## OP-SF WEB

### Extract from OP-SF NET



Topic #7  ------------    OP-SF NET 6.5   -----------   September 15, 1999
~~~~~~~~~~~~~
From: OP-SF NET Editor
Subject: Hong Kong Panel Discussion

This is a report on the Panel discussion held on Thursday, June 24, 1999,
5.00-6.30 p.m.  during the International Workshop on Special Functions,
Asymptotics, Harmonic Analysis, and Mathematical Physics at the City
University of Hong Kong.

The basic account was written by Tom Koornwinder modified by input from
the participants and from Daniel Lozier, Martin Muldoon and Andre
Ronveaux.

The session was chaired by Charles Dunkl. He stressed the importance of
a discussion of important directions so that the rest of the world would
know what we were up to. This could be useful both to funding agencies
and to inform young people of directions in research. He proposed a
discussion of research perspectives for, successively:

1. Asymptotics
2. Harmonic analysis
3. Classical special functions
4. Mathematical physics

In the ensuing discussion, classical special functions were skipped as a
separate item, but they were covered by the item on harmonic analysis.

Mourad Ismail raised the issue of future meetings in the series
Fields-Toronto (1995) - CRM-Montreal (1996) - Mount Holyoke (1998) - Hong
Kong (1999)  - Arizona (2000) ... with the suggestion that there should be
more than one year between meetings and that perhaps there should be a
joint meeting with the (mainly European) series on Orthogonal Polynomials.
There was unanimous agreement that the series should continue and support
for the idea of a meeting in the Netherlands in 2002 to be organized by
Tom Koornwinder, Nico Temme and Erik Koelink.  Luc Vinet proposed that
there should be some kind of coordinating group for these meetings.  There
was general agreement that the SIAM Activity Group on Orthogonal
Polynomials and Special Functions might provide this coordination. It was
stressed that this did not include the actual organization of meetings.

1. Asymptotics

1.1. Frank Olver
For ordinary differential equations it is time to write another
book to replace W. Wasow's "Asymptotic expansions for ordinary
differential equations", Wiley, 1965. The new book should be more
practical, giving simpler proofs, examples, error bounds (where
possible), increased regions of validity in the complex plane, and a
description of the different types of asymptotic solutions (explicit
and implicit). Some of the recent work on re-expansions of the
remainder terms (hyperasymptotics) should also be included.

For difference equations, the whole asymptotic theory needs reworking
in a much simpler and more readily applicable form than in the
classic (and almost impenetrable) papers of G. D. Birkhoff and W.J.
Trjitzinsky.  Proofs can be given (by use of boundary-value type
methods) that obviate the need to pass from the set of integers on
which the solutions actually live, into the complex plane--with all
its attendant problems of analyticity.  The two 1992 papers of R.
Wong and H. Li are a good beginning.  Expansions should be sought in
inverse factorial series as well as, or in place of, conventional
expansions in inverse powers.

1.2. Nico Temme
For asymptotics of integrals, the theory and construction of error bounds
lags behind the corresponding results for differential equations.

1.3. Roderick Wong
He agreed with Temme.  In addition to the problem of error bounds, for
asymptotics of integrals, regions of validity are usually smaller than the
corresponding ones that can be established by using the differential
equation theory.  This situation needs to be improved.  For asymptotic
solutions of difference equations, Airy-type expansions should be
considered.  He referred to a paper (by Costin and Costin) in SIAM J Math
Anal.  In singular perturbations, derivations of asymptotics are mostly
formal; rigorous proofs may give more insight.  For the nonlinear
Klein-Gordon equation, even to show that the solution is uniformly bounded
for all time is not a simple matter.  For specific problems even with two
terms, it is difficult to show that the remainder is of the right order.

1.4. Robert O'Malley
Parabolic p.d.e.'s sometimes have travelling wave solutions (like
hyperbolic tangent in Burgers' equation). To obtain such solutions one has
to go back to solutions of very special o.d.e.'s and their asymptotics.
