Table Of ContentSPECIAL
FUNCTIONS
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Proceedings of the International Workshop
SPECIAL
FUNCTIONS
Hong Kong 21-25 June 1999
Editors
Charles Dunkl
University of Virginia
Mourad Ismail
University of South Florida
Roderick Wong
City University of Hong Kong
V fe World Scientific
Singapore • New Jersey • London • Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
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British Library Cataloguing-in-Publication Data
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SPECIAL FUNCTIONS
Proceedings of the International Workshop
Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd.
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ISBN 981-02-4393-6
Printed in Singapore.
Preface, IWSF '99 Proceedings
These proceedings consist of refereed and selected papers which were pre
sented at the International Workshop on Special Functions - Asymptotics,
Harmonic Analysis and Mathematical Physics that was held June 21-25,
1999 at the City University of Hong Kong. This workshop was one of the
largest international meetings on these topics. A total of 130 participants
from 26 countries, including 109 participants from outside China and Hong
Kong, were in attendance. In addition to the plenary, invited, and con
tributed talks, there was a panel discussion on future directions and open
problems in the various areas represented at this workshop. These areas in
cluded Asymptotics, Classical Special Functions, Combinatorics, Harmonic
Analysis and Quantum Groups, Mathematical Physics, Partial Differential
Equations, and Orthogonal Polynomials. A report on this panel discussion
is also included.
Investigations of uniform asymptotic expansions are still going strong.
Wong's survey article summarizes some of the recent results in this area
concerning orthogonal polynomials such as Charlier, Meixner, Pollaczek,
Meixner-Pollaczek, and Krawtchouk polynomials. Olver addresses the more
recent topic of "exponential asymptotics," illustrating some of the new con
cepts in this theory with results obtained for linear difference equations of the
second order. Hermite polynomials occur frequently in the study of asymp
totics. Temme and Lopez present three different instances in which these
polynomials are used as main approximants. A completely different type of
asymptotics is discussed by O'Malley who gives an elementary introduction
to the method of multiple scales and its relation to the newer renormaliza-
tion group method and the invariant condition method. A brief summary of
different methods of evaluation of the Riemann zeta function on the critical
line is given by Paris.
One area which lies within special functions in a broad sense is the the
ory of polynomials orthogonal with respect to exponential weights, especially
those orthogonal with respect to exp (—v (x)). Chin and Rizdal have devised
a method to generate the polynomials when the orthogonality is on a finite
interval and v is a polynomial while Kaminski studies the recursion coeffi
cients for v(x) — xA. Martinez-Finkelshtein, Martinez-Gonzales, and Orive
study the asymptotic zero distribution of Jacobi polynomials P„ " '(x) as
n —> oo. Kwon, Yoon and Littlejohn survey orthogonal polynomials satis
fying higher order differential equations. The paper by Koekoek discusses
v
VI
inversion formulas, which are powerful tools for computing coefficients in
higher order differential equations satisfied by specific polynomials.
Spirodonov surveys novel applications and developments in special func
tions related to integrable systems. Audrey Terras surveys work done on the
statistical behavior and spacings of zeros of zeta functions and eigenvalues
of Laplacians and group Laplacians. Rangarajan and Suryanarayan discuss
a class of matrices and apply it to tiling. Kalnins, Miller, Torres de Castillo,
and Williams tackle the difficult task of solving the perturbation problem
for massive charged rotating black hole using a certain third order symmetry
operator. Gerardin and Lai study harmonic analysis on homogeneous trees
and describe the asymptotic behavior of members of the eigenspaces of a
certain type (A) near the boundary of the tree.
Product formulas and integral representations are hard to come by and
their ^-analogues are even harder to find. Rahman provides a very elaborate
g-analogue of Bailey's formula reducing a special F -series as a product of
4
two 2F\-series. Aomoto gives an integral representation for a multivariate
analogue of Wright functions. Numerical computations of special functions
have always been important and difficult, and Reinsch's paper presents a
method for computing elliptic integrals.
The function expressing the number of partitions of the positive integer n
into four squares, as a function of n, satisfies very few arithmetic properties.
Hirschhorn and Sellers's contribution contains three new relations for this
function. Baker develops Robinson-Schensted-Knuth insertion and sliding
schemes for the tensor and spmor representations of the Lie algebra b .
n
The last twenty five years have seen many contributions to the question
of recovering orthogonality measures from three term recurrence relations for
specific sets of orthogonal polynomials. This amounts to solving a moment
problem. Berg's paper ties moment problems to convolution semigroups
with applications to the case of indeterminate moment problems like the
Q-Laguerre and lognormal moment problems.
The old idea of product formulas and addition theorems for orthogonal
polynomials has led to the hypergroup formalism; the product formulas lead
to convolution structures in Banach spaces of series (orthogonal expansions).
