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.