SIAM AG on Orthogonal Polynomials and Special Functions


Extract from OP-SF NET

Topic #12  ------------   OP-SF NET 4.6  ------------ November 15, 1997
From: OP-SF NET Editor (
Subject: Recent International Workshop at RIMS, Kyoto

At RIMS (Research Institute of Mathematical Science, Kyoto, Japan) an
international workshop on "Invariant Differential Operators, Special
Functions and Representation Theory" was held during October 20-31, 1997.
The organizer was Toshio Oshima (University of Tokyo).  I take the
following information from the URL
In the next Topic Mathijs Dijkhuizen, who participated, will give a

One half of the workshop (the second week) was devoted to
"Integrable systems of difference and differential equations"
The main speakers, with series of 3 or 4 lectures each, were

Eric M. Opdam (Leiden Univ.): Dunkl Operators
In these lectures I will give an overview of results on Dunkl's
"differential-reflection" operators, up to the most recent developments.
Mainly I will concentrate on the (differential)  trigonometric case, the
case of the Dunkl-Cherednik operators, because in this case the theory has
reached the most mature level at present. And also there are several older
theorems and applications whose proofs can be polished by modern methods,
but many of these things were never written. So I feel that giving such a
series of lectures can be rewarding, and I am happy to embark on such a
project. Roughly, I have in mind to treat the following subjects: 

Knizhnik-Zamolodchikov connection, the Harish-Chandra system, monodromy
representation, the shifting principle, asymptotic expansions, the Gauss'
summation formula.
  2. ALGEBRAIC PROPERTIES. Nonsymmetric orthogonal polynomials, the graded
Hecke algebra, (affine) intertwiners, the recursion formula of Knop and
  3. HARMONIC ANALYSIS. The Fourier transform for the Dunkl-Cherednik
operators, the Paley Wiener theorem, the action of the affine Weyl group.
  4. RESIDUE CALCULUS FOR ROOT SYSTEMS. The Plancherel measure for the
attractive case; classification of all square integrable eigenfunctions,
and their explicit norms.

S. Ruijsenaars (CWI, Amsterdam):
Special functions solving analytic difference equations

   * I Generalized gamma functions
   * II A generalized hypergeometric function
   * III Generalized Lame functions

I. We discuss a new solution method for difference equations of the form
F(z+ia/2)/F(z-ia/2) = Phi(z), with Phi(z) meromorphic and free of zeros
and poles in a strip |Im(z)| < C. The method gives rise to generalized
gamma functions of hyperbolic, elliptic and trigonometric type (Euler's
gamma function being of rational type), whose properties we sketch.

II. The hyperbolic gamma function can be used as a building block to
construct a novel generalization of the hypergeometric function _2 F_1 . 
The new function is a simultaneous eigenfunction of four independent
hyperbolic difference operators of Askey-Wilson type. The integral
representation through which this joint eigenfunction is defined
generalizes the Barnes representation for _2 F_1. It is meromorphic and
has various remarkable symmetry properties that are not preserved for its
q -> 1 ( or `nonrelativistic') limit _2 F_1. 

III. The `q=1/nonrelativistic' Lame differential operator can be
generalized to a `q \ne 1/relativistic' difference operator. (The latter
may be viewed as the Hamiltonian defining the elliptic relativistic
Calogero-Moser N-particle system for N=2.) We present eigenfunctions of
this operator. They are in fact joint eigenfunctions of three independent
difference operators. The functions are used to define the Hamiltonian as
a self-adjoint operator on a Hilbert space. Their asymptotics is governed
by a c-function that is a quotient of two elliptic gamma functions. 

Topic #13  ------------   OP-SF NET 4.6  ------------ November 15, 1997
From: Mathijs S. Dijkhuizen (
Subject: Report on RIMS workshop, Kyoto


The international workshop at RIMS late October was part of the scientific
activities organized in the framework of a special Research Project on
Representation Theory.  Most of the activities are concentrated in the
autumn. The project includes the invitation of a number of foreign
researchers to RIMS for a more or less extended period.  The two
distinguished visitors this autumn are Prof. Grigori Olshanski (Moscow)
and Prof. Eric Opdam (Leiden), who are staying here for four months. 

