Obtaining Generating Functions

108 Pages·1971·1.968 MB·English

by  Elna Browning McBride (auth.)

Springer Tracts in Natural Philosophy
Volume 21
Edited by B. D. Coleman
Co-Editors:
R. Aris . L. Collatz . J. L. Ericksen' P. Germairi
M. E. Gurtin . E. Sternberg . C. Truesdell
Bina B. McBride
Obtaining
Generating Functions
Springer-Verlag New York Heidelberg Berlin 1971
Elna Browning McBride
Professor of Mathematics
Memphis State University
Memphis, Tennessee 38111
AMS Subject Classifications (1970):
33-00, 33-02
ISBN-13: 978-3-642-87684-4 e-ISBN-13: 978-3-642-87682-0
DOl: 10.1007/978-3-642-87682-0
This work is subject to copyright. All rights are reserved, whether the whole or part of the
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by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1971. Library of Congress Catalog Card Num-
ber 72-138811. . Typesetting, printing, and binding: Universitats
Softcover reprint of the hardcover lst edition 1971
Preface
This book is an introduction to the study of methods of obtaining
generating functions. It is an expository work at the level of the beginning
graduate student.
The first part of Chapter I gives the reader the necessary definitions
and basic concepts. The fundamental method of direct summation is
explained and illustrated.
The second part of Chapter I deals with the methods developed by
Rainville. These methods are based principally on inventive manipulation
of power series.
Weisner's group-theoretic method is explained in detail in Chapter II
and is further illustrated in Chapter III. When this method is applicable,
it yields a set of at least three generating functions. In Chapter II for the
Laguerre polynomials six generating functions were found.
Truesdell's mane thod is studied in Chapter IV. For a given set of
functions {fez, the success of this method depends on the existence
of certain transformations. If fez, a) can be transformed into F(z, a)
such that
a
a-;
F(z, a)=F(z, a+ 1),
or if fez, a) can be transformed into G(z, a) such that
a
a-;
G(z, a)=G(z, a-I),
then from each transformed function a generating function can be
obtained. Truesdell's method for obtaining the transformed functions
does not require any ingenuity on the user's part. Truesdell has shown
how these simple results may be exploited to generate more complicated
results by means of specified, systematic, and general processes. His
method of obtaining generating functions is only one of these results.
Although the principal objective of this exposition is to bring to the
reader's attention the methods developed by Rainville, Truesdell, and
VI Preface
Weisner, there are other methods in the literature which deserve con
sideration. Some of these are presented in Chapter V.
The author is especially grateful that Professor Truesdell, Professor
Weisner, and the late Professor Rainville read and made valuable
suggestions concerning the parts of the original manuscript dealing with
the method developed by each of them.
Contents
Chapter I. Series Manipulation Methods . 1
First Part: Underlying Ideas 1
1. Introduction. . . . . 1
2. The factorial function and the generalized hypergeometric
functions ..... . 7
3. Obtaining generating functions from expansions in powers of x 10
Second Part: Rainville's Methods. 13
4. Using an auxiliary variable 13
5. A bilinear generating function 15
6. Bilateral generating functions 18
7. Summary of results .... 22
Chapter II. The Weisner Method 25
1. Introduction. . . . . . . 25
2. The differential equation 25
3. Linear differential operators 26
4. Group of operators. . . . 29
5. The extended form of the group generated by Band C 30
6. Generating functions . . . . . . . . . . . . 34
7. Summary ................ . 42
Chapter III. Further Results by the Weisner Method 43
1. Introduction. . . . . . . . . . . 43
2. The modified Laguerre polynomials . 43
3. The simple Bessel polynomials 47
4. The Gegenbauer polynomials 50
Chapter IV. The Truesdell Method 57
1. Introduction. . . . . . . . 57
2. The ascending equation . . . 57
3. The Hermite polynomials {Ha+nex)} . 62
4. The descending equation . . . . . 64
5. The Hermite polynomials {Ha-n(x)} . 67
6. The Charlier polynomials . . . . . 68
VIII Contents
Chapter V. Miscellaneous Methods . 72
1. Introduction. . . . . . . . . 72
2. Classes of generating functions . 72
3. Natural pairs of generating functions 77
4. Generating functions in differentiated form or in integrated
form . . . . . . . . . . . . . . . . . . . . . . 81
5. Generating functions related by the Laplace transform 88
6. The contour integral method. 91
7. Recent developments 96
Bibliography . 97
Index. . . . 99
Chapter I
Series Manipulation Methods
First Part. Underlying Ideas
1. Introduction. The purpose of this study is to describe and to make
