Mathematical Sciences: q-Orthogonal Polynomials and Special Functions

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ABSTRACT Proposal: DMS-9625459 PI: Ismail Ismail plans to investigate certain indeterminate moment problems and multivariate analogues of the Askey-Wilson operators and polynomials. The Askey-Wilson operators act very nicely on theta functions, so it is expected that the multivariate operators (the Dunkl operators) will act nicely on higher dimensional theta functions. He also plan to study integral operators whose kernels are Poisson and related kernels of q-orthogonal polynomials, their adjoints, spectral properties, inversion formulas as well as transform properties. In addition, Ismail will spend some time on the management of the Askey-Bateman Project. This project of this proposal concerns theoretical and computational problems involving areas of mathematics called orthogonal polynomials and special functions. It has potential applications to random matrix models for conduction and insulation in solid state physics, as was demonstrated in earlier published research of Ismail in collaboration with physicists from the University of Florida and Imperial College of London. It also deals with asymptotics and bulk scaling techniques which approximate efficiently complex systems by picking up the bulk contribution of the states of the system with knowing the explicit solution. This is particularly useful because in many cases one cannot write down the solution explicitly. Ismail's work deals with one and several variables so it corresponds to systems with one or several components. He also plan to investigate inversion problems of integral transforms, which tell us the input needed to reach any specific output.