Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

Abstract

We construct a commutative algebra A z \mathcal A_{z} , generated by d d algebraically independent q q -difference operators acting on variables z 1 , z 2 , … , z d z_1,z_2,\dots ,z_d , which is diagonalized by the multivariable Askey-Wilson polynomials P n ( z ) P_n(z) considered by Gasper and Rahman (2005). Iterating Sears’ 4 ϕ 3 {}_4\phi _3 transformation formula, we show that the polynomials P n ( z ) P_n(z) possess a certain duality between z z and n n . Analytic continuation allows us to obtain another commutative algebra A n \mathcal A_{n} , generated by d d algebraically independent difference operators acting on the discrete variables n 1 , n 2 , … , n d n_1,n_2,\dots ,n_d , which is also diagonalized by P n ( z ) P_n(z) . This leads to a multivariable q q -Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.