FoCM IMA

Workshop on Special Functions, FoCM'02, IMA, Minneapolis, 5-7 August 2002

This is one of the 19 workshops during the conference FoCM'02 at the IMA, University of Minnesota, Minneapolis, MN, USA, 5-14 August 2002. The workshop will run for 3 successive afternoons during 5-7 August (Monday - Wednesday).

The workshop will be held immediately after the IMA 2002 Summer Program Special Functions in the Digital Age at the IMA in Minneapolis, 22 July - 2 August 2002.

Workshop organizers
Plenary speaker on workshop theme
Dan Lozier (NIST, Gaithersburg, Maryland, USA), Development of a new handbook of properties of special functions, see abstract.
Semiplenary speakers during the workshop
  • Peter A. Clarkson (University of Kent, Canterbury, UK),
    Connection formulae for the Painlevé equations, see abstract in html. Expanded version in pdf.
  • Mourad E.H. Ismail (University of South Florida, Tampa, Florida, USA),
    Solutions of Bethe Ansatz equations in some physics models, see abstract in html.
  • Nico M. Temme (CWI, Amsterdam, Netherlands),
    Two-point Taylor expansions for convergent asymptotic expansions, see abstract in html. Expanded version in pdf.
Other workshop speakers

workshop schedule (see also the full conference schedule)
Monday 5 August
2:40pm - 3:30pm   Petkovsek
4:00pm - 4:50pm   Schlosser
4:50pm - 5.40pm   Spiridonov
5:40pm - 6:30pm   Clarkson   (semi-plenary)
Tuesday 6 August
1:50pm - 2:40pm   Ismail   (semi-plenary)
2:40pm - 3:30pm   Terwilliger
4:00pm - 4:50pm   Miller
4:50pm - 5.40pm   Berkovich
5:40pm - 6:30pm   Maier
Wednesday 7 August
1:50pm - 2:40pm   Koornwinder
2:40pm - 3:30pm   Temme   (semi-plenary)
4:00pm - 4:50pm   Howls
4:50pm - 5.40pm   Olde Daalhuis
5:40pm - 6:30pm   Garrett

Links

Abstracts

Abstract of Berkovich' lecture
         In this talk I explain how the Dyson adjoint of a partition can be used in order to derive a binary tree of polynomial analogs of Euler's pentagonal number theorem. This tree contains a well- known polynomial identity due to Schur. At present only one of those new analogs has a q-hypergeometric explanation. Next, I show how one can extend Dyson's adjoint of partitions to deal with MacMahon's mod 2 graphs. This way one is led to the new combinatorial proof of the Gauss identity. This talk is based on my joint work with Frank Garvan.

(canceled) Abstract of Bogatyrev's lecture
         Many mathematical and physical theories (i.e. conformal field theory, string theory, final gap solutions for completely integrable systems, extremal polynomials) essentially use function theory on Riemann surfaces. Such objects as abelian differentials, their integrals and periods, quadratic differentials, Eichler periods, meromorphic sections of vector bundles over the curves etc. often appear in the formulas of theoreticians. Sometimes it is necessary to know how the above entries vary with the deformation of the curve, say if you have to solve some equations in the moduli space of the curves. The latter problem arises in the algebro-geometric approach to the Chebyshev optimization problems.
         Every real algebraic curve admits Schottky uniformization with convergent linear Poincaré theta series (A.I. Bobenko, 1986). This theorem becomes a clue to the numerical analysis on Riemann surfaces and their deformation spaces. We discuss effective calculation of Schottky automorphic functions and effective solution of abelian equations in Teichmüller space with application to the extremal polynomials.

Abstract of Clarkson's lecture
         The six Painlevé equations were first discovered around the turn of the century by Painlevé and his colleague in an investigation of nonlinear second-order ordinary differential equations. Recently there has been considerable interest in the Painlevé equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering. Consequently the Painlevé equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations, despite being nonlinear equations. Although first discovered from strictly mathematical considerations, the Painlevé equations have arisen in a variety of important physical applications including statistical mechanics, plasma physics, nonlinear waves, quantum gravity, quantum field theory, general relativity, nonlinear optics and fibre optics.
         In this talk I shall discuss connection problems for the Painlevé equations. Such problems can be characterized by the question as to how the asymptotic behaviours of solutions are related as the independent variable is allowed to pass towards infinity along different directions in the complex plane. Connection problems have been previously tackled by a variety of methods. Frequently these are based on the ideas of isomonodromic deformation and the matching of WKB solutions, which is a nonlinear version of the classical steepest descent method for oscillatory Riemann-Hilbert problems and is rather complex. I shall describe a uniform approximation method developed in collaboration with Bassom, Law & McLeod. This procedure, which is rigorous, removes the need to match solutions and can lead to simpler solutions of connection problems.