illustrative use of some effective methods for obtaining generating func
tions. We define a generating function for a set of functions {f,,(x)} as
follows: Let G (x, t) be a function that can be expanded in powers of t
such that
00
G(x, t)= ~>n f,,(x) tn,
n=O
where Cn is a function of n that may contain the parameters of the set
{f,,(x)}, but is independent of x and t. Then G(x, t) is called a generating
function of the set {f,,(x)}.
To illustrate we generate the set offunctions {l, x, x2, ... , xn, ••• }. We
know that
1
00 00
exp{xt}= L(xtt!n!= L -, xntn.
n.
n=O n=O
Then corresponding to the notation in our definition of a generating
function we have
G(x, t)=exp {x t}, Cn= lin!, and fn(x)=xn.
By the above definition a set of functions may have more than one
generating function. However, if
00
G(x, t)= L hn(x) tn
n=O
then G(x, t) is the unique generator for the set {hn(x)} as the coefficient
set. F or example, the set of functions {xn} is generated as a coefficient set
only by (l-x t)-l.
2 Chapter 1. Series Manipulation Methods
We use the symbol {fn(x)} to indicate the infinite set {fo(x), fl(X),
f2 (x), ... ,fn(x), ... }. If fn(x) is also defined for negative integral n, we
would like to find a function H(x, t) having a Laurent series expansion
of the form
co
n= - co
Presently, we will extend our definition of generating function to include
functions whose expansions are Laurent series.
We define a formal power series as one for which the radius of
convergence is not necessarily greater than zero. When a function H(x, t)
has a power series expansion in t, then H(x, t) determines the coefficient
set {hn(x)} even if the series is divergent for t=l=O. The relation between
the generating function and the coefficient set is a qualitative relation
whose validity does not depend on the length of the radius of convergence.
In 1923 Eric T. Bell [1] presented a paper in which he established the
validity of" results obtained by equating coefficients after formal manip
ulation of series". (Also see Bell [2] and [3].) Accordingly, we do not
consider it necessary to determine the radius of convergence for the
power series representation of each generating function. However, if the
generating function has a power series expansion which is obviously
divergent for t =1= 0, we will use the following notation to indicate diver-
gence:
co
H(x, t)~ L hn(x) tn.
n=O
We now extend our definition of a generating function to include
functions with a Laurent series expansion, functions whose expansions
have a zero radius of convergence, and finally functions which generate
functions of more than one variable. (See Erdelyi [3; p.228].) Let
G(XI' x2, ... , xp; t) be a function of p+l variables. Suppose
has a formal expansion in powers of t such that
co
G(XI,X2, ... ,xp; t)= L cnfn(xl,x2, ... ,xp)tn
n= - co
where cn is independent of the variables Xl' x2, ••• , xp' and t. Then
we shall say that G(XI' X2, •.• , xp; t) is a generating function for the
fn(xI,x2, ••• ,xp) corresponding to nonzero cn' In particular, if
co
G(x,y;t)= Lcnfn(x)gn(y)tn,
n=O
1. Introduction 3
the expansion determines the set of constants {cn} and the two sets of
functions Un(x)} and {gn(Y)}. Then G(x, Y; t) is to be considered as a
generator of anyone of these three sets and as the unique generator of
the coefficient set {cn j,,(x) gn(Y)}.
A generating function may be used to define a set of functions, to
determine a differential recurrence relation or a pure recurrence relation,
to evaluate certain integrals, etc. We will use generating functions to
define the following special ·functions: the Bessel functions and the
polynomials of Legendre, Gegenbauer, Hermite, and Laguerre.
The Legendre polynomials {P.(x)} were introduced by Legendre [1]
in 1785. He defined them by means of the generating relation:
00
(1-2xt+t2)-t= LP'(x)tn. (1)
n=O
In 1874 Gegenbauer [1; pp.6-16] generalized the Legendre poly
nomials and used the notation {C~(x)} for the set which satisfies the
generating relation:
00
(1-2xt+t2)-v= L C~(x)tn. (2)
n=O
These polynomials are now called the Gegenbauer polynomials. In this
book we adopt Legendre's definition of P'(x) and Gegenbauer's definition
of C~(x).
We define the Hermite polynomials {Hn(x)} by means of the generat
ing relation
(3)
When Hermite [1; p.294] introduced these polynomials in 1864, he
used the symbol Un and defined Un by what we call a Rodrigues-type
relation:
The functions Hn(x) and Un differ only in sign, i.e., Hn(x)=(-ltUn.
However, the twentieth century notation is not uniform. Magnus and
Oberhettinger [1; p.80] use a different symbol, Hen(x), to warn us
of a different definition:
1
00
exp{xt-t2/2}= L -Hen(x) tn. (4)
n=O n!
xV2,
If in (4) we replace t by q/i and x by we get
(5)