On another characterization of Askey-Wilson polynomials

Abstract: In this paper we show that the only sequences of orthogonal polynomials $(P_n){n\geq 0}$ satisfying \begin{align*} \phi(x)\mathcal{D}_q P{n}(x)=a_n\mathcal{S}q P{n+1}(x) +b_n\mathcal{S}q P_n(x) +c_n\mathcal{S}_q P{n-1}(x), \end{align*} ($c_n\neq 0$) where $\phi$ is a well chosen polynomial of degree at most two, $\mathcal{D}_q$ is the Askey-Wilson operator and $\mathcal{S}_q$ the averaging operator, are the multiple of Askey-Wilson polynomials, or specific or limiting cases of them.