffer for ( y ( r ) , f ( t ) ) . Suppose that Qk is a sequence of linear operators on P for which Qksn(.x)
= rk(n)Sn+@(.x),
where r n ( x )is Sheffer for ( h ( r ) ,/ ( r ) )and p is an integer. Suppose further that T is an operator on P for which TsnW= u W n+ i(.4,
where u ( x )is a polynomial and i.is an integer. We wish to expand T i n terms of the operators Q k . By the polynomial expansion theorem,
If O : s n ( x )-+
sn+ l(x)is
the Sheffer shift for sn(.x) and if
2 min{O,i.}, then
and so the desired expansion is (5.4.2)
We have not taken the most general approach to deriving a formula of the type (5.4.2).In Roman [9] we consider more general operators Q k , where r n ( x )
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OPERATIONAL FORMULAS
may not be Sheffer. However, to avoid details that tend to obscure the main idea, we shall confine ourselves here even to the case E. = p = 0. Theorem 5.4.1 Let s,(x) be Sheffer for (y(r), f ( t ) ) and let r,(x) be Sheffer for ( h ( t ) ,l ( t ) ) .Then if Qk and T are operators defined by Q k s , ( x ) = r k ( n ) s f l ( . ~ L Ts,(x) = u(n)s,,(x),
where u ( x ) is a polynomial, we have
Let us consider a simple example. If we take Qk
=
@Y(r)k,
then Q k s , ( . ~=) O ~ f ( f ) k ~ ~ = , l (( .n~) ), s , , ( ~ ) , where, of course, if k > n, then r,,(x) = (x),,,
= 0. Thus
h ( t ) = 1,
I ( t ) = e'
-
1,
and so we have
Taking
7-= [c.r.f'tt)l"> we have
Ts,(.w) = [O.f'(t)]"s,,(.u)
=
nms,(x),
and so u ( s )=
X"l.
Thus we obtain the operational formula
or, since( l/k!)((e' kind,
I xm) = S(ni,k ) are the Stirling numbers of the second m
[O,f(t)]"
=
2 S(m,
k=O
k)Pf'(t)k.
(5.4.3)
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Here 8:sn(.u) + s,+ ,(x) is the Sheller shift for s,(.x), which is Sheffer for a pair whose delta series is f ( t ) . Let us give some instances of (5.4.3).
Example 1 Lets&)
= ( x - a)", which
is Sheffer for (eat,?). By Theorem 3.7.1,
and (5.4.3) becomes
5
1c.u - a ) t l m =
S(rn,k)(.x - a ) k t k ,
k=O
which, for c i
= 0,
is (5.4.1).
Example 2 Let s,(x) = (.Y/u)(.Y/(u + l))...(.x/(a + n - l)), which is associated to the backward difference f ( t ) = 1 - e-"'. Then 0
= .u[.f'(t)]
= a - '.war,
and so we get m
[a-'.u(e"' - l)]"
C S(tii.k)a-k(.xye")k(f
=
-
e-"'lk.
k=O
Using the notation
for the forward difference, backward difference, and translation operators, we have m
(.xA,)"'
S(rn,k)am-k(xEP)kV~.
= k-0
Example 3
Let
S,,(.Y)
= H f ) ( . x )be the Hermite polynomials. Then
and so we get rn
[(.Y
+ V f ) t l r n = 1 S ( r n , k ) ( x+ L't)ktk. k=O
Other examples of Theorem 5.4.1are just as easily obtained.
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INVERSE RELATIONS
5. INVERSE RELATIONS
An inverse relation is a pair of equations of the form
(5.5.1)
where xn and yn are variables. One of the simplest examples of an inverse relation is
There is a vast literature on inverse relations, and Riordan [2] devotes no less than 85 pages to the subject. The relations (5.5.1) are equivalent to a single orthogonality relation, obtained by substitution, n hn.j
=
un.kbk,j.
(5.5.2)
k=j
From our point of view, we observe that if s,(x) and rn(x)are sequences of polynomials for which
and
then an.k and b n , k satisfy (5.5.2), and so also (5.5.1). Now suppose that s,(x) is Sheffer for ( g ( t ) , f ( t ) )and r,(x) is Sheffer for ( h ( t ) ,[ ( t ) ) .Then by the polynomial expansion theorem
and
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This gives the inverse pair
Actually, this pair can be simplified somewhat. For if p,(x) is associated to f ( t ) and q n ( x )is associated to I(?), then " 1 yn =
,(g(?)t=ok. "
' h ( t ) l ( t ) k1 P n ( x ) ) x k ,
1
Since y(t) and h ( t ) are arbitrary (invertible) series, we have proved the following result. Theorem 5.5.1 Let p,(x) be associated to f ( t ) and let qn(x)be associated to I ( [ ) . Then for any invertible series h ( t )we have the inverse pair
In case j ( r ) = I ( [ ) . this theorem can be simplified. Corollary 5.5.2 Let p , ( x ) be an associated sequence. Then for any invertible h ( t )we have the inverse pair
Corollary 5.5.3 we have
Let p,(x) be an associated sequence. Then for any constant y
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INVERSE RELATIONS
Let us give a few examples. Example 1 If p,(x) = x", we get
Example 2 If p n ( x ) = (x)", then we have
Example 3 If pn(x) = G,,(x;a;h ) are the Gould polynomials, we get
'"
n
=k
y
z o y -
a(n - k )
(
-
a(n - k ) b
)"-Pk'
n
x n =
C
k=Oy
-k a('
-
k)
)n -k
yk '
Example 4 The Fibonacci numbers F, can be defined by the generating function
f Fktk
=
k=O
1 1-t-f2'
If pn(x)is the associated sequence for which
then, for x = I , we get
and, for x
=
-
1, we get
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5. TOPICS
Thus, taking J’ = - 1 in Corollary 5.5.3, we get the inverse pair y,
= x, -
nx,-
-
n(n - 1 ) ~ , - ~ ,
Replacing x, by x,/n! and y, by y J n ! gives V ” = x, - x,_ I
- x,-2,
kyO
Next, let us take h ( t ) = u ( t ) l ( t ) Ain Corollary 5.5.2, giving
(5.5.3)
where s!,~)(x)= I(t)’p,(?r) is, in the language of Section 3, a cross sequence. Taking u ( t ) = eyt, we get
We continue the examples. Example 5 Let s!,”(x)
=
H!,”(x) be the Hermite polynomials. Then
Since (n - k)! H1;“!,(0)=
(+(n - k ) ) !’
lo,
n -k
even,
n -k
odd,
5.
151
INVERSE RELATIONS
we get the inverse pair
n -k even
I n - k)12
(t(n
k=O n-keven
Example 6 If sl;"(x)
=
(n - k)! - k ) ) ! J'k'
B!,"(x) are the Bernoulli polynomials, we have
For A = 1 and y = 0, since BiLi)(O)=
e' - 1
Xn-k
)
=
1
n-k+l'
we have Yn
= k = O ( " k) B n - k ( o ) x k ,
Yk.
Example 7 The Laguerre polynomials satisfy L ~ ) ( x= ) (1
-
t)"+'L,(x),
and so if we set u ( t ) = ey'(l - t ) and s!,"(x) yn
= (1 - t)'L,,(x) in
(5.5.3),we get
= k = o ( nk) L ? l k ( J ' ) x k ,
x, = k = O (")L:;',"k(-y)yk.
Example 8 Recalling that the Laguerre polynomials are self-inverse, that is, L p ( L ' y x ) )= xn,
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TOPICS
we get the orthogonality relation n! I + n
x n = ~ o ~ ( l . + k( - ) l)kLF)(x)
which gives the inverse pair
Let us consider a somewhat more delicate corollary of Theorem 5.5.1. Namely, we take f ( t ) = ea'u(t)
and
I(t) = eb'u(t),
where u ( t ) is a delta series and a # b. Then Theorem 5.5.1 gives the inverse pair whose coefficients are 1
-(h(t)ebk'U(r)klPn(x)), k! 1 -(h(t)-'eak'u(t)k k!
This is equivalent to the pair of coefficients
5. INVERSE RELATIONS
153
or
+ (b - a)k))?
(i)(h(t)lpn-k(x
(5.5.4)
(i)(h(t)-l
Iqn-k(X + ( a
-
b)k)).
Now if u,(x) is associated to u(t)%then, according to Corollary 3.8.2,
= xe-anrx-lun(x) --
x
and, similarly, qn(x) =
x -
an
u,(x - an),
a X
U,(X -
bn).