Dumb computing won't work there. Applied people have to learn about the
powerful tools of solutions by special functions and their asymptotics.

1.5. Eric Opdam
People in this workshop include, on the one hand, researchers with very
detailed knowledge of one-variable theory and, on the other hand, workers
in higher rank theory (i.e. more variable theory associated with root
systems). They should join forces in order to do asymptotics in the higher
rank case.  Olver commented that in asymptotics the difficulty goes up by
an order of magnitude as each new variable is added.  Moreover, because of
the increasing complexity a point of diminishing returns is soon reached.
Koelink responded that the symmetry available in the higher rank case may
help in doing asymptotics.

2. Harmonic analysis

2.1. George Gasper
A big problem is that of convolution structures.
He suggested a study of nonnegative kernels, as done for q-Racah
polynomials in the paper
G. Gasper and M. Rahman, "Nonnegative kernels in product formulas
for q-Racah polynomials", J. Math. Anal. Appl. 95 (1983), 304-318
They were able to handle only the case with symmetry; the non-symmetric
case is still open. Many other cases (of convolution structures) are
still unknown.
Dunkl remarked that all positivity results are important.

Be aware of possible applications of special functions in signal analysis,
stochastic processes, financial mathematics.
Furthermore, a lot of inspiration for further work can be obtained from
the work by Percy Deift on Riemann-Hilbert problems and random matrix theory,
see for instance
P. Deift, "Orthogonal polynomials and random matrices: a
Riemann-Hilbert approach", Courant Lecture Notes in Math., Vol. 3,
Courant Institute of Math. Sciences, 1999.
See also the topics covered in this fall's special semester on
Foundations of Computational Mathematics in Hong Kong
(http://www.damtp.cam.ac.uk/user/na/FoCM/HK.html),
in particular the workshop on Minimal energy problems
(http://www.math.usf.edu/FoCM99/).
Finally, KZ equations and elliptic hypergeometric functions are promising
fields.

Terwilliger's results on Leonard pairs (see Terwilliger's lecture in this
workshop) gives a new approach to Leonard's characterization of q-Racah
polynomials and some related polynomials in
D.A. Leonard, "Orthogonal polynomials, duality and association schemes",
SIAM J. Math. Anal. 13 (1982), 656-663.
Terwilliger's approach works for any field. What are the implications
for special functions for other fields than the reals?
Next, Askey suggests finding an integral equation for the
so-called Barnes G-function, which satisfies G(x+1)=gamma(x)*G(x), G(1)=1.
There is a problem on this function in Whittaker and Watson with a reference
to late 19th century work by a Russian.  Barnes has a number of papers
on this function, in the early part of this century.
There might be an infinite dimensional integral
which represents this function, based on the form Selberg's integral
takes.  The G-function shows up in a number of places.
Andrew Lenard used it in a paper (J. Math. Phys. 5 (1964), 930-943) on the
strong Szego limit theorem.  Szego pointed out to him how some of
his results could be stated using the G-function.  It also shows up
in K-theory.
Finally, for 9j-symbols, a representation as a double sum should be found.
Such a representation can be expected, since they have a two variable
orthogonality.

2.4. Christian Berg
1. Concerning the indeterminate moment problem one should try to find a
closer relationship between the growth rate of the coefficients of the
three terms recurrence relation for the orthonormal polynomials and the
growth properties of the entire functions in the Nevanlinna matrix. In
particular one should relate the order of these functions to the growth
rate of the coefficients.
In a special case of birth and death rates being a specific polynomial in
n of degree four, it turned out that the entire functions have order 1/4,
see C. Berg and G. Valent, "The Nevanlinna parametrization for some
indeterminate Stieltjes moment problems associated with birth and death
processes", Meth. Appl. Anal 1 (1994), 169-209. In a recent manuscript by
G. Valent,"Indeterminate moment problems and a conjecture on the growth of
the entire functions of the Nevanlinna parametrization", there is an
example of birth and death rates being polynomials of degree 3 and the
order of the entire functions are 1/3. The paper also formulates a
conjecture.
