Q-classical Orthogonal Polynomials: a General Difference Calculus Approach

It is well known that the classical families of orthogonal polynomials are characterized as the polynomial eigenfunctions of a second order homogenous linear differential/difference hypergeometric operator with polynomial coefficients. In this paper we present a study of the classical orthogonal polynomials sequences, in short classical OPS, in a more general framework by using the differential (or difference) calculus and Operator Theory. The Hahn's Theorem and a characterization Theorem for the q-polynomials which belongs to the q-Askey and Hahn tableaux are proved. Finally, we illustrate our results applying them to some known families of orthogonal q-polynomials.