Putting this into (5.5.4) gives the following result. Corollary 5.5.4 Let u,(x) be an associated sequence. Then for any invertible h ( t ) we have the inverse pair
Corollary 5.5.5 we have
Let u,(x) be an associated sequence. Then for any constant y
Let us consider some examples of Corollary 5.5.5. Exnmple 9 If we take u,(x) = (x), and a = 0, we get
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TOPICS
and when b = 0,
Example I0 If we take u,(x) = (x), and b = - 1, we get n
n!y-(a+
yn=kzOk?
xn=
y-an-k
l)k y - a n - k n-k
(
)xk'
n ! -y + ( a + l ) k -y + n + ak 1) n-k k=ok! - y + n + ak (
Yk,
which is equivalent to a class of inverse relations given by Could (Riordan [2, P. 50, Eq. (7)l). Riordan [2, pp. 92-99] also gives four inverse relations associated with the name of Abel. If we take u,(x) = X" in Corollary 5.5.5, we have
x, = t ( n ) ( - y k=O
k
+ ( a - b ) k ) ( - y - bn
Example 11 I f a = b = - 1, we get
which is Riordan's first Abel inverse relation. Example 12 If a = 0 and b = - 1, we have
which is Riordan's third Abel inverse relation.
+ Uk)"-k-'yk.
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155
INVERSE RELATIONS
Example 23 If a =
-
1 and b
=
1, we get
which is Riordan's fourth Abel inverse relation. To obtain Riordan's second Abel inverse relation, it is more instructive to invert one of the two relations rather than to prove that the given pair of relations are inverse to each other. In fact, we can generalize a bit and consider the problem of inverting the relation (5.5.5)
(When x
=0
and
fi = 1 , we get the first equation of Riordan's pair.) If we set s,(x) = (x
+ + fin)", ci
then, according to Theorem 3.8.3, s,(x) is Sheffer for (e-a'(l - fit),te-P').
Thus
s,(x) = ea'(l - f i t ) - A , ( x ; - fi), where A , ( x ; - f i ) = x ( x written as
+ @I)"- ' are the Abel polynomials. Now (5.5.5)can be
and by Corollary 5.5.2 its inverse is
It is straightforward to compute that (1 - fit)A,(x;
-fi)
= (1 - fit)x(x
and thus we get the inverse pair
+ @rn)"'-'
= ( x 2 - b2rn)(x
+ firn)m-2,
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5. TOPICS
Setting c(
=0
and
fl
=
I , we get Riordan’s second Abel inverse relation
( y 2- n
+ k)( -y + n - k ) ” - k - 2 y k .
The examples in this section are only a small sampling of the power of Theorem 5.5.1. 6. SHEFFER SEQUENCE SOLUTIONS TO RECURRENCE RELATIONS
The umbral calculus can be used to determine the Sheffer sequence solutions to certain recurrence relations. One o f the most important nontrivial examples is that o f the familiar three term recurrence sn + 1
(XI= (.%
- bn)sn(x) - d n s n
~
1 (x)
(5.6.1)
for n 2 0, with so(x) = 1 and S L ~ ( X= ) 0, which is known to characterize polynomial sequences that are orthogonal with respect to a moment functional. For details on orthogonality we refer the reader to Chihara [l]. The Sheffer sequence solutions to (5.6.1)were first determined by Meixner [l] in 1934 and then by Sheffer [I], using a different method, in 1939. The modern umbral method gives a very elementary solution to this and other recurrence relations. The basic idea is that a Sheffer sequence satisfies a certain recurrence relation if and only if its associated pair of linear functionals satisfies a certain pair of formal differential equations.
Theorem 5.6.1 Let h(r) be an invertible series and let I ( [ ) be any series. Then the recurrence relation (5.6.2) for n 2 0, with sk(x) = 0 for k < 0, has a solution s,(x), which is Sheffer for (q(r),f ( t ) ) ,if and only if
(i) (ii) (iii) (iv)
o(y(t)) = O,o(f’(t)) = 1, ~ ~ + ~ , ~ = ( l / k ! ) ( k c ~ I+) ~~ ,~ ~. ~- )( fk2 o-r 1,k f ‘ ( t ) = (l/h(r))[I - Clm_,(cj+ I , j - ~ ~ . ~ ) j ( t ) ’ + and ’], ( i ( t ) / q ( t ) )= -(l/h(r))[l(O + Cj”_ocj.jf(t)’I.
6.
157
SHEFFER SEQUENCE SOLUTIONS TO RECURRENCE RELATIONS
and (g(t)f(t)‘ln! c , , ~ s ~ - ~ ( . x=) )t ~ ! c , , ~dkn !. n - j = (k!ck+j,jg(t)f(r)‘+’Is,(x)),
which holds even for n ( g ( t ) ,f ( t ) ) ,if and only if
-
j < 0. Thus (5.6.2) holds for s,(x), Sheffer for m
for all k 2 0, along with o ( g ( t ) )= 0 and o(f’(t)) = 1. Taking the indicated derivative and canceling a factor f ( t ) ‘ - give k g ( t ) = h ( t ) g ’ ( t ) f ’ ( t ) + k h ( t ) d t ) f ’ ( t )+ I(t)g(t)f(t)
(5.6.3) for all k 2 0. For k = 0 this is
0 = h ( t ) g ’ ( t ) f ( t+ ) i(t)g(t),f(t) +
2 cj.j g ( t ) f ( t ) j +
1.
(5.6.4)
j=O
By canceling a factor f ( t )and rearranging, we get (iv).Thus (5.6.2)holds if and only if (i), (5.6.4),and (5.6.3)hold. Next we subtract (5.6.4)from (5.6.3),cancel a factor g ( t ) , and rearrange to get h ( t ) f ’ ( t )= 1 -
m
1
- ( k ! c k + j . j- e j . j ) f ( t ) J + L j=ok
( 5.6.5)
for all k > 0. Thus (5.6.2)is equivalent to (i), (5.6.4),and (5.6.5)for all k > 0. But (5.6.5)has a solution f ( t ) for all k > 0 if and only if it has a solution when k = 1, which gives (iii), and ( I / k ) ( k ! C k + j . j - C j , j ) = C l + . I . J. - C . 1.1.
for all k > 0, which gives (ii). Thus (5.6.2)is equivalent to (i)-(iv) and the proof is complete.
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TOPICS
Condition (ii) is subject to separation of variables and can be solved in a variety of cases. Once f ( t ) is determined, (iv) can be solved and, since ( d t ) I so(x)) = 1, we get
We may also obtain differential equations for f ( t ) and g ( f ( t ) ) - ’ , for use in determining the generating function for the solution s,(x). Condition (iii) is equivalent to
(5.6.7) and from this and (5.6.6) we get
Corollary 5.6.2 The recurrence relation (5.6.2) has a solution s,(x), which is Sheffer for (g(r),f(t)),if and only if Eqs. (i), (ii), (5.6.7),and (5.6.8)hold. Let us give two simple examples.
Example I
The recurrence relation
with so(x) = 1, is well known to characterize the Hermite polynomials. Pretending ignorance of this fact and referring to Theorem 5.6.1, we have
h ( t ) = 1,
cn.k= 0,
i(2) = -vr,
and so (iii) and (iv) give f ’ ( t )= 1
and
g ’ ( t ) / g ( t )= vt.
f ( t )= f
and
y ( t ) = ev’’’*,
Hence which is indeed the pair for the Hermite polynomials.
Example 2
For the recurrence S , + ~ ( X )=
(2n
+ cx + I
-
x)s,(x) - n(n
+ Z)S,-~(X)
with so(x) = I , we have
co, 0 = 0,
2
en, = ___
( n - I)!
( n > O),
c,., =
n+IX -~
( n - I)! ’
6.
SHEFFER SEQUENCE SOLUTIONS TO RECURRENCERELATIONS
159
Thus (iii) is and so and (iv) is g'(t) -= (a
s(t)
a+l + l ) ( f ( t )- 1) = t-1'
and so g(t) = (1 - t ) - u - l ,
reminding us that the Laguerre polynomials satisfy this recurrence. From these examples the reader should have no difficulty solving other recurrence relations (whose solutions are not known in advance). Finally, let us consider (5.6.1), which is only a little less trivial than the preceding examples. In this case we have I ( t ) = 0,
h ( t ) = 1, C,J)
Equation (ii) gives, for j
=
bn --
b,
CnJ =
n!'
= 0 and
rn
then j
=
= k(b1 - bo)
= 2,
dn
--
n!'
