A characterization of continuous $q$-Jacobi, Chebyshev of the first kind and Al-Salam Chihara polynomials

Abstract

The purpose of this note is to characterize those orthogonal polynomials sequences $(P_n{)}_{n\ge 0}$ for which $$\pi (x){\mathcal{D}}_q{P}_n(x)=(a_{n}x+b_n)P_n(x)+c_n{P}_{n-1}(x),n=0,1,2,\dots ,$$ where ${\mathcal{D}}_q$ is the Askey-Wilson operator, $\pi$ is a polynomial of degree at most 2, and $(a_n{)}_{n\ge 0}$, $(b_n{)}_{n\ge 0}$ and $(c_n{)}_{n\ge 0}$ are sequences of complex numbers such that $c_n\ne 0$ for $n=1,2,\dots$.