Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

TL;DR

This paper constructs two dual commutative algebras of q-difference operators diagonalized by multivariable Askey-Wilson polynomials, revealing a bispectral structure and duality between variables and indices.

Contribution

It introduces a dual algebraic framework for multivariable Askey-Wilson polynomials, extending bispectral theory to a multivariable q-Askey scheme.

Findings

Construction of algebra A_z diagonalized by polynomials

Dual algebra A_n acting on discrete variables

Establishment of a multivariable bispectral orthogonal polynomial scheme

Abstract

We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper and Rahman [6]. Iterating Sears' transformation formula, we show that the polynomials P_n(z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra A_n, generated by d algebraically independent difference operators acting on the discrete variables n_1, n_2,..., n_d, which is also diagonalized by P_n(z). This leads to a multivariable q-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.