1,
+ bo
and
Equations (5.6.7) and (5.6.8)are '(')
=
1+ 1
and 1 g ( m ) = exp
s
dt ( b , - bo)t ( d J 2 - d l ) t 2
+
1
+ (b,
-
-bo - d l t dt. b0)t ( d 2 / 2 - d l ) t 2
+
These two integrals are routinely evaluated by elementary techniques and we omit the details. Suffice it to say that one obtains, as Sheffer did, four cases, depending on whether the denominator 1 + ( b , - bo)t + ( d J 2 - d l ) t 2 is constant, linear, quadratic with distinct roots, or quadratic with confluent roots. Meixner considers five cases according to whether the denominator is constant, linear, or quadratic with negative discriminant, zero discriminant, or
5.
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TOPICS
positive discriminant. Chihara [ l , p. 1641 gives the explicit solutions, which are in terms of the Hermite, Laguerre, Poisson-Charlier, and Meixner (of both kinds) polynomials (see also Erdelyi [ I , Vol. 3, p. 2731). The technique used in the proof of Theorem 5.6.1 works for types of recurrence relations other than the one given in that theorem. Thus if the reader has a particular recurrence relation he wishes to solve for Sheffer sequence solutions, this approach might prove useful. 7. BINOMIAL CONVOLUTION
We know that if f ( t ) and g ( t ) are delta series, then so is f ( g ( t ) ) . Theorem 3.4.6 gives us the relationship between the associated sequences for f ( t ) ,g ( t ) and f ( g o ) ) . Now if f ( t ) and g ( t ) are delta series satisfying f ' ( 0 ) + g'(0) # 0, then f ( t ) + g ( t ) is also a delta series. In this section we briefly consider the relationship between the associated sequences for f ( t ) ,g ( t ) , and f ( t ) + y(t). Theorem 5.7.1 Let p,(x) be associated tof(t) and let q n ( x ) be associated to g ( t ) . If f ' ( 0 )+ y'(0) # 0, then f ( t ) + g ( t ) is a delta series whose associated sequence is
Proof
This follows from
We call the sequence r n ( x ) in Theorem 5.7.1 the binomial convolution of p n ( x )and qn(x),and write
If we introduce the notation P , ( x ) for the inverse, under umbra1 composition, of pn(x), then Theorem 5.7.1 has the following corollary.
Corollary 5.7.2 Let p n ( x ) be associated to f ( t ) and let q.(x) be associated to g(t). If f ' ( 0 )+ g'(0) # 0, then the delta series f ( t ) + g ( t ) has associated
7.
161
BINOMIAL CONVOLUTION
sequence P-,(.u)0 q n ( X ) .
Theorem 5.7.1 can easily be extended to arbitrary Sheffer sequences, although the notation is a bit cumbersome. Theorem 5.7.3 Let s,(x) be Sheffer for ( g ( t ) ,f ( t ) )and let r,(x) be Sheffer for ( h ( t ) I(r)). , If f'(0) g'(0) # 0, then the binomial convolution
+
is the Sheffer sequence for ( g ( f ( J ( f )+ m ) ) h ( w = ( f ) +
w,70)+ W)).
In the Appell case Theorem 5.7.3 is a bit more manageable.
Corollary 5.7.4 Let s,(x) be Appell for g ( t ) and let r n ( x ) be Appell for h(t). Then the binomial convolution
is Sheffer for ( g ( t / 2 ) h ( t / 2 ) t, / 2 ) .
Thus s,(x).r,(x) = 2 " g ( t / 2 ) - ' h ( t / 2 ) - ' x n .
As a simple example, the Bernoulli polynomials B?'(x) are Appell for (e' - l)"/t" and the Euler polynomials E F ' ( x )are Appell for (e' + 1)"/2".Thus
BIp'(x)@ E)P'(x) = 2" = 2"-)
t
e' - 1 =
That is,
2"B?'(x).
aX"
CHAPTER 6
NONCLASSICAL UMBRAL CALCULI
1. INTRODUCTION
In this chapter we shall discuss briefly the nonclassical umbral calculi. It is our intention to give only an introduction to the theory. For a more detailed discussion we refer the reader to Roman [6-8, lo]. Let c, be a sequence of nonzero constants. If n! is replaced by c, throughout the preceding theory, then virtually all of the results remain true, mutatis mutandis. In this way each sequence c, gives rise to a distinct umbral calculus. Actually, Ward [2] seems to have been the first to suggest such a generalization (of the calculus of finite differences) in 1936, but the idea remained relatively undeveloped until quite recently, perhaps due to a feeling that it was mainly generalization for its own sake. Our purpose here is to indicate that this is not the case. In the next section we list the principal results. In Section 3 we discuss a particular delta series, which for certain values of c, gives rise to the Gegenbauer and Chebyshev polynomials. In the final section we discuss a particular nonclassical umbral calculus, which is known as the q-umbra1 calculus. Sections 3 and 4 are independent of each other. 2. THE PRINCIPAL RESULTS
In this section we consider the effect of replacing n! by c, in the preceding theory. Since this chapter is intended only to be isagogic, we merely give a summary of the main results. In most cases the proofs follow lines identitical to those of the classical case, and for these we refer the reader to Roman [6-8, lo]. 162
2.
163
THE PRINCIPAL RESULTS
Let c, be a fixed sequence of nonzero constants. Then we may map the algebra .F of all formal power series in t isomorphically onto the vector space P* of all linear functionals on P, in a manner completely analogous to that of Section 2.1, by setting ( f k I X " ) = Cfl8,,k.
Again we obscure this isomorphism and think of 4 as the algebra of all linear functionals on P. If
then we have
If we define the continuous operator a, on B by
then Theorem 2.1.10 has the following analog. Theorem 6.2.1 If f ( t ) is in 4,then
for all polynomials p ( x ) . The evaluation functional is k = O ck
which is the analog of the exponential series for the c,-umbra1 calculus. As one might expect, this series lies at the heart of the theory. We have (E,(f)'l
P(X))
= P(Y)
for all polynomials p(x). With regard to linear operators, it is clear that the statement of Theorem 2.2.5,
(f(t)df) I P ( 4 ) = (f(t)I d t ) P ( X ) ) , is all but indispensable. Guided by this requirement, we are forced, quite willingly, to adopt the definition t k x n= ( C , / C , _ k ) X n - k
and, more generally,
6.
164
The operator
E$)
NONCLASSICAL UMBRAL CALCULI
satisfies
this sum being the analog of (x + y)". In fact, following old-style umbra1 notation, one would write this sum as (x + y)". A sequence of polynomials s,(.Y) is Shefleer for the pair ( g ( t ) , f ( t ) )if ( g ( t ) f ( t ) kI s n ( x ) ) = en8n.k
for all n, k 2 0. As usual we require that o(y(t)) = Oand o ( f ( t ) )r= 1. Associated and Appell sequences are defined as expected, that is, with g ( t ) = 1 and f ( t ) = t , respectively. All of the principal results of Section 2.3 have analogs for this definition. Theorem 6.2.2 (The Expansion Theorem) Let s,(x) ( y ( t ) , f ( t ) ) Then . for any h ( t ) in 3,
be Sheffer for
Theorem 6.2.3 (The Polynomial Expansion Theorem) Let s,(x) be Sheffer for ( y ( t ) ,f ' ( t ) ) .Then for any polynomial p ( x ) ,
Theorem 6.2.4 The sequence s,(x) is Sheffer for ( g ( t ) ,f ( t ) )if and only if
for all constants y in c. Boas and Buck [l], in their work on polynomial expansions of analytic functions, define a sequence of polynomials to be of generalized Appell type if it has the above generating function, but without the factor l / c , on the right. Theorem 6.2.5 (The Conjugate Representation) The sequence s,(x) is Shefferfor ( y ( t ) , f ( t ) if) and only if
Theorem 6.2.6 The sequence s,(x) is Sheffer for ( g ( t ) , f ( t ) )if and only if y(t)s,(x) is the associated sequence for f ( t ) .
2.
