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**Lecture 1. The Askey-Wilson Transform Scheme**

**Abstract**

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**

**Abstract**

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.