$q$-Classical orthogonal polynomials: A general difference calculus approach

Abstract

It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a more general context 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 are deduced. A more general characterization Theorem that the one given in [1] 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. [1] R. Alvarez-Nodarse. On characterization of classical polynomials. J. Comput. Appl. Math., 196:320{337, 2006. [2] M. Alfaro and R. Alvarez-Nodarse. A characterization of the classical orthogonal discrete and q-polynomials. J. Comput. Appl. Math., 2006. In press.

18 pages

References in corpus (2)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• The structure relation for Askey-Wilson polynomials

Cited by in corpus (6)

• Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy
• A Survey on q-Polynomials and their Orthogonality Properties
• Structure Relations of Classical Orthogonal Polynomials in the Quadratic and $q$-Quadratic Variable
• The complementary polynomials and the Rodrigues operator. A distributional study
• The Laguerre constellation of classical orthogonal Polynomials
• Characterizations of classical orthogonal polynomials on quadratic lattices