165
THE PRINCIPAL RESULTS
Theorem 6.2.7 A sequence s,(.Y) is Sheffer for (y(r),f(t)), for some invertible g ( t ) , if and only if cn
f ( t ) s , ( u ) = -sn-l(x). Cn- I
Theorem 6.2.8 (The Sheffer Identity) A sequence s,(x) is Sheffer for ( g ( t ) , f ( t ) )for , some invertible g ( t ) , if and only if
for all constants y, where p , ( x ) is associated to f ( t ) . If p,(x) is the associated sequence for , f ( r ) in the cn-umbra1calculus, then the transfer operator for p , ( . ~ ) ,or for f ( r ) ,is the operator defined by
I.: xn + p , ( x ) . In the classical case, a transfer operator is an umbral operator. Sheffer operators are defined in an analogous manner. As before, the adjoint map is a bijection between the set of all transfer operators on P and the set of all automorphisms of .F(Theorem 3.4.2). The theory of umbral shifts takes on a slight complication in the nonclassical case. Let p,(x) be associated to f ( t ) . In order to preserve the characterization of umbral shifts given in Theorem 3.6.1, we must take the umbral shift Of,,,to be
Of course, in the classical case, for which cn = n ! , we get the classical umbral shift of Section 3.6. Thus 8 is the umbral shift for f ( t ) if and only if 8*is the derivation i;,,) on 3. Notice that
0,:.Y"
+
(n
+ l)c, xn
+
Cn+ I
is the analog of multiplication by 8,t.x" =
Y.
I t is interesting to observe that
Cn of.^"-
c"-
= nx",
1
and so 8,t = sD,
where D is the ordinary derivative. We still have a very useful recurrence formula.
166
6.
NONCLASSICAL U M B R A L CALCULI
Theorem 6.2.9 (The Recurrence Formula)
If s,,( u) is Sheffer for ( y ( t ) , f ' ( t ) ) ,
then
Finally, we mention the transfer formulas. Theorem 6.2.10
(iij p,(x)
If p,(.x) is associated to / ( t ) ,then
c',
= __
nc,-
("I")
or
-"
.xn-'
> 0).
(i7
I
3. A PARTICULAR DELTA SERIES AND THE CECENBAUER POLYKOMIALS
In this section we briefly discuss the delta series '1
.f(O =
-
t' - 1
This will lead us eventually to the Gegenbauer and Chebyshev polynomials. We begin by observing that
-
-
=
f(f)
f'(f
2t
1 +t"
~
j ==
f(r)
tJ1
.
-f2
Also, since
(I
+JG) z=2-2
x+2;-
(see, for examplc, Rainville [ I , p. 701j, we have .f'(fjk = ( - z j - k [ f k
+ J1k(+. 2;
-
.I - 1
I
1
>i. 1. k
rzj+k
(6.32)
3.
167
A PARTICULAR DELTA SERIES AND THE GEGENBAUER POLYNOMIALS
The associated sequence p,(x) for
J ' ( t ) has
the generating function
An explicit expression for p , ( x ) comes from the conjugate representation
But
b
n
-
k
odd,
and so
We shall concentrate mostly on recurrence formulas in this brief discussion, but let us give one example of the polynomial expansion theorem, namely,
which, from (6.3.2), is
Let us turn to some recurrence formulas. Theorem 6.2.7 implies that
which is equivalent to (6.3.3)
6.
168
NONCLASSICAL UMBRAL CALCULI
But also.
which, for n replaced by n
+ 1, is equivalent to
Equating these two expressions for J m p n ( x ) gives the recurrence ('n
p t P n +
1(x) + P
Cn+ I
Cn
t P n -
('n-
1(x) + 2pn(x) = 0.
(6.3.4)
1
We remark that (6.3.4) holds for any Sheffer sequence that uses f ( r )as its delta series. The recurrence formula for p,(x) is
Writing p,(.u) = (cn/cn+ ) f ( t ) p n + ,(x), substituting into the above equation, and replacing n 1 by n give
+
n p n ( x )= e,t J r 7 p n ( x ) .
Employing (6.3.3),we obtain
and recalling that Olr
=
.uD,we have
(.xD - n)p,,(x)
+ --.YDtp,('n
1 (x)
=
0.
(6.3.5)
(', 1-
Equations (6.3.4) and (6.3.5) can be used to derive an operator equation involving pn(.u)only. First, we observe that
r0, - flft = 1. Multiplying by t on the right and replacing O f t by x D , we get r x D - .uDr
=
t.
(6.3.6)
3.
A PARTICULAR DELTA SERIES AND THE GEGENBAUER POLYNOMIALS
169
(Of course, this could easily be verified directly.) Applying X D to (6.3.4),we get
+ -XDtp,-,(x) Cn
Cn
-xDtp,+,(x) Cn+ 1
+ 2 x D p , ( x ) =O.
C,-1
I ) x D t p n -I(.x) in (6.3.5) and substitute t o get
We then solve for (c,/c,-
+ ( x D + n)p,,(x) = 0.
Cn
--xDtp,,+
I(x)
Cn+ 1
If ( x D - n) is applied t o this, we obtain Cn
-xD(xD
- n ) t p n +I ( . ~ )
+ ( X D- n ) ( x D + n)p,(x) = 0,
Cn+ 1
and using (6.3.6),we have Cn
---xDt(xD
-
n - l)pn+I ( x )
+ ( x D - n ) ( x D + ~ ) P , ( X ) = 0.
Cn+ 1
+ 1, we get n ) ( x D + n)p,(x) = 0
Again, using (6.3.5),but with n replaced by n
+ (xD
-(xDt)’p,(x)
-
or [(XDt)’
-
(xD)’
+ n 2 ] p n ( x )= 0.
Finally, since (XDt)’
= X D t x D t = .YD(xDr
+ t ) t = x D ( x D + l)f’,
we have [xD(xD
+ l)t2
-
(XD)’
+ n 2 ] p n ( x )= 0.
Now let us consider the Sheffer sequence sF’(x) for the pair
In the language of Section 5.3 s F ’ ( x ) = (1
is a cross sequence.
+
.;Iyp.(.)
(6.3.7)
170
6. NONCLASSICAL UMBRAL CALCULI
It is easy to see that s ( f ( t )= ) (1
+ t2)',
and so the generating function for s(,")(.x) is
(6.3.8) To obtain the conjugate representation we observe that (y(f:(t))- l J ( t ) k1 x n > = (( -2)ktk(1
+
('n
=(-2)k7-((1
t2)-k-p
+
1 xn>
t2)-k-p(.xn-k)
cn-k
I
Hence
n- k
0,
odd.
According to (6.3.l), y(t).f.(t)k
= 2"(-t)k(1
+
Ji-7-v-k
and, after a few simple manipulations, we arrive at the expansion of x n in terms of sy(.Y),
Let us turn to recurrence formulas. We recall that (6.3.4)holds for st'(x), Cn
-t.s;;
I(.Y)
Cn+ I
The recurrence formula is
+ -t.s:1 Cn
Cn- I
l(x)
+
2Sf'(X)
= 0.
(6.3.9)
3.
171
A PARTICULAR DELTA SERIES AND THE GEGENBAUER POLYNOMIALS
Now
and
and so
=e
ttJT7sFj
,(XI
cn + 1 + -prskv)(x). fn
Replacing n
+ 1 by n and using (6.3.3),which holds for s;’”(x),
+
C
(xD - ~ ) s ? ) ( x ) “(xD
+ p)t$!
1 ( ~= )
we have
(6.3.10)
0.
cn- 1
Again, we can use (6.3.9)and (6.3.10) to obtain an operator equation for p ) to (6.3.9),
s:)(x). First, we apply (xD
‘ (Cx D
+
+ p ) t ~ r i + -(xD + p)ts!!’! cn
1 ( ~ )
L(x)
+ ~ ( x +D ~ ) s ? ) ( x=) 0.
Cn- 1
cn+1
Substituting into (6.3.lo), we have, after collecting like terms, c, -(xD en+ I
+ p ) t s W l(x) + ( x D + n + 2p)sp’(x)= 0.
Applying ( x D - n) to this and using (xD - n)t = t(xD - n C
‘(XD Cn+ I
+ p)t(xD - n
-
1)s$i
+ (xD
1 ( ~ )
-
n)(xD
-
I), we get
+ n + Zp)s?’(x) = 0.
172
6.
NONCLASSICAL UMBRAL CALCULI
+ 1, gives - ( x D - n)(xD + n + 2p)]sF'(x) = 0.
Using (6.3.10), with n replaced by n [((xD
+ p)t)'
Finally, from (6.3.6)we have [(xD
+ p ) ( x D + p + l ) t 2 - ( x D - n)(xD + n + 2p)]st'(x) = 0.
(6.3.1 1)
Let us now specialize the sequence c,. The case c, = n! is the classical one, and we shall not discuss it except to remind the reader of the example given in 4.12, at the end of Section 4 of Chapter 4. Let us take
where j . is not a negative integer. Then
and tx" =
n -i.-n+l
xn-
1
,
from which it follows that f =
-(IL
+ xD)-'D.
