8 Lectures on quantum groups and q-special functions

Abstract

Lecture notes for an eight hour course on quantum groups and $q$-special functions at the fourth Summer School in Differential Equations and Related Areas, Universidad Nacional de Colombia and Universidad de los Andes, Bogot\'a, Colombia, July 22 -- August 2, 1996. The lecture notes contain an introduction to quantum groups, $q$-special functions and their interplay. After generalities on Hopf algebras, orthogonal polynomials and basic hypergeometric series we work out the relation between the quantum SU(2) group and the Askey-Wilson polynomials out in detail as the main example. As an application we derive an addition formula for a two-parameter subfamily of Askey-Wilson polynomials. A relation between the Al-Salam and Chihara polynomials and the quantised universal enveloping algebra for $su(1,1)$ is given. Finally, more examples and other approaches as well as some open problems are given.