Theory and Application of Special Functions. Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, the University of Wisconsin–Madison, March 31–April 2, 1975

562 Pages·1975·16.73 MB·English

by  Richard A. Askey

Theory and Application
of Special Functions
Edited by Richard A. Askey
Proceedings of an Advanced Seminar
Sponsored by the Mathematics Research Center
The University of Wisconsin-Madison
March 31-April 2, 1975
Academic Press
New York · San Francisco · London 1975
A Subsidiary of Harcourt Brace Jovanovich, Publishers
COPYRIGHT © 1975, BY ACADEMIC PRESS, INC.
ALL RIGHTS RESERVED.
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Library of Congress Cataloging in Publication Data
Advanced Seminar on Special Functions, Madison, Wis.,
1975.
Theory and application of special functions.
(Publication of the Mathematics Research Center, the
University of Wisconsin ; no. 35)
Bibliography: p.
Includes index.
1. Functions, Special-Congresses. I. Askey,
Richard. II. Wisconsin. University-Madison. Mathe-
matics Research Center. III. Title. IV. Series:
Wisconsin. University-Madison. Mathematics Research
Center. Publication ; 35.
QA3.U45 no. 35 [QA351] 510'.8s [515\5] 75-30406
ISBN 0-12-064850-4
PRINTED IN THE UNITED STATES OF AMERICA
Foreword
This volume contains the Proceedings of the Advanced Seminar on Special
Functions held in Madison, Wisconsin, March 31-April 2, 1975, sponsored by
the Mathematics Research Center, University of Wisconsin-Madison,with finan-
cial support from the National Science Foundation under Grant MP575-05571
and the United States Army under Contract No. DA-31-124-ARO-D^62. The
five sessions were chaired by:
B. C. Carlson, Iowa State University
Yudell Luke, University of Missouri, Kansas City
A. Erdélyi, University of Edinburgh
Richard Askey, University of Wisconsin, Madison
W. J. Cody, Jr., Argonne National Laboratory
The program committee consisted of Professors L. Durand, J. Hirschfelder,
and F. W. J. Olver, with the editor as chairman. Mrs. Gladys Moran handled all
the organizational details in her usual efficient and pleasant way, yet even she
could not make the beautiful weather of Monday last until Wednesday, when it
was replaced by a spring snow storm which will be remembered by all those
who spent an extra day in Madison or in a motel in the vicinity of Chicago.
Mrs. Doris Whitmore did the typing of the manuscripts, and her skill will be
evident to the reader.
VII
Preface
Twenty years ago it was widely believed that the existence of large, fast,
computing machines spelled the end of the study and use of special functions.
Differential equations would be solved numerically, integrals would be evalu-
ated numerically, and special functions would become a fossil. Many mathema-
ticians acted on this belief, and special functions disappeared from their tradi-
tional place in the curriculum as part of a course in complex variables. They
were often replaced in the mathematical physics course by a section on Hubert
space and functional analysis.
To paraphrase Mark Twain, reports of the death of special functions were
premature. A sociologist of science would accept the following two facts as
proof. Over 150,000 copies of the Handbook of Mathematical Functions have
been,sold. Two of the five most widely cited mathematics books (as measured
by the citations in Science Citation Index) are this book and Higher Transcen-
dental Functions by A. Erdélyi et al. Most of us are more concerned with
quality than quantity, and so would not be convinced by these two facts. A
more compelling proof is given in this book. Some interesting results, conjec-
tures, and problems are given. In addition there are surprising connections with
other fields.
J. J. Seidel told me about Delsarte's work on coding theory and Krawtchouk
polynomials. My initial reaction was: What is coding theory? I still do not
really understand it, but Sloane's article has given me some insight into a
fascinating set of problems. These and related combinatorial problems will lead
to important results on orthogonal polynomials in the next few years. Dunkl
has used a finite group and the associated homogeneous space to find a deep
addition formula for Krawtchouk polynomials, and in this formula there is a
new set of orthogonal polynomials in two variables which is analogous to the
disc polynomials described in Koornwinder's paper. Delsarte has made a start
on the problem of finding an analogous space on which some of the q-
orthogonal polynomials live. While some very deep results have been proved for
q(ox basic)-hypergeometric functions (see Andrews' paper) the results are not
the type that come from considering orthogonal polynomials as spherical
harmonics.
IX
PREFACE
There are at least three ways to look at the classical orthogonal polynomials.
One is to consider them as a special case of general orthogonal polynomials.