Actually, the motivation for choosing c, as we did is that
We consider first p,(x). The generating function is
= (1
+ tZ)"1
-
2xt
+ t2)-!
(The reader may begin to notice a Gegenbauer flavor here.) The conjugate representation is
Dividing (6.3.4) by c, gives
3.
173
A PARTICULAR DELTA SERIES AND THE GEGENBAUER POLYNOMIALS
Since
-(A
+ xD)t = D, we get (6.3.12)
which holds also for any Sheffer sequence using f ( t ) as its delta series. xD), Equation (6.3.5)is, after dividing by c, and applying -(A
+
+ I ) ( x D n)p,,(x)- n - 1 Finally, since D(J + xD)-' = (1+ 1 + .wD)-'D, Eq. (6.3.7) is [xD(xD + l)(xD + j L ) - ' ( x D+ 1 + 1)-'D2 - ( ~ 0+ n2]p,(x) ) ~ = 0. (;')(xD
-
Next, let us consider the sequence s!,'"(x) for the above choice of c,. The generating function is (6.3.8),
Thus if p
=
E., the polynomials
are the Gegenbauer polynomials (Erdelyi [ l , Vol. 2, p. 1771). Taking p = i= 1, we get the Chebyshev polynomials of the second kind. The Chebyshev polynomials of the first kind are Sheffer for a different g(t). The conjugate representation is
and for p = 2 we get
which is a well-known expression for the Gegenbauer polynomials (Erdelyi [1, Vol. 2, p. 1751). Our example of the polynomial expansion theorem is '"-"(p j= 1
I)(
-1 )p n - 2j
+ n - 2j st2 2j(x). j
Let us turn to the recurrence formulas. Equation (6.3.9) is
6.
174
When p
= j,,
NONCLASSICAL UMBRAL CALCULI
we get DCVJ 1
+
( ~ ~ )DCV!
l ( x )- 2 0 . + xD)C!,')(x)= 0,
which is also a well-known formula for the Gegenbauer polynomials (Erdelyi [l, Vol. 2, p. 1761). Equation (6.3.10)is
In the Gegenbauer case, since p
= 1,we
may apply ( x D + i - ' to get
+
(n - xD)C!,"(X) DC!,'! ' ( x ) = 0.
Finally, Eq. (6.3.I 1 ) is [(.yo
+ p ) ( x D + p + l)(xD + i . ) - ' D ( x D + E,)-'D
- ( x D - n)(xD
+ n - 2p)]s!,'"(x)= 0,
and since D ( x D + ).)-I = (xD+ 1. + l ) - ' D , we get [(xD
+ p ) ( x D + p + l ) ( x D + i ) - ' ( x D + 1 + 1)-'D2 - (XD n)(xD + n 2 p ) ] s f ' ( x )= 0. -
-
In the Gegenbauer case, when p = i., we get [ D 2 - ( x D - n)(xD + n
-
2i.)]C!,')(x)= 0
or, using (xD)' = x 2 D 2 + x D , [(I - .x2)DZ- ( 1
+ 2 j . p + n(n + 2 j b ) J C ! , a )=( ~0,)
which is the well-known differential equation for the Gegenbauer poIynomials. In this section we have shown that the Gegenbauer polynomials fit nicely into the umbral catalog. There are indeed other important polynomial sequences that share this property, but further discussion would be inappropriate here. For a short discussion of the Jacobi polynomials we refer the reader to Roman [ 6 ] . 4. THE q-UMBRAL CALCULUS
One of the most interesting of the nonclassical umbral calculi is defined by setting c, =
( 1 - q ) ( 1 - q2)-(1
(1
-
4)"
- 4") 3
4.
I75
THE 4-UMBRAL CALCULUS
where 4 # 1. This is the q-umbra1 culrulus. We have
and 1-4"
t x " = _ _ .xn-
I-q
' = x"
x
-
(qx)"
-
4x
7
and so
The operator t is known as the q-deriuariue, although some use this term for the operator (1 - 4 ) t . The q-binomial coeflcient is
-
( 1 - q)...(l - 4 " ) (1 - 4)...(1- q k ) ( l - 4)...(1- 4 n - k ) '
This is also known as the Gaussian coefficient and has important combinatorial significance. In Goldman and Rota [l] it is shown that when 4 is a prime power, (& is the number of vector subspaces of dimension k of a vector space of dimension n over the finite field G F [ q ] . We also have
and
It is very easy to see that
or E&t)
=(I
-
(I
-
q)yt)c,(t).
Recalling that o,:t" -+ ( c n / c n -, ) YI , we have
176
6.
NONCLASSICALUMBRAL CALCULI
The q-derivative operator f, and so also a,,satisfies a q-Leibniz formula
For an umbral proof of this see Roman [6]. As Andrews [ I ] has observed, the series e y ( t )can be expressed as an infinite product
that is valid for 141 < 1 and Irl < 1/(1 - 4 ) . Many of the combinatorial properties of the q-binomial coefficientscan be derived easily by umbral means. We observe first that
(;j
('n
q
=-
('k('n-k
1
= - ( E ~ ( t )I t k . x " ) ('k
From this we obtain some of the simplest properties of (&, for example,
and, by using the y-leibniz formula.
4. THE 9-UMBRAL
177
CALCULUS
Let us give a somewhat more delicate example. It is easy to see, by using the q-Leibniz formula, that atE,(t)
=Y q )
and a,&,(t)&;ft) =
( v+ z - ( 1 - 4)YZt)EY(t)EZ(t).
Then
I EY(t).Yn) = (cy(t)cz(t) I .Y") = (atE,(t)E,(t) 1 xn- ') = (EJt)
=((y
+2
-
(1
-
q)yzt)Ey(t)Ez(t)Ix"-')
'> k=O
Now if we take y
=z =
ykZn-2-k
4
1, then
is the total number of subspaces, of all dimensions, of a vector space of dimension rn over the field G F [ q ] . Denoting this by G,, we have the recurrence formula G, = 2G,_ 1 - (1 - q"-')G,-,. Taking y
=
-
1 and z
=
1, we obtain, for the alternating sum
the recurrence A, = ( 1 - q " - ' ) A , - , .
Combinatorial considerations aside, the q-umbra1 calculus, and other related umbra1 calculi, plays a definite role in the theory of basic hypergeometric series and hence in the theory of numbers. However, this is not the place to enter into a general discussion, and we refer the reader to Slater [I] and to Hardy and Wright [l] for the q-analog of the hypergeometric series.
178
6.
NONCLASSICAL UMBRAL CALCULI
We shall content ourselves with a brief discussion of two important Sheffer sequences, namely, the Appell sequences for ~ , . ( tand ) ~,,(t)Let us use the notation =
sn(x)
L ~ 1 y . n
for the Appell sequence for
d f )= q t ) . In order to determine
[ X ] ~ . ,we
first observe that
C ~ l y . n + 1 = (~y(t)l
CxIy.n+~>
= Cn+ldn+l.O = 0,
and so we may set CxIy,n+ 1
= (X - Y)rn(-xh
The q-Leibniz formula gives C k + 1 6n
I [xly.n+ 1 ) I (x - y ) r n ( x ) )
+ 1. k + I = ( E y ( t ) t k ' = (c,(r)tk+' =
(((7,
- Y)Er(t)fk+'
I rJ-4)
ck+l
=-<~y(qt)t'I
rn(x)),
Ck
and since c y ( q t ) = ~ ~ , ( twe ) , get (Eqy(t)tk
I rn(-y))
= ckdn,k.
Thus r n ( x ) = [x]~)..~ and 1
[x]y.n+
= (X - ~)[xIqy.n,
which leads to [.XlY."
= (x
-
y ) ( s - qy)...(x - q n - l y ) .
The generating function for [ x ] ~is. ~ Cxly.k k=O
'
t k =('1 A.
ck
E,(t)
Setting .x = 0 and observing that cOl,.k
=(-Y)
k
4 ($1
7
4. THE q-UMBRAL
179
CALCULUS
we have
This gives an explicit expression for [ x ] ~ , ~ ,
Incidentally, in view of the expression
=fjj=o
1 1 - qjzt'
which holds for 141 < 1 and I t ( < 1/(1 - q), the generating function becomes, after replacing t by t/(l - q),
Using the notation of basic hypergeometric series (Slater [l]), cn(1 - 4)" = (4;q)n
and
this becomes
=
,4oCY/x;q;xtl?
which is Heine's theorem (Slater [ l , p. 923). Since [x],, is Appell, we have tCxl,,,
cn
= --L-xly,.-l~ cn- 1
180
6.
