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

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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.