Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

Abstract

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 (2005). Iterating Sears' 4 φ 3 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.

A bispectral q-hypergeometric basis for a class of quantum integrable models

“…First, let us recall the definition of the multivariable polynomials introduced by Gasper and Rahman in [1] using the notations of [2]. As multivariable generalizations of the well-known Askey-Wilson polynomials, these polynomials are obtained by applying a GramSchmidt process.…”

“…In Section 2, the orthogonal system of multivariable polynomials of Gasper and Rahman [1] is recalled. Then, we introduce q−difference and difference operators that essentially follow from Iliev's work [2], and are diagonalized by Gasper-Rahman polynomials. Studying the structure of the operators, new infinite dimensional modules of the q−Onsager algebra are exhibited in terms of these polynomials.…”

“…Following [2], we define ν + j = max(ν j , 0) and ν − j = −min(ν j , 0). We denote I ν − the composition of the involutions I j corresponding to the non-zero coordinates of ν − .…”

“…For the class of quantum integrable models generated from the q−Onsager algebra, a basis of bispectral multivariable q−orthogonal polynomials is exhibited. In a first part, it is shown that the multivariable Askey-Wilson polynomials with N variables and N + 3 parameters introduced by Gasper and Rahman [1] generate a family of infinite dimensional modules for the q−Onsager algebra, whose fundamental generators are realized in terms of the multivariable q−difference and difference operators proposed by Iliev [2]. Raising and lowering operators extending those of Sahi [3] are also constructed.…”

Theq-Onsager algebra and multivariableq-special functions

“…q−difference operators [Il08]. Another approach to connect q−special functions to the q−Onsager algeba is considered here.…”