This field of investigation is represented here in the papers by Filbir, Lasser
and Obermaier, Trimeche, and Xu. Another approach to the harmonic anal
ysis of special functions is by way of Lie groups, homogeneous spaces, and
root systems. In a sense the Lie groups came first and then the homogeneous
spaces (spaces on which a Lie group acts), although surely the sphere was
VII
studied much earlier. Eventually it was found that a considerable amount
of the work relating special functions and Lie groups can be carried out
just with the machinery of root systems- The papers by Ding, Dunkl, and
Rosier are on this topic. Calogero-Sutherland systems comprise a research
area in mathematical physics that has recently found significant applications
for orthogonal polynomials of several variables. These systems are quantum
mechanical models of identical particles in a one-dimensional space, that is,
the circle or the line, with certain types of interactions. One important ex
ample is 1/r2, where each pair of particles produces a potential varying as
the inverse square of the distance between them. The wave functions of these
systems (eigenfunctions of the associated Schrodinger equation) are products
of base states and orthogonal polynomials of the Jack family (symmetric or
non-symmetric). Kakei and Kato and Ujino and Nishino wrote papers on
this area for these proceedings. The article by Hiro-Fumi Yamada concerns
vertex operator algebras and Schur functions; this is another active area of
research in the intersection of algebra and mathematical physics.
The Handbook of Mathematical Functions, published in 1964 by the Na
tional Bureau of Standards (NBS), is undoubtedly one of the most frequently
cited books in scientific literature. Recently, the National Institute of Stan
dards and Technology (formerly, National Breau of Standards) has decided
to produce a successor to this Handbook. This successor will be in the form
of a Digital Library of Mathematical Functions (DLMF). The details of this
project such as origins, contents, and authorship are presented in the paper
by Lozier.
On behalf of all the attendees, we express our appreciation to Ms. Colette
Lam for her untiring and marvelously efficient administration of the many
tasks that made the workshop such a success. We would also like to take this
opportunity to thank the following organizations for their generous support:
• EPSON Foundation,
• K C Wong Education Foundation,
• Lee Hysan Foundation Limited,
• Liu Bie Ju Centre for Mathematical Sciences (at City University of
Hong Kong),
• National Science Foundation (USA),
VIM
Royal Plaza Hotel (Hong Kong).
Charles Dunkl, University of Virginia
Mourad Ismail- University of South Florida
Roderick Wong, City University of Hong Kong
CONTENTS
Preface v
Integral Representations of Quasi Hypergeometric Functions 1
K. Aomoto
Combinatorics of Crystals for Tensor and Spinor
Representations of U (B ) 16
q n
T. H. Baker
On Infinitely Divisible Solutions to Indeterminate Moment Problems 31
C. Berg
Generating Orthogonal Polynomials for Exponential Weights on
a Finite Interval 42
R. C. Y. Chin and D. Ridzal
Spherical Multipliers and Spherical Multiplier Transformations 57
H. Ding
Generating Functions Associated with Dihedral Groups 72
C. F. Dunkl
Existence of Approximate Identities in Maximal Ideals
of Hypergroup-Algebras 88
F. Filbir
Zeros of Jacobi Polynomials with Varying Non-Classical Parameters 98
A. Martinez-Finkelshtein, P. Martinez-Gonzalez and R. Orive
Asymptotic Behavior of Eigenfunctions for the Hecke Algebra
on Homogeneous Trees 114
P. Gerarding and K. F. Lai
Some Relations for Partitions into Four Squares 118
M. D. Hirschhorn and J. A. Sellers
Algebraic Aspects of Quantum Calogero Models 125
S. Kakei and Y. Kato
Special Functions and Perturbations of Black Holes 140
E. G. Kalnins, W. Miller Jr., G. F. Torres Del Castillo
and G. C. Williams
ix
X
On a Nonlinear Recurrence Related to Nevai Polynomials 152
D. Kaminski
Inversion Formulas Involving Orthogonal Polynomials and
Some of Their Applications 166
R. Koekoek
Bochner-Krall Orthogonal Polynomials 181
K. H. Kwon, G. J. Yoon and L. L. Littlejohn
Orthogonal Expansions for U>- and C-Spaces 194
R. Lasser and J. Obermaier
The DLMF Project: A New Initiative in Classical
Special Functions 207
D. W. Lozier
Resurgence in Difference Equations, with an Application
to Legendre Functions 221
F. W. J. Olver
On Exorcising Secular Terms 236
R. E. O'Malley Jr.
New Asymptotic Formulas for the Riemann Zeta Function
on the Critical Line 247
R. B. Paris
A q- Analogue of a Product Formula of Bailey and
Related Results 262
M. Rahman
The Brahmagupta Matrix and its Applications to Tiling
282
R. Rangarajan and E. R. Suryanarayan
Elliptic Integrals of the First and Second Kind — Comparison of
Bulirsch's and Carlson's Algorithms for Numerical Calculation
293
K.-D. Reinsch and W. Raab
Short-Time Estimates for Heat Kernels Associated
with Root Systems
309
M. Rosier