From this point of view they can suggest interesting questions about general
orthogonal polynomials. Case's paper can be looked at as showing that from
the point of view of recurrence relations, general orthogonal polynomials look
surprisingly like the classical polynomials. To be more precise, consider the
Pollaczek polynomials which are described in the appendix of Orthogonal Poly-
nomials by G. Szegö. If the Pollaczek polynomials which are orthogonal on
(-1,1) are written in monic from (highest coefficient is one), then their recur-
rence relation takes the form xp(x) = p ](x) + a(n)p (x) + b(n) P .\ (x\
n n+ n n
where a(n) and b(n) are certain rational functions satisfying a(n) = An~l +
0{n"2), bn = lA +Bn~x + 0(n~2). These Pollaczek polynomials are Jacobi poly-
nomials if and only if A = B = 0 , and it is exactly in this case when the weight
function with respect to which they are orthogonal satisfies t\ !og 2\i dx ^
oo. Case shows that for an arbitrary a(n) = 0(n~2) and positive b(n)=xA +
0(n"2), a similar conclusion is true.
A second way of looking at the classical orthogonal polynomials is to con-
sider them as hypergeometric functions. Much of the work in Gasper's paper is
of this sort. Many of the raising and lowering operators which played a prom-
inent part in Miller's early work came from the differential-difference equations
satisfied by hypergeometric functions. The limits of generality of this method
have not been reached, especially when multiple series are considered, and the
degree of complexity of some of the transformation formulas is almost over-
whelming, yet some of these very complicated formulas are exactly what is
needed to solve some interesting problems. Not only do these methods give
results for the classical polynomials of Hermite, Laguerre, and Jacobi, which
satisfy second order differential equations in x, but also for the classical dis-
crete polynomials of Charlier, Krawtchouk, Meixner, and Hahn, which satisfy
second order additive difference equations in χ and a class of ^polynomials
studied by Hahn, which satisfy second order multiplicative difference equations
in x But the deeper formulas associated with the addition formlas do not arise
in a natural way in this approach.
The third way to treat the classical orthogonal polynomials is to consider
them as spherical functions on a homogeneous space. This is a very powerful
method as is seen by the ease with which very deep addition formulas are
obtained. However to get started one must find the right space on which the
functions live, and this has not been done for all of these functions. There is
much work to be done here, especially for the additive and multiplicative
difference type orthogonal polynomials.
One striking fact about the talks at this meeting was the number of talks on
orthogonal polynomials in several variables. When the Bateman Project was in
operation twenty-five years ago there was very little known about orthogonal
polynomials in several variables, and much of the chapter on orthogonal poly-
nomials in several variables in Higher Transcendental Functions, Vol. II, deals
x
PREFACE
with biorthogonal sets of polynomials. It now looks like genuinely orthogonal
sets of polynomials will be more interesting and useful than the classical
biorthogonal sets, and it is clear that many decades will pass before the prob-
lems which have already come up are solved, and it is likely that only a start
has been made in uncovering the right questions in this area.
The remaining papers are not as closely tied together as those mentioned
above. Computational problems are still as improtant and interesting as when
Jacob Bernoulli used the coefficients in a certain power series, which we now
call Bernoulli numbers, to replace an elaborate set of tables of sums of the first
six powers prepared by Ismail Bullialdus by a single page of work, or when
Gauss calculated the zeros of low degree Lengendre polynomials to use them to
approximate some integrals by a method we call Gaussian quadrature.
Gautschi's survey gives an indication of the current interest in this field. It is
extensive.
Berndt shows some of the consequences that can be obtained by use of
Bernoulli numbers and some of their generalizations. This paper illustrates one
surprising fact of special functions. They are often useful in areas disjoint from
the one in which they were originally introduced. Bernoulli did not have
number theory in mind when he introduced Bernoulli numbers, yet that is one
of the places where they have been very useful. They have recently been used
in algebraic topology, and the slight generalization to Bernoulli polynomials has
been very useful in studying spline functions.
Olver's paper complements his recent book, Asymptotics and Special Func-
tions in a very nice way. The book provides an up to date treatment of the
methods used to find asymptotic expansions of special functions,and this paper
contains many problems which need to be worked on.
There are many other uses of special functions which could not be surveyed in
a meeting of three days, but a reader with an open eye will see them in most
areas where mathematics is used. There was one other paper presented at this
meeting which is not included here, since the author was unable to find time in
his busy schedule to pull his notes together to make a paper he was willing to
publish now. The talk was "Some relationships between commutative algebra
and special functions" by Gian-Carlo Rota. I am particularly sorry about this
omission, since the talk was very interesting and so are the notes which the
author has sent me. I hope his paper will appear elsewhere in the not too
distant future.