NONCLASSICAL UMBRAL CALCULI
and since t is the q-derivative, we obtain
From the polynomial expansion theorem we get
The Appell sequence for g(t) = Ey(t)-
'
is
which is the q-analog of (x + y)". The polynomials H,,(x; 1) are known, however, as the q-Hermite polynomials. (One reason for this is that the polynomials H,,(x;y) have some algebraic properties that are analogous to those of the Hermite polynomials; for example,
for another reason see Allaway [2].) The generating function for the q-Hermite polynomial is
and if 141 < 1 and
It1 < 1/(1 - q),
fi
f_ H k t ; Y )_ (
1 j=O(l - q'yt)(l
_) k = ~
1 -q
k=O
-
q'xt)'
The fact that H J x ; y) is inverse to [x],,~under umbra1 composition can be expressed as .Y"
= k
f (") (-~')"~~q("Z~)H~(x;y), k =
~
4. THE q-UMBRAL
181
CALCULUS
Since H n ( x ;y ) is Appell, we have cn
tH,(x;y) = -Hn-l(x;y), cn-
1
which is equivalent to
Finally, from the polynomial expansion theorem we have
and so
We have scratched only the surface of the q-umbra1 calculus in briefly discussing these two Appell sequences. For a somewhat more detailed discussion along these lines we refer the reader to Roman [6, lo]. There have been several works on the subject of the q-umbra1 calculus in the past decade. Andrews [I] was the first to make a detailed study. However, his theory differs from the one we have just outlined. For example, the role of the Sheffer identity
is taken by the identity
182
6.
NONCLASSICAL UMBRAL CALCULI
Without going into any detail, it happens that Andrews' theory is actually the theory of Appell sequences in a whole class of umbral calculi, all of which are related to the q-umbra1 calculus. This has opened up a study of the relationship among the various related umbral calculi. For example, the umbral calculus defined by setting
or, more generally, 1
c, = _ _
(1 - q)-..(l
Caly.,
-
4")
(1 - 4)"
has strong ties to the q-umbra1 calculus. The study of these relationships has barely begun and promises to be quite fruitful (Roman [lo]).
REFERENCES
This list contains those references used in the text as well as a few others. It is intended only as a guide to further study of the umbral calculus and not as a bibliography on Sheffer sequences. Allaway, W. R. [I] Extensions of Sheffer polynomial sets, S I A M J . Math. Anal. 10 (1979), 38-48. [2] Some properties of the q-Hermite polynomials. Canad. J . Math. 32 (1980), 686-694. 131 A comparison of two umbral algebras, J . Math. Anal. A p p l . 85 (1982), 197-235. Allaway, W. R., and Yuen, K. W. [l] Ring isomorphisms for the family of Eulerian differential operators, J . Math. Anal. A p p l . 77 (1980). 245-263. Al-Salam, W. A. [I] q-Appell polynomials, Ann. Mar. Pura AppL 57 (1967). 31-46. Al-Salam, W. A., and Ismail, M. E. H. 111 Some operational formulas, J . Moth. Anal. Appl. 51 (1975),208-218. Al-Salam, W. A,, and Verma, A. [I] Generalized Sheffer polynomials, Duke Math. J . 37 (1970). 361-365. Andrews, G. E. [ I ] On the foundations of combinatorial theory V, Eulerian differential operators, Stud. A p p l . Math., 50 (1971),345-375. Bateman, H. [I] The polynomial of Mittag-Leffer, Proc. N o t . Acad. Sci. U.S.A. 26 (l940), 491-496. Bell, E. T. [I] Postulational bases for the umbral calculus, Amer. J . Math. 62 (1940),717-724. Boas, R. P., and Buck, R. C. [I] “Polynomial Expansions of Analytic Functions,” Academic Press, New York, 1964. Boole, G . [ I ] “Calculus of Finite Differences,” Chelsea, Bronx, New York, 1970. Brown, J. W. [ I ] A note on generalized Appell polynomials, Amer. Math. Monthly 75 (1968). [2] Generalized Appell connection sequences 11, J . Math. Anal. Appl. 50 (l975),458-464 183
184 [3] [4]
REFERENCES
On orthogonal sheffer sequences, Glas. M a t . Ser. / / I 10, 30 (1975), 63-67. Steffensen sequences satisfying a certain composition law. J . Math. Anal. Appl. 81 (1981), 48-62. Brown, J. W.. and Goldberg, J. L. [I] Generalized Appell connection sequences, J. Math. Anal. Appl. 46 (1974), 242-248. Brown, J. W., and Kuezma, M. [l] Self-inverse Sheffer sequences. S I A M J. Math. Anal. 7 (1976), 723-728. Brown, J. W., and Roman, S. [I] Inverse relations for certain Sheffer sequences, S I A M J . Math. Anal. 12 (1981), 186-195. Brown, R. B. [ I ] Sequences of functions of binomial type, Discrete Math. 6 (1973), 313-331. Burchnall, J. L. [I] The Bessel polynomials. Canad. J . Math. 3 (1951). 62-68. Carlitz, L. [ I ] Some polynomials related to theta functions, Ann. M a t . Pura Appl. (4)41(1955),359-373. [2] A note on the Bessel polynomials, Duke Math. J. 24 (1957). 151- 162. Chak, A. M. [ l ] An extension of a class of polynomials I, Rir. Mat. Unio. Parma (2) 12 (1971). 47-55. Chak, A. M.. and Agdrwal, A. K. [ I ] An extension of a class of polynomials 11, S I A M J. Math. Anal. 2 (1971). 352-355. Chak, A. M., and Srivastava, H. M. [ I ] An extension of a class of polynomials 111, Ric. M a t . Unio. Parma (3) 2 (1973). 11-18. Chihara, T. S. [ l ] “An Introduction to Orthogonal Polynomials,” Gordon & Breach, New York, 1978. Cigler, J. [ I ] Some remarks on Rota’s umbral calculus. Inday. Math. 40 (1978), 27-42. [Z] Operatormethoden fur q-ldentitaten, Monatsh. Math. 88 (1979). 87-105. [3] Operatormethoden fur q-Identitaten 11: y-Laguerre-Polynome, Monatsh. Math. 91 (1981). 105-117. [4] Operatormethoden fur q-Identitaten 111: Umbrale Inversion und die Lagrangesche Formal. Arch. Math. 35 (l980),533-543. Comtet. L. [ I ] “Advanced Combinatorics,” Reidel, Boston, 1974. Erdelyi, A. (ed.) [ I ] “Higher Transcendental Functions,” The Bateman Manuscript Project, Vols. 1-111, McGraw-Hill, New York, 1953. Fillmore, J. P.. and Williamson. S. G. [ I ] A linear algebra setting for the Mullin-Rota theory of polynomials of binomial type, Linear and Mulrilinear Alyehra 1 (1973).67-80. Freeman, J. M. [ I ] New solutions to the Rota/Mullin problem of connection constants, Proc. Southeastern Con/. Comhinatorics Graph Theory Cornput.. 9th. Bocu Raton, 1978, pp. 301 -305. Garsia, A. [ I ] An expose of the Mullin-Rota theory of polynomials of binomial type, Linear and Multilinear Alyehra 1 (1973).47-65. Garsia, A,, and Joni, S. A. [ I ] A new expression for umbral operators and power series inversion, Proc. Amer. Math. Soc. 64 (1977). 179- 185.
