Topic #8 ------------ OP-SF NET 7.4 ------------- July 15, 2000
~~~~~~~~~~~~~
From: Walter Van Assche
Subject: Future Directions in Special Functions
[From the June Newsletter]
On the last day of the NATO Advanced Study Institute on "Special functions
2000: Present Perspectives and Future Directions" (Tempe, Arizona, May 29
- June 9, 2000) there was a session on Future Directions, chaired by
Richard Askey. The following is an attempt to summarize what was said.
First Askey gave some _advice_.
- Ramanujan is still a very big source of future research, especially
regarding congruences for the partition function. Exciting new results
have been found by Ken Ono, but what has been found is probably only a
hint of what else will be discovered.
- Other indications that there is still a lot to be learned from a study
of Ramanujan is the recent work on elliptic functions with different
bases, which was probably the first use of cubic transformations of
hypergeometric functions, Ramanujan's wonderful series for 1/\pi and the
remarkable identities found in the lost notebook (including results on
mock theta functions).
- A second source of problems is in the work of David and Gregory
Chudnovsky. They have mentioned many problems and results, some of which
are eventually published, but many have not been published. Their papers
are worth studying, although this is not easy.
- Work of Rodney Baxter led to the discovery of quantum groups and was one
of the sources for elliptic hypergeometric functions. There is much more
there which needs to be understood.
- Bill Gosper has sent e-mail containing many interesting formulas to many
people. Some e-mails have been understood, but many still are full of
mysteries.
He then continued with some _safe_predictions_:
- Special functions of several variables will be studied extensively
(orthogonal polynomials, hypergeometric and basic hypergeometric
functions, elliptic hypergeometric functions).
- Cubic transformations will get more attention (see, e.g., Bressoud's
treatment of alternating sign matrices).
- There will be much more combinatorial work.
- Computer algebra will become important but will not replace thinking.
- Nonlinear equations and special functions (Painleve) will receive more
attention.
- Regarding asymptotics, there will be a deeper understanding in one variable,
there will be much more on difference equations, and asymptotics for several
variables will be developed more fully.
Askey then expressed some _hopes_:
- Special functions in infinite dimensional spaces will appear.
- Linear differential equations with more than three regular singular points
will be understood better than at present.
- Special functions over p-adic and finite fields become more popular.
- Orthogonal (and biorthogonal) rational functions will start to have more
applications.
- Understanding mock theta functions via mock modular functions will partly
succeed.
- The location of zeros of _2F_1(a,b;c;z) on (-\infty,0), (0,1), and
(1,\infty) in the terminating case is known (also in the complex plane).
We need extensions to _3F_2 and _2\phi_1 and other (basic) hypergeometric
functions.
Finally Askey mentioned some _wild_guesses_:
- Cubic transformations for hypergeometric functions really live in double
series associated to G_2 and we are only seeing one dimensional parts of
this.
- The function G satisfying the relation G(x+1) = \Gamma(x) G(x) has an
integral representation, probably an infinite dimensional one (a limit of
Selberg's integral?).
- 9-j symbols as orthogonal polynomials in two variables can be
represented as a double series.
Some other participants added some other interesting observations
and suggestions for future work.
Tom Koornwinder:
- Matrix valued special functions. An obvious source of such functions are
the generalized spherical functions associated with Riemannian symmetric
pairs (G,K) and higher dimensional representations of K. See Grunbaum's
lecture at this meeting for the example (SU(3),SU(2)).
- Orthogonal polynomials depending on non-commuting variables naturally
occur in connection with quantum groups, see for instance the q-disk
polynomials studied by Paul Floris, which are spherical functions for the
quantum Gelfand pair (U_q(n), U_q(n-1)). More examples should be obtained
and a general theory of such polynomials should be set up.
- Special functions associated with affine Lie algebras. Remarkable
interpretations of special functions have already been found on affine Lie
algebras (see the book by Victor Kac), but much more should be possible
here. The lecture by Paul Terwilliger at this meeting gives some hints in
this direction.
- The work of Borcherds: generalized Kac-Moody algebras, vertex algebras and
lattices in relationship with automorphic functions.
