Classical Orthogonal Polynomials: a General Difference Calculus Approach

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator with polynomial coefficients. In this paper we present a study of classical orthogonal polynomials in a more general framework by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthermore, some well known results related to the Rodrigues Operator, introduced in Section 3, are deduced. A more general Characterization Theorem than the one given in [4] and [2] for the q-polynomials of the q-Askey and Hahn Tableaux, respectively, is established. Finally, the families of Askey-Wilson polynomials, q-Racah polynomials, Al-Salam & Carlitz I and II, and q-Meixner are considered.