Lecture 1. The Askey-Wilson Transform Scheme
Askey's scheme of hypergeometric orthogonal polynomials is the collection of orthogonal polynomials for which we have explicit expressions together with all its interconnections and limit transitions. A typical entry is the set of Jacobi polynomials. But there are also connections to well known function transforms, such as the Hankel transform, which can be obtained as a limit case of the Jacobi polynomials, and to the Fourier-cosine transform, Mehler-Fock transform and its generalisation to the Jacobi function transform. We discuss a far-reaching q-analogue of this scheme having the Askey-Wilson function transform on top. We discuss some limiting cases including several kinds of q-Hankel transforms and some related indeterminate moment problems.
Lecture 2. Special Functions and Dynamical Quantum Groups
In the last decade it has become clear that special functions of basic hypergeometric type can be associated to quantum groups. The notion of quantum groups, in the sense of a deformation of the function algebra on a group, has recently been extended to that of a dynamical quantum group by Etingof and Varchenko. The dynamical quantum groups are obtained from solutions to the dynamical Yang-Baxter equation. In this lecture we discuss how Askey-Wilson polynomials can be related to one of the simplest examples, the dynamical quantum SL(2) group obtained from the trigonometric solution to the dynamical Yang-Baxter equation. Some direct applications for special functions and future directions will be discussed.
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