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

The purpose of this note is to characterize all the sequences 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 \(\,\mathrm {I}\) identity operator, x defines a class lattices with, generally, nonuniform step-size, and \(\triangle f(s)=f(s+1)-f(s)\). proposed method can be applied similar more general problems involving mentioned operators, in order obtain new characterization theorems for some specific families classical on lattices.