Bispectral extensions of the Askey–Wilson polynomials

Following the pioneering work of Duistermaat and Grünbaum, we call a family {pn(x)}n=0∞ of polynomials bispectral, if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators: one consisting of difference operators acting on the degree index n, and another one of operators acting on the variable x. The goal of the present paper is to construct and parametrize bispectral extensions of the Askey–Wilson polynomials, where the second algebra consists of q-difference operators. In particular, we describe explicitly measures on the real line for which the corresponding orthogonal polynomials satisfy (higher-order) q-difference equations extending all known families of orthogonal polynomials satisfying q-difference, difference or differential equations in x.