Bispectrality of multivariable Racah-Wilson polynomials

TL;DR

This paper constructs dual commutative algebras of difference operators for multivariable Racah and Wilson polynomials, revealing their bispectral properties and dualities in a multivariable setting.

Contribution

It introduces a new framework for bispectrality of multivariable Racah-Wilson polynomials via dual commutative algebras of difference operators.

Findings

Established a duality between variables n and x for Racah polynomials.

Constructed bispectral pairs of difference operators for multivariable Racah polynomials.

Extended the bispectral framework to multivariable Wilson polynomials.

Abstract

We construct a commutative algebra A_x of difference operators in R^p, depending on p+3 real parameters which is diagonalized by the multivariable Racah polynomials R_p(n;x) considered by Tratnik [27]. It is shown that for specific values of the variables x=(x_1,x_2,...,x_p) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra A_n in the variables n=(n_1,n_2,...,n_p) which is also diagonalized by R_p(n;x). Thus R_p(n;x) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grunbaum [8]. Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials [26], this change of variables and parameters in A_x and A_n leads to bispectral commutative algebras for the multivariable Wilson polynomials.