Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

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.

References in corpus (2)

• Bispectrality of multivariable Racah-Wilson polynomials
• Discrete orthogonal polynomials and difference equations of several variables

Cited by in corpus (20)

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• The generic quantum superintegrable system on the sphere and Racah operators
• Higher Rank Relations for the Askey-Wilson and $q$-Bannai-Ito Algebra
• Orthogonal Polynomials in Mathematical Physics
• Orthogonality of Macdonald polynomials with unitary parameters
• On $6j$-symbols for symmetric representations of $U_q({\mathfrak{su}}_N)$
• Meixner polynomials in several variables satisfying bispectral difference equations
• Bispectrality and biorthogonality of the rational functions of $q$-Hahn type
• The $q$-Bannai-Ito algebra and multivariate $(-q)$-Racah and Bannai-Ito polynomials
• Coupling coefficients of $s{u}_q(1,1)$ and multivariate $q$-Racah polynomials
• The $q-$Onsager algebra and multivariable $q-$special functions
• A quantum algebra approach to multivariate Askey-Wilson polynomials
• $q$-Rotations and Krawtchouk polynomials
• The multivariate Hahn polynomials and the singular oscillator
• A bispectral q-hypergeometric basis for a class of quantum integrable models
• Bivariate Bannai-Ito polynomials
• $3nj$-symbols and identities for $q$-Bessel functions