A quantum algebra approach to multivariate Askey-Wilson polynomials

Abstract

We study matrix elements of a change of base 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.

26 pages

References in corpus (2)

• Coupling coefficients of $s{u}_q(1,1)$ and multivariate $q$-Racah polynomials
• $3nj$-symbols and identities for $q$-Bessel functions

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