The workshop was split into two parts. The first week was devoted to
representation theory and featured two series of lectures by Olshanski (on
combinatorial and probabilistic aspects of harmonic analysis on big, i.e.
infinite-dimensional, groups) and Prof. Michael Eastwood (Adelaide, on
invariant differential operators on homogeneous spaces)  plus a number of
other talks. During the second week the topic was integrable systems of
difference and differential equations. Since I did not attend the first
part of the workshop, I will restrict myself to some comments about the
second part. As suggested by the topic, most of the talks were somehow
concerned with systems of commuting operators. The two main speakers were
Eric Opdam and Prof. Simon Ruijsenaars (Amsterdam), who each gave three or
four one-hour talks. 

Opdam talked about trigonometric Dunkl operators and their use in the
study of multivariable hypergeometric functions associated with root
systems. Hypergeometric functions in one variable have been known for a
very long time; the earliest indications how to generalize them to many
variables came from representation theory, where they arise as zonal
spherical functions on Riemannian symmetric spaces. The notion of Dunkl
operator, however, is something completely new which is not all hinted at
by the connection with Riemannian symmetric spaces. Dunkl wrote down his
original differential-reflection operators with rational coefficients
around 1989.  One of their main properties is that they commute with each
other. A trigonometric version of these operators was introduced by
Cherednik who related them to the degenerate affine Hecke algebra. The
importance of Dunkl operators for the theory of hypergeometric functions
is explained by the fact that they allow one to give an elementary
algebraic construction of the commuting system of hypergeometric
differential operators for arbitrary values of the coupling constants. The
existence of this hypergeometric system was established earlier by Heckman
and Opdam using analytic methods. Over the last couple of years a whole
body of theory has developed around Dunkl operators. 

Opdam's talk was phrased in the elegant language of arbitrary reduced root
systems and Weyl groups. This was in quite some contrast with Ruijsenaars'
series of lectures, which, though certainly no less interesting, was
characterized by a rather down-to-earth approach to a one-variable
problem, namely the study of meromorphic solutions of certain types of
analytic difference equations. Ruijsenaars actually started by remarking
that an analyst from the late nineteenth century would have had no problem
following at least the first part of his talks. Analytic difference
operators were studied by several distinguished mathematicians until less
than a hundred years ago, but later they failed to attract much interest. 
This seems to be changing now.  Due to a notable lack of general theory
about solutions of analytic difference equations most results have to be
proved "by hand". One striking feature is the (rather obvious) fact that
the solution space is usually infinite-dimensional. By imposing certain
conditions on the asymptotic behaviour of the solution one can, however,
arrive at certain uniqueness results. Ruijsenaars' motivation for studying
these analytic difference equations partly comes from relativistic
analogues of the quantum integrable Calogero-Moser-Sutherland models for N
interacting particles on the real line. The trigonometric versions of
these relativistic quantum models may be regarded as a q-analogue of the
hypergeometric system discussed by Opdam. As shown by Cherednik, their
algebraic properties are also amenable to a Dunkl operator approach. The
polynomial solutions of these systems have been studied by Macdonald and
others (for reduced root systems) and Koornwinder, Van Diejen, Noumi and
others in the BC_n case (in the one-variable case they reduce to
well-known families of q-hypergeometric polynomials of Askey-Wilson type).
These polynomials are also known to occur in connection with quantum
groups. As for non-polynomial solutions of these systems, not much is
known at this time. As is apparent from Ruijsenaars' talks, a lot of
interesting work in this direction is still waiting to be done.

In short, this was a very stimulating workshop with some very
interesting mathematics.

Mathijs S. Dijkhuizen 

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