Askey-Wilson polynomials and the quantum SU(2) group: survey and applications

Abstract

Generalised matrix elements of the irreducible representations of the quantum $SU(2)$ group are defined using certain orthonormal bases of the representation space. The generalised matrix elements are relatively infinitesimal invariant with respect to Lie algebra like elements of the quantised universal enveloping algebra of $sl(2)$. A full proof of the theorem announced by Noumi and Mimachi [Proc. Japan Acad. Sci. {\bf 66}, Ser. A (1990), pp. 146--149] describing the generalised matrix elements in terms of the full four-parameter family of Askey-Wilson polynomials is given. Various known and new applications of this interpretation are presented.