REFERENCES
185
Goldman, J.. and Rota, G.-C. [I] The number of subspaces of a vector space, in "Recent Progress in Combinatorics" (W. T. Tutte, ed.), Academic Press, New York, 1969. [2] O n the foundations of combinatorial theory IV: Finite vector spaces and Eulerian generating functions, Stud. Appl. Math. 49, (1970). 239-258. Could, H. W. [I] A series transformation for finding convolution identities, Duke Math. J . 28 (l96l), 193202. [Z] A new convolution formula and some new orthogonal relations for inversion of series, Duke Math. J . 29 (1962), 393-404. Guinand, A. [ l ] The umbral method: A survey of elementary mnemonic and manipulative uses, Amer. Math. Monthly 86 (1979). 187-195. Hardy. G. H.. and Wright, E. M. [l] "An Introduction t o the Theory of Numbers," Oxford Univ. Press, London and New York, 1979. Henrici, P. [I] An algebraic proof of the Lagrange-Burmann formula, J . Math. Anal. Appl. 8 (1964), 21 8-224. Ihrig, E. C., and Ismail, M. E. H. [ I ] O n an umbral calculus, Proc. Southeastern Con/. Combinatorics. Graph Theory Comput., loth, Boca Raton, 1979, pp. 523-528. 121 A q-umbra1 calculus, J . Math. Anal. Appl. 84 (1981). 178-207. Ismail, M. E. H. [I] Polynomials of binomial type and approximation theory, J . Approx. Theory 23 (1978), 177- 186. Joni, S . A. [I] Lagrange inversion in higher dimensions and umbral operators, Linear and Multilinear Algebra 6 (1978). 11 1-121. 121 Expansion formulas 1: A general method. J . Math. Anal. Appl. 81 (1981), 364-377. [3] Expansion formulas 11: Variations on a theme, J . Math. Anal. Appl. 82 (1981). 1-1 3. Joni, S. A,, and Rota, G.-C. [I] "Umbra1 Calculus and Hopf Algebras," Amer. Math. Soc., Providence, Rhode Island, 1978. Jordan, C . [ I ] "Calculus of Finite Differences," Chelsea, Bronx, New York, 1965. Krall, H. L., and Frink, 0. [I] A new class of orthogonal polynomials: The Bessel polynomials, Trans. Amer. Math. Soc. 65 (1948). 100- 1 15. Krouse, D., and Olive, G. [I] Binomial functions with the Stirling property, J . Math. Anal. Appl. 83 (1981). 110126. Labelle, G . [ I ] Sur I'inversion et I'iteration continue de series formelles, European J . Combin. I (2),(1980), 1 13- 138. Meixner, J. [ 13 Orthogonale Polynomsystem rnit linern besonderen Gestalt der eryengenden Funktion, J . London Math. Soc. 9 (1934). 6- 13.
186
REFERENCES
Milne-Thomson, L. M. [ I ] “The Calculus of Finite Differences,” Chelsea. Bronx, New York, 1960. Morris, R. A. [ I ] Frobenius endomorphisms in the umbral calculus, Stud. Appl. Math. 62 (1980), 85-92. Mullin, R., and Rota. G.-C. [I] On the foundations of combinatorial theory 111: Theory of binomial enumerations, in “Graph Theory and Its Applications” (B. Harris, ed.), Academic Press, New York, 1970. Niederhausen, H. [I] Sheffer polynomials for computing exact Kolmogorov-Smirnov and Renyi type distributions, Tech. Rep. No. 6, M.I.T., Cambridge, Massachusetts, 1979. Niven, 1. [ I ] Formal power series, Amer. Math. Monthly 76 (1969). 871-889. Olive, G. [ I ] The b-transform, J . Math. Anal. Appl. 60 (1977). 755-778. [2] Binomial functions and combinatorial mathematics, J. Math. Anal. Appl. 70 (1979),460473. [3] Acombinatorial approach to generalized powers,./. M a t h . Anal. Appl. 74(1980),270-285. Rainville, E. [ I ] ”Special functions,” Chelsea, Bronx, New York, 1960. Reiner, D. L. [l] Sequences of polynomials of fractional binomial type. Linear and Multilinear Algebra 5 (1977). 175-179. [2] The combinatorics of polynomial sequences, Stud. Appl. Math. 58 (1978), 95-1 17. [3] Eulerian binomial type revisited. Stud. Appl. Math. 64 (1981). 89-93. Riordan, J. [ I ] Inverse relations and combinatorial identities, Amer. Math. Monthly 71 (1964), 485-498. [2] “Combinatorial Identities,” Wiley, New York, 1968. [3] “An Introduction to Combinatorial Analysis.” Princeton Univ. Press, Princeton, New Jersey, 1980. Roman,, s. [I1 The algebra of formal series, Adu. in Math. 31 (1979), 309-339; Erratum 35 (1980),274. P I The algebra of formal series 11: Sheffer sequences, J. Math. Anal. Appl. 74 (1980). 120143. ~ 3 1The algebra of formal series 111: Several variables, J. Appro-x. Theory 26 (l979), 340-38 I. [41 The formula of Faa di Bruno, Amer. Math. Monthly 87 (1980). 805-809. ~ 5 1 Polynomials. power series and interpolation, J . Math. Anal. Appl. 80 (1981). 333-371, The theory of the umbral calculus 1, J. Math. Anal. Appl. 87 (1982), 58-1 15. The theory of the umbral calculus 11, J . Math. Anal. Appl. 89 (1982), 290-314. The theory of the umbral calculus 111, J. Math. Anal. Appl., to appear. Operational formulas. Linear and Multilinear Algebra 12 (1982). 1-20. More on the umbral calculus, with emphasis on the q-umbra1 calculus, J . Math. Anal. Appl.. to appear. Roman, S.. and Rota. G.-C. [ I ] The umbra1 calculus, Adc. in Math. 27 (1978), 95-188. Rota, G.-C. [l] “Finite Operator Calculus,” Academic Press. New York, 1975 Rota, G.-C.. Kahaner, D.. and Odlyzko, A [ 13 On the foundations of combinatorial theory VIII: Finite operator calculus,J. Math. Anal. Appl. 42 (1973),684-760.
REFERENCES
187
Sheffer. 1. M. [I] Some properties of polynomial sets of type zero. Duke Math. J . 5 (1939). 590-622. . 51 (1945). 739-744. [2] Note on Appell polynomials. Bull. Amer. M ~ i hSOC. Shohat, J . [I] The relation of the classical orthogonal polynomials to the polynomials of Appell. Amer. J . Math. 58 (1936).453-464. Slater. L. J. [ 11 "Generalized Hypergeometric Functions," Cambridge Univ. Press, London and New York. 1966. Steffensen, J. F. [I] The poweroid, an extension of the mathematical notion of power, Acta Math. 73 (l941), 333-336. [2] "Interpolation," Chelsea, Bronx. New York. 1950. Szego, G . [I] "Orthogonal Polynomials," Amer. Math. SOC..Providence, Rhode Island, 1978. Ward. M. [ I ] A certain class of polynomials, Anti. of Mnrh.31 (1930),43-51. 121 A calculus of sequences, Amer. J . Marh. 58 (1936),255-266. Whittaker, E. T., and Watson. G. N. [I] "A Course of Modern Analysis," Cambridge Univ. Press, London and New York, 1978. Yang, K.-W. [ I ] Integration in the umbral calculus, J . M o t h . Anal. Appl. 74 (1980). 200-21 I. Zeilberger, D. [ I ] Some comments on Rota's umbral calculus. J . Math. Anal. A p p l . 74 (1980). 456-463.