Abstract of Garrett's lecture
We will give a general theorem that arises from combinatorial consideration of certain polynomials. We will state several corollaries. These corollaries will be polynomial genralizations of many of the Rogers-Ramanujan type identities from L.J. Slater's list.

Abstract of Howls' lecture
         The Digital Library of Mathematical Functions is currently under construction at NIST. One of the new types of special functions to be included is that arising from canonical integrals with coalescing saddles. Although studied previously as independent functions, such integrals are the basis of applications of Thom's catastrophe theory to a wide variety of problems, notably caustic diffraction phenomena in wavefield (optics, quantum mechanics, waterwave, ...) and uniform asymptotic analysis. These integrals fall into two types.
         The first type are single dimensional integrals with (K+2)th order polynomial phases (K saddlepoints) in the integrand, depending on real parameters {x1, x2, ..., xK}. They form a natural hierarchy of special functions with the most famous defined by K=2 being the Airy function that decays exponentially at +infinity, and K=3 the Pearcey function. A consequence of this hierarchy is that large-K functions can be expressed asymptotically in terms of those with lower K-values.
         The second type of canonical integrals studied are the elliptic and hyperbolic umbilics, which take the form of double integrals and are used to model more complicated diffraction phenomena.
         In this talk we shall describe analytical properties of these classes of functions and demonstrate methods to calculate such functions to high numerical accuracies.

Abstract of Ismail's lecture
         The Bethe Asatz equations are nonlinear algebraic equations satisfied by the eigenvalues of a physical system. Stieltjes solved these equations for the Coulomb gas model. This work is also connected to earlier work of Heine who counted the number of polynomial solutions to second order differential equations with polynomial coefficients. q-Analogues of these results will be described and I will show the connection with Bethe Ansatz equations for the XXX and XXZ models. In doing so one needs to develop a new theory of singularities of second order equations in the Askey-Wilson operators.
         This is based on joint work with S. S. Lin and S.S. Roan from Taipei.

Abstract of Koornwinder's lecture
I will present some items from my miscellaneous (but relevant) work on special functions. These will include, among others:
  • New examples of Paule's creative symmetrizing in connection with Zeilberger's algorithm;
  • Higher derivatives of the Gauss hypergeometric functions rewritten as iterated derivatives, and a fractional generalization.

Abstract of Maier's lecture
         The theory of quadratic and cubic hypergeometric transformations was systematized by Goursat, who built on an earlier insight of Riemann. A hypergeometric equation is a second-order Fuchsian equation on the Riemann sphere with no more than three singular points, and a rational transformation between two such equations is possible only if it takes singular points to singular points, and appropriately transforms the characteristic exponents of each singular point, i.e., the local monodromy.
         The Heun equation (including its special case the Lamé equation) is a Fuchsian equation with four rather than three singular points. The coefficients of its series solutions, which may be called Heun series, satisfy three-term rather than two-term recurrence relations. Also, these solutions depend on an accessory parameter, besides the characteristic exponents. So the Heun equation and its solutions are hard to handle. We shall explain how, nonetheless, the Heun equation can be transformed to the hypergeometric equation by a rational change of its independent variable, so long as its parameters satisfy any of a family of equations. On the level of series, this is equivalent to the existence of a family of Heun-to-hypergeometric identities. The possible rational transformations are reminiscent of the morphisms that occur in Klein's theory of pullbacks of the hypergeometric equation, but Klein's theory is specific to hypergeometric equations that have a full set of algebraic solutions, and the new Heun-to-hypergeometric transformations are valid even in the absence of algebraicity.

Abstract of Miller's lecture
         We outline the basic ideas relating to the notion of superintegrable potentials and how they are related to separability in multiple coordinate systems. We give examples and indicate how superintegrability can be of use, particularly in relation to resolving the degeneracy problem for multiply degenerate bound states. Virtually all of the classical special functions of mathematical physics (in one and several variables) arise in this study, and formulas expanding one type of special function as a series in another type emerge as a byproduct. We describe how one can, in principle, classify all such systems, through the use of symbolic algebra programs. This is joint work with E.G. Kalnins, J. Kress and G. Pogosyan.

