A quantum algebra approach to multivariate Askey-Wilson polynomials

TL;DR

This paper links multivariate Askey-Wilson polynomials to quantum algebra representations, deriving their orthogonality and bispectral properties through algebraic methods involving $U_q(su(1,1))$.

Contribution

It provides a quantum algebraic framework for understanding multivariate Askey-Wilson polynomials and their bispectral properties, which is a novel approach.

Findings

Identifies matrix elements with multivariate Askey-Wilson polynomials.

Derives orthogonality relations algebraically.

Shows these polynomials satisfy a multivariate bispectral $q$-difference equation.

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.