Some Appell-type orthogonal polynomials on lattices

Abstract

We investigate some Appell-type orthogonal polynomial sequences on q-quadratic lattices and we provide some entirely new characterizations of some special cases of the Al-Salam–Chihara polynomials (including the Rogers q-Hermite polynomials). The corresponding regular forms are well described. We also show that the Rogers q-Hermite polynomials constitute a nice orthogonal polynomial base to use when dealing with problems related with the Askey-Wilson and the averaging operators. The proposed method can be applied to similar and to more general problems involving the Askey-Wilson and the Averaging operators, in order to obtain new characterization theorems for classical and semiclassical orthogonal polynomials on lattices.