Abstract of Olde Daalhuis' lecture
         In this talk we will present asymptotic expansions for the Gauss hypergeometric function F(a+e1 p,b+e2 p;c+e3 p;-z), as |p| goes to infinity, where ej are elements of {-1,0,1}. The expansions hold for fixed values of a, b, c, and are uniformly valid for z in the domain |ph z|<pi.

Abstract of Petkovsek's lecture
         Wilf and Zeilberger conjectured in 1992 that a hypergeometric term is proper-hypergeometric if and only if it is holonomic. Their conjecture concerns hypergeometric terms which depend on several discrete and continuous variables. Jointly with S.A. Abramov, we have proved a slightly modified version of their conjecture in the multivariate discrete case, namely that every holonomic hypergeometric term is conjugate to a proper term (meaning that they have the same certificates). This modification is necessary as shown, e.g., by the bivariate hypergeometric term T(n,k) = |n-k| which is holonomic but not proper. Our proof is based on the Ore-Sato Theorem which states essentially that for every hypergeometric term T there is a rational function R and a proper term T' such that T and RT' have the same certificates. This was proved in the bivariate case by Ore using elementary means, and in the multivariate case by Sato using homological algebra. We give an elementary proof of the multivariate Ore-Sato Theorem. The necessary tools that are useful also for other purposes are normal forms of rational functions and several notions of shift-invariance for multivariate polynomials. We have also shown that a rational sequence is holonomic if and only if its denominator factors into integer-linear factors.

Abstract of Schlosser's lecture
         We present a new matrix inverse with applications in the theory of bilateral basic hypergeometric series. Our matrix inversion result is directly extracted from an instance of Bailey's very-well-poised 6psi6 summation theorem, and involves two infinite matrices which are not lower-triangular. We combine our bilateral matrix inverse with known basic hypergeometric summation theorems to derive, via inverse relations, several new identities for bilateral basic hypergeometric series.

Abstract of Spiridonov's lecture
         An elliptic extension of the WP-Bailey chain of Andrews is constructed. This leads to an infinite sequence of identities for very-well-poised elliptic hypergeometric series. In particular, a new proof of Frenkel-Turaev's elliptic analogue of the Bailey's identity is obtained. Elliptic extention of a Bressoud's Bailey pair results in a new elliptic hypergeometric series identity representing a generalization of one of the identities for very-well-poised basic hypergeometric series derived recently by Andrews and Berkovich.

Abstract of Temme's lecture
         A new method will be presented for obtaining convergent expansions for Bernoulli, Charlier, Laguerre and Jacobi polynomials (and perhaps other special functions). The expansions have an asymptotic character for large values of the degree of the polynomials. The expansions are obtained from Cauchy-type integrals, that follow from the generating functions. For some of the results we need two-point Taylor expansions for analytic functions. A few properties of these expansions will be discussed as well. This research is done together with José L. Lopez (Pamplona, Spain).

Abstract of Terwilliger's lecture
         Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean a pair of linear transformations A : V -> V and B : V -> V which satisfy conditions (i), (ii) below:
         (i)   There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal.
         (ii)   There exists a basis for V with respect to which the matrix representing B is irreducible tridiagonal and the matrix representing A is diagonal.
         (A tridiagonal matrix is said to be irreducible whenever each entry on the superdiagonal or subdiagonal is nonzero.) The Leonard pairs are closely related to the q-Racah polynomials and related polynomials in the Askey Scheme. Roughly speaking, it is appropriate to think of a Leonard pair as a sequence of q-Racah polynomials in disguise. In this talk we discuss several "canonical forms" for Leonard pairs. We call these the TD-D canonical form and the LB-UB canonical form. In the TD-D canonical form, the Leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix, subject to a certain normalization. In the LB-UB canonical form, the Leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix, subject to a certain normalization. We show each Leonard pair is isomorphic to a unique Leonard pair which is in TD-D canonical form and a unique Leonard pair which is in LB-UB canonical form. We give a detailed description of each of these forms.
References
         [1]   T. Ito, K. Tanabe and P. Terwilliger, Some algebra related to P- and Q-polynomial association schemes, in Codes and Association Schemes (Piscataway NJ, 1999), Amer. Math. Soc., 2000.
         [2]   P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear Algebra Appl. 330 (2001), 149-203.
         [3]   P. Terwilliger, Two relations that generalize the q-Serre relations and the Dolan-Grady relations, in Proceedings of Nagoya 1999 Workshop on Physics and Combinatorics (Nagoya, Japan 1999), World Scientific, 2000.
         [4]   P. Terwilliger, Leonard pairs from 24 points of view, accepted by Rocky Mountain J. (special issuse, Special functions, Tempe, Arizona, 2000).



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