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Bispectral extensions of the Askey-Wilson polynomials

Following the pioneering work of Duistermaat and Grünbaum, we call a family $\{p_n(x)\}_{n=0}^{\infty}$ 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 $x$.