A Quantum Algebra Approach to Multivariate Askey-Wilson Polynomials

Abstract

We study matrix elements of a change of basis between two different bases of representations of the quantum algebra U q(su(1, 1)). The two bases, which are multivariate versions of Al-Salam Chihara polynomials, are eigenfunctions of iterated coproducts of twisted primitive elements. The matrix elements are identified with Gasper and Rahman s multivariate Askey Wilson polynomials, and from this interpretation we derive their orthogonality relations. Furthermore, the matrix elements are shown to be eigenfunctions of the twisted primitive elements after a change of representation, which gives a quantum algebraic derivation of the fact that the multivariate Askey Wilson polynomials are solutions of a multivariate bispectral q-difference problem.