On Another Characterization of Askey-Wilson Polynomials

Abstract

In this paper we show that the only sequences of orthogonal polynomials $(P_n)_{n\ge 0}$ satisfying

$$\begin{aligned} \phi (x){\mathcal {D}}_q P_{n}(x)=a_n{\mathcal {S}}_q P_{n+1}(x) +b_n{\mathcal {S}}_q P_n(x) +c_n{\mathcal {S}}_q P_{n-1}(x), \end{aligned}$$

($c_n\ne 0$) 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.