q-Classical Orthogonal Polynomials: A General Difference Calculus Approach

Abstract

It is well known that the classical families of orthogonal polynomials are characterized as the polynomial eigenfunctions of a second order homogeneous 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.

On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices

“…This basis allows, for example, to recover from the divided-difference Equation (15), without the need of any further transformation, the defining representation of the Askey-Wilson polynomials given by (22).…”

“…Costas-Santos and Marcellán [15], using the corresponding weight function, gave four equivalent characterizations for classical orthogonal polynomials on nonuniform lattices: More precisely, they proved the equivalence between the second-order divided-difference Equation (8), the orthogonality of the divided-difference derivatives, the Rodrigues-type formula, as