Remarks on Askey–Wilson polynomials and Meixner polynomials of the second kind

The purpose of this note is to characterize all the sequences of orthogonal polynomials $$(P_n)_{n\ge 0}$$ such that $$\begin{aligned} \frac{\triangle }{\mathbf{\triangle } x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm {I})P_n(x(s-1/2)), \end{aligned}$$ where $$\,\mathrm {I}$$ is the identity operator, x defines a class of lattices with, generally, nonuniform step-size, and $$\triangle f(s)=f(s+1)-f(s)$$ . The proposed method can be applied to similar and to more general problems involving the mentioned operators, in order to obtain new characterization theorems for some specific families of classical orthogonal polynomials on lattices.