xi
Computational Methods in Special Functions-A Survey
Walter Gautschi
Introduction
§1. Methods based on preliminary approximation
1. 1 Best rational approximation
1.1.1 Best uniform rational approximation
1.1.2 A list of available Chebyshev approximations
1.1.3 Computation of Chebyshev approximations
1.2 Truncated Chebyshev expansion
1.2.1 Convergence
1.2.2 Relation to best uniform approximation
1.2.3 Calculation of expansion coefficients
1. 2.4 Tables of Chebyshev expansions and computer programs
1. 3 Taylor series and asymptotic expansion
1. 3. 1 Computational uses
1. 3. 2 An example
1.4 Pade and continued fraction approximations
1.4.1 Padé table
1.4.2 Corresponding continued fractions
1.4. 3 Relation between Pade table and continued fractions
1.4.4 Algorithms
1.4. 5 Applications to special functions
1.4.6 Error estimates
1.4.7 Generalizations
1.4. 8 Other rational approximations
1
WALTER GAUTSCHI
1. 5 Representation and evaluation of approximations
1. 5. 1 Polynomials
1. 5.2 Rational functions
1. 5. 3 Orthogonal sums
§2. Methods based on linear recurrence relations
2. 1 First-order recurrence relations
2. 1. 1 A simple analysis of numerical stability
2. 1.2 Applications to special functions
2.2 Homogeneous second-order recurrence relations
2.2.1 Minimal solutions
2.2.2 Algorithms for minimal solutions
2.2.3 Applications to special functions
2. 3 Inhomogeneous second-order and higher-order recurrence
relations
2. 3. 1 Subdominant solutions of inhomogeneous second-order
recurrence relations
2.3.2 Higher-order recurrence relations
§3. Nonlinear recurrence algorithms for elliptic integrals and elliptic
functions
3. 1 Elliptic integrals and Jacobian elliptic functions
3. 1. 1 Definitions and special values
3. 1.2 Gauss transformations vs. Landen transformations
3. 2 Gauss' algorithm of the arithmetic-geometric mean
3. 3 Computational algorithms based on Gauss and Landen
tr a n s for m at io n s
3. 3. 1 Descending Gauss transformation
3. 3. 2 Ascending Landen transformation
3. 3. 3 Ascending Gauss transformation
3. 3.4 Descending Landen transformation
3.4 Complete elliptic integrals
3. 5 Jacobian elliptic functions
2
COMPUTATIONAL METHODS
§4. Computer software for special functions
4. 1 NATS software for special functions
4.2 NAG software for special functions
4. 3 Other software for special functions
Introduction
Scientific computing often requires special functions. In the past,
the need for numerical values was partly satisfied by extensive mathe-
matical tables. Today, w'ith powerful digital computers available, such
values are obtained almost invariably by direct computation. We wish to
review here the principal methods used in computing special functions.
We may group these methods into two large classes, namely those
based on direct approximation, and those based on functional equations.
Among the former, we consider only rational approximation methods (§1).
We thus leave aside a multitude of possible expansions in terms of other
special functions. These expansions, indeed, while often helpful, still
leave us with the problem of evaluating the special functions involved.
Among the functional equations most useful for computation are linear and
nonlinear recurrence relations. These are discussed in §§2 and 3. We omit
references to other functional equations, such as differential and integral
equations, since we consider them of secondary importance in our context.
In §4 we give a brief account of the current state of computer software
development for special functions.
Due to limitations in time and space, a number of important topics
are omitted in this survey. Nothing is said, e. g., about elementary
functions and special computational techniques related to them. Good
accounts of this can be found in Lyusternik, Chervonenkis and Yanporskii
[I965] and Fike [1968]. Other topics not covered include methods based on
numerical quadrature and on Euler-Maclaurin and Poisson summation for-
mulas, the computation of zeros of special functions and of inverse func-
tions, and the computation of special constants to very high precision.
3
WALTER GAUTSCHI
Few references are given to computer algorithms for special functions, as
they can be retrieved from the indices in the journal "Communications of
the ACM" and in "Collected Algorithms from CACM" (a looseleaf collec-
tion issued by ACM of all algorithms published in Comm. ACM since
I960). Another topic dealt with only superficially are asymptotic methods,
as these are discussed more fully elsewhere in this volume.
There are not many general references on computational methods
for special functions. The only book devoted entirely to this subject is
Hart et al. [I968]. The two volumes of Luke [1969] also contain much
relevant material, and informative survey articles have been written by
Bulirsch and Stoer [1968] and Thacher [1969].
As to notations for special functions, we try to be consistent with
Abramowitz and Stegun [1964]. With regard to bibliographic references,
we give special emphasis to the literature of the past twenty years or so.
Little attempt has been made to trace all results back to the original
sources.
§1. Methods based on preliminary approximation
Our concern in this paragraph is with the approximation of a given
function of a real or complex variable by means of "simpler" functions.
Most attractive among these simpler functions are polynomials and
rational functions, since they can be evaluated by a finite number of
rational operations. Hence we restrict ourselves to polynomial and
rational approximation. One should keep in mind, however, that other
means of approximation, e.g., expansions in special functions like
Bessel functions, can be equally effective if one takes advantage of ap-
propriate recursive schemes of computation. (cf., e.g., 1.5.3, 2.2.2.)
The selection of a particular rational approximation depends on a
number of circumstances. If the region of interest is an interval on the
real line and our objective is to produce an approximation of high effici-
ency, and if we are prepared to expend the necessary effort, then we may
seek to obtain a best rational approximation, i. e., one whose maximum
4