- Algebraic and combinatorial techniques in contrast with analytic
techniques have quickly gained importance in work on (q-)special functions
during the last few decades. Algebra often gives rise to quick and easy
formal proofs of, for instance, limit results. Usually, a rigorous
analytic proof is much longer, while it does not give new insights. In
fact, the rigorous proof is often omitted. There is need for a meta-theory
which explains why formally obtained results are so often correct results.
Vyacheslav Spiridonov:
- It is likely that important special functions are hidden in some of the
work on differential-delay and differential-difference equations.
- Development of elliptic special functions (elliptic beta integral, elliptic
deformations of Painleve).
- Connections of our work with other fields (biology, economy, etc.).
- Wavelets could be studied as special functions.
- Ismail's q-discriminant needs an interpretation in statistical mechanics.
Stephen Milne and Tom Koornwinder:
- The lecture by Jan Felipe van Diejen and the discussion after Stephen
Milne's last lecture at this meeting made clear that several different
types of multivariable analogues of one-variable (q-)hypergeometric series
have been studied extensively, but that their mutual relationship is
poorly understood. The three most important types are:
1. Explicit series associated to classical root systems
(Biedenharn, Gustafson, Milne),
2. Hecke-Opdam hypergeometric functions and Macdonald polynomials
associated to any root system ((q-)differential equations, usually no
explicit series),
3. Gelfand hypergeometric functions (again (q-)differential equations,
usually no explicit series).
Van Diejen, in his lecture, added to this list:
4. hypergeometric sums of q-Selberg type,
5. hypergeometric sums coming from matrix inversion.
Koornwinder would like to add:
6. Solutions of KZ(B) and q-KZ(B) equations,
7. 3-j, 6-j and 9-j symbols for higher rank groups.
- Elliptic generalizations of one and multivariable hypergeometric
functions are also coming up now. Stephen Milne added that it is likely
that the concept "very well poised" ties these various types of
multivariable functions together.
- Applications in combinatorics and number theory are welcome.
Sergei Suslov:
- One needs to understand the classical q-functions, beginning with the
q-exponential and q-trigonometric functions.
- Orthogonal q-functions (also the non-terminating series) and
special limiting cases are useful.
- Biorthogonal rational functions are a rich source of research problems.
Mourad Ismail:
- There is still a lot of work to be done in moment problems and continued
fractions, in particular indeterminate moment problems.
- Discriminants, lowering operators and electrostatics, such as the Coulomb
gas model.
- Multivariate extensions.
Walter Van Assche: There is still quite some work in orthogonal polynomials:
- The asymptotic zero distribution and logarithmic potential theory (with
external fields and constraints) has been worked out in quite some detail now.
For some q-polynomials one seems to need circular symmetric weights. We don't
know how to handle big q-Jacobi, big q-Laguerre, q-Hahn and q-Racah yet.
- There is a well established theory for strong asymptotics of orthogonal
polynomials on the unit circle and on the interval [-1,1] (Szego's
theory). The analog of this theory for the infinite interval (e.g., Freud
weights) is starting to become clear. So far there is no theory for
orthogonal polynomials on a discrete set (such as the integers). The
Riemann-Hilbert technique may be useful here.
- Multivariate orthogonal polynomials need more attention.
- Multiple orthogonal polynomials (one variable but several weights) may
be a rich source of nice research. Some of these multiple orthogonal
polynomials can be written in terms of nice special functions (generalized
hypergeometric functions, hypergeometric functions of several variables,
etc.). The analysis involves Riemann surfaces with several sheets,
equilibrium problems for vector potentials, banded non-symmetric
operators. We already know some nice applications in number theory and
dynamical systems. Other applications would be nice.
- Higher order recurrence relations and asymptotics for solutions of
difference equations are useful.
George Gasper:
Positivity proofs and proofs that certain functions only have real zeros
are very useful.
Erik Koelink
- The _8\phi_7 basic hypergeometric is very nice and the multivariate
case would be even nicer.
- Where do the elliptic hypergeometric functions of Frenkel and Turaev live?
- Is there a way to use Riemann-Hilbert problems for quantum groups?
- Applications of multivariate orthogonal polynomials in probability theory.
This is just a brief description and a personal account of what was said
during the session on future directions. Some other participants added
some open problems, but it would take too much space to report on these in
the newsletter.