2. It was proved by C. Berg and W. Thill, "Rotation invariant moment
problems", Acta Math. 167 (1991), 207-227, that a moment problem in
R^n, n > 1, can have a unique solution mu (i.e. mu is a determinate
measure on R^n) and yet the polynomials are not dense in the Hilbert space
L^2(R^n,mu).  Apart from the rotationally invariant case no criterion
seems to be known for this phenomenon to happen, and this kind of question
needs further study.

2.5. Eric Opdam
Special function theory reflects deep properties in group theory,
mathematical physics and number theory. For advances in special functions
one should better understand these three fields.  In mathematical
physics the Calogero-Moser system is a deformation of the boson gas. Find
correlation functions for the Calogero-Moser system as generalizations of
such functions for the boson gas.  Macdonald theory corresponds to the
boson gas with periodic constraints.  In getting rid of these constraints
one would arrive at Macdonald functions for the noncompact case (see
lectures by Koelink and Stokman at this workshop for the rank one case),
which might be studied from the point of view of double affine Hecke
algebras.  In representation theory one should not restrict oneself to the
spherical case.  Askey added that we need to get past page 2 of Zygmund
and start solving hard problems with real applications.

2.6. Yuan Xu
Most of the L^p theory for orthogonal polynomials in several variables
is not yet understood.
There is the question of finding explicit formula for orthogonal
polynomials associated to a weight function that is invariant under an
octahedral group. For example, Dunkl's h-harmonics associated to the type
B weight. Such a basis may be useful in studying cubature formulae
(numerical integration formulae). There is a possible connection between
common zeros of invariant orthogonal polynomials for the weight function
(x1^2 - x2^2)^2(x2^2 - x3^2)^2(x3^2 - x1^2)^2 on the sphere S^2 and a
family of cubature formulae on S^2 conjectured by V. I. Lebedev.

2.7. Dennis Stanton
Macdonald's conjecture about the positivity of the coefficients in
the polynomial expansion of the Macdonald-Kostka coefficients and
the Garsia-Haiman n factorial conjecture
are at present the most important conjectures in algebraic combinatorics,
see for instance
A.M. Garsia and M. Haiman, Some natural bigraded $S_n$-modules and
$q,t$-Kostka coefficients. The Foata Festschrift. Electron. J. Combin.
3 (1996), no. 2, Research Paper 24,
http://www.combinatorics.org/Volume_3/Abstracts/v3i2r24.html
Furthermore, problems associated with graph spectra for finite matrix groups
(A. Terras, lecture at this workshop) are very important.

2.8. Audrey Terras
Graph spectra could lead to some interesting special functions.

2.9. Charles Dunkl
1) I suggest the study of orthogonal polynomials (special functions,
transforms) associated with the non-crystallographic reflection groups,
that is, H3 and H4; of icosahedral type. These groups are related to
quasicrystals; an area of research in both physics and mathematics.
2) Alberto Gruenbaum (referring to his work with Duistermaat on bispectral
problems) suggested to me earlier, the idea of finding bispectral
differential-difference operators (eigenfunctions of one operator satisfy
another d-d equation with respect to the eigenvalue). Alberto and I worked
out a small example (a new way of looking at a known situation) which
involved a perturbed one-variable differential-difference operator related
to the group Z2. Thus I speculate there may be perturbations for a limited
set of parameter values (more speculation: those called singular by de
Jeu, Opdam and myself).
3) I speculate that the phenomenon called super-integrability (e.g.,
Konstein) may apply to algebras of differential-difference operators
with integer parameter values.

Study asymptotics of

P_n(x):= c_n \int_a^b \ldots \int_a^b \prod_{j=1}^n (x-\lambda_j)
\prod_{1\le i