Askey–Wilson Polynomials as Zonal Spherical Functions on the ${\operatorname{SU}}(2)$ Quantum Group

Abstract

On the $SU(2)$ quantum group the notion of (zonal) spherical element is generalized by considering left and right invariance in the infinitesimal sense with respect to twisted primitive elements of the $sl(2)$ quantized universal enveloping algebra. The resulting spherical elements belonging to irreducible representations of quantum $SU(2)$ turn out to be expressible as a two-parameter family of Askey–Wilson polynomials. For a related basis change of the representation space a matrix of dual q-Krawtchouk polynomials is obtained. Big and little q-Jacobi polynomials are obtained as limits of Askey–Wilson polynomials.