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INDEX
A
Abel function, 1 1, 72. 85 Abel operator, 14 Abel polynomials. 72-75 binomial identity, 73 expansion theorem. 74 generating function. 72 inverse under umbral composition, 75 transfer formula. 72 Actuarial polynomials, 123- 125. 140 generating function. 123 recurrence relation, I24 Sheffer identity, 123 umbral composition. 125 Adjoint. of linear operator on P, 34 Appell cross sequence, 140 Appell identity. 27. see also specific polynomial sequence Appell sequence, 17,26-28. 164 Appell identity. 27 conjugate representation, 27 expansion of x.T,,(x),27 expansion theorem, 26 generating function, 27 multiplication theorem, 27 polynomial expansion theorem. 26 Associated sequence(s), 17. 25-26. 48. 50-51. 164 action of operator h(f) on. 26 binomial identity, 26 conjugate representation. 25 expansion ofp;(x), 26 expansion of xpn(x),26 generating function. 25
operator characterization of, 25 polynomial expansion theorem, 25 recurrence formulas, 48 relationship between, 5 I transfer formulas, 50 Automorphisms of P*. 33-36.38
B Backward difference functional, 63 Bell polynomials, 82-86 connection with Abel function. 85 exponential polynomials, 85 idempotent numbers, 85 Laguerre polynomials. 86 Lah numbers, 86 Stirling numbers. 85 Bernoulli numbers, 12, 94, 98- 100 connection with harmonic series. 99- 100 Stirling numbers, 99 Bernoulli numbers of second kind. 1 14 Bernoulli polynomials, 12, 93-100, 105. 117-118. 129, 135, 140. 151 Appell identity, 94 complementary argument theorem. 100 connection with Bernoulli polynomials of second kind. 117. 1 I8 Euler polynomials, 105, 135 lower factorial polynomials, 93 Stirling polynomials. 129
189
190
INDEX
Euler-Maclaurin expansion. 97 expansion theorem, 96 generating function, 94 multinomial theorem, 97 polynomial expansion theorem, 93 recurrence formula, 95 transfer formula, 96, 100 Bernoulli polynomials of second kind, 113-119 connection with Bernoulli polynomials, 117-118 expansion theorem, 116 generating function, I 16 Gregory’s formula. 1 17 polynomial expansion theorem. I 17 Sheffer identity. I16 symmetry of, I 19 Bessel polynomials, 78-82 binomial identity. 79 generating function, 79 polynomial expansion theorem, 80 recurrence formula, 80 transfer formula, 78-79 Binomial convolution, 160- 16 1 Binomial identity, 26. w e also Sheffer identity Binomial type, sequence of. 26 Boole polynomials, I27 Boole’s summation formula, 104
C
c; 3 Central difference series. 68. scc ulso Could polynomials Chebyshev polynomials. 162. 166, I73 Complementary argument theorem for Bernoulli polynomials, 100 for Euler polynomials, 106 Conjugate representation, s w a l . specific ~ polynomial sequence of Appell sequences. 27 of associated sequences. 25 of Sheffer sequences. 20 Connection constants problem. I3 I - I38 Cross sequences. I40
D Delta functional. 10 Delta operator. I3
Delta series, 4 generic. 82 Derivations on P*.34 chain rule for, 46 Dobinski‘s formula, 66 Duplication formulas, 132, I37 - I38 for Hermite polynomials, 137 for Laguerre polynomials, I37 for Mittag-Leffler polynomials. 138
E Euler numbers, I02 Euler polynomials, 12. 100- 106. 135-136, 161 Appell identity, 102 Boole’s summation formula, 104 complementary argument theorem, 106 connection with Bernoulli polynomials. 105, 135-136 expansion theorem, 104 generating function, 102 multiplication theorem, 105 Newton’s expansion of, 10I recurrence formula, 103 Evaluation functional. 7, I I , 163 Expansion theorem for associated sequences, 25 for Appell sequences, 26 for Sheffer sequences, 18. I64 Exponential polynomials. 63 -67. 85 binomial identity. 64 connection with actuarial polynomials. 123 Laguerre polynomials, 134 Poisson -Charher polynomials, I22 Dobinski’s formula. 66 expansion theorem, 65 generating function. 64 recurrence formula, 64
F
I.; 3. 7. 12 Falling factorial polynomials. s w Lower factorial polynomials Fibonacci numbers, 149 Formal power series. 3-4 Forward difference functional. i I . 56 Forward difference operator. I4
191
INDEX
C Gaussian coefficient, see q-Umbra1 calculus Gegenbauer polynomials, 162, 166, 173 Generating function. see also specific polynomial sequence for Appell sequences, 27 for associated sequences. 25 for Sheffer sequences, 18- 19 Generalized Appell type, 19, 164 Generic associated sequence, 82 - 84, s w also Bell polynomials binomial identity, 83 conjugate representation, 82 generating function, 83 recurrence formula, 83-84 Gould polynomials, 67-72, 149 binomial identity, 69 expansion theorem, 70 generating function, 68 recurrence formula, 69 Stirling’s formula, 70 Vandermonde’s convolution formula, 69 Gregory’s formula, 59, I I7
H Heine’s Theorem, see q-Umbra1 calculus Hermite polynomials, 87-93, 135, 137, 140, 150, 158 Appell identity, 88 classical Rodrigues formula, 89 duplication formula, 137 generating function, 88 multiplication theorem, 93 polynomial expansion theorem, 89 -90 q-Hermite polynomials, see q-Umbra1 calculus recurrence formula, 89 recurrence relation for, I58 squared Hermite polynomials, I28 umbral composition of, 92 Weierstrass operator, 88
I Idempotent numbers, 85 Integration by parts, 1 18 Inverse relations, 147- 156 Invertible functional, 10 Invertible operator, I3
K Krawtchouk polynomials, 126
L Lagrange inversion formula, 138- 140 Laguerre polynomials, 1 I, 108-1 13. 120, 133-134, 137. 140, 151, 158-159 classical Rodrigues formula, 109 connection with exponential polynomials, I34 Poisson -Charher polynomials, I20 duplication formula, I37 generating function, 1 10 Lah numbers, 109 polynomial expansion theorem, 1 12 recurrence formula, 110 recurrence relation for, 158- 159 second order differential equation for, 110-1 I 1 Sheffer identity, I10 transfer formula, 109 umbral composition of, I I2 - I I3 Lah numbers, 86, 109 Linear operator, continuous. 33 Lower factorial polynomials. 56-63 binomial identity, 57 conjugate representation, 6 1 -62 expansion theorem, 58 generating function, 57 Gregory’s formula, 59 Newton’s forward difference interpolation formula. 58 polynomial expansion theorem, 59 recurrence formula, 56, 58 Vandermonde convolution formula, 57
M Mahler polynomials, 129- I30 Meixner polynomials of first kind, 125- I26 Meixner polynomials of second kind, 126 Mittag-Leffler polynomials, 75-77, 138 binomial identity, 76 duplication formula, I38 expansion theorem, 77 generating function, 76 recurrence formula, 77 Mott polynomials, I30
192
INDEX
Multiplication theorem, 27, 93. 97-98. 105 for Bernoulli polynomials, 97-98 for Euler polynomials, 105 for Hermite polynomials, 93 N
Narumi polynomials, I27 Newton’s forward difference polynomials, 58
0 Operational formulas, I44 - I47 Orthogonal polynomials, three term recurrence relation for, 156, 159- I60
P
P. 6 Peters polynomials, I28 Pidduck polynomials, 126- 127 Pincherele derivative. 49 Poisson-Charlier polynomials. 119- 123, 136, 140 connection with Laguerre polynomials. 120
generating function. 120 recurrence formula. 12 I Sheffer identity. I2 I umbral composition of, I22 Poisson distribution. I19 Polynomial expansion theorem, see also specific polynomial sequence for Appell sequences, 26 for associated sequences, 25 for Sheffer sequences, 18. 164 Poweroid. 19
Q q-Binomial coefficient. see q-Umbra1 calculus q-Derivative, s w q-Umbra1 calculus q-Hermite polynomials. see q-Umbra1 calculus q-Leibniz formula. see q-Umbra1 calculus q-Umbra1 calculus, 174- 182 Gaussian coefficients, I75 Heine’s theorem, 179 q-binomial coefficients, 175- 177 qderivative, I75
q-Hermite polynomials, 180- I8 I generating function, 180 inverse under umbral composition, 180 polynomial expansion theorem, 18I q-Leibniz formula, 176
R Recurrence formulas for associated sequences, 48 for Sheffer sequences, 50, I66 Rising factorial polynomials, 63
S Sheffer identity, 2 I , see ulso specific polynomial sequences Sheffer operators, 42 - 44 adjoint of, 42 group of, 43 Sheffer sequence. 17-24. 164- 165 action of operator of form h ( ~on, ) 2I characterization by Sheffer identity, 2 I , 165 conjugate representation of, I9 - 20, I64 expansion of sA(x), 24 expansion of xsn(x), 24 expansion theorem, 18, 164 generating function, 18- 19. 164 operator characterization of, 20, 165 performance with respect to multiplication in umbral algebra, 23 polynomial expansion theory, 18. 164 recurrence formulas, 50 Sheffer shift, 49 - 50 Signless Stirling numbers of first kind, 63 Squared Hermite polynomials. 128 Steffensen sequences, 140- 143 Stirling’s formula, 70 Stirling numbers of first kind, 57-59. 61 -62,66-67, 85,99, 115, 129, 134 connection with Stirling polynomials, 129 signless, 63 Stirling numbers of second kind. 59-60, 62-64.66-67.85-99. 129, 144 connection with Stirling polynomials, 129 Stirling polynomials, 128- 129 connection with Bernoulli polynomials, 129 Stirling numbers, 129 generating function, 128
INDEX
193 T
Transfer formulas, 50, 166 Transfer operator, I65 Translation operator. I4 U Umbral algebra. 7 Umbral composition, 4 1 Umbral operator, 37-42, see also Transfer operator
adjoint of, 38 group of. 40 Umbral shift(s).45-49, 165 adjoint of, 46 formula for. 48 relationship between, 47 V
Vanderrnonde convolution formula. 57. 69
Pure and Applied Mathematics A Series of Monographs and Textbooks Editors
Samuel Eilenberg and Hyman Bass Columbia University, New Y o r k
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