Bispectral commuting difference operators for multivariable Askey-Wilson polynomials
Abstract
We construct a commutative algebra Az, generated by d algebraically independent q-difference operators acting on variables z1, z2,..., zd, which is diagonalized by the multivariable Askey-Wilson polynomials Pn(z) considered by Gasper and Rahman [6]. Iterating Sears' transformation formula, we show that the polynomials Pn(z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra An, generated by d algebraically independent difference operators acting on the discrete variables n1, n2,..., nd, which is also diagonalized by Pn(z). This leads to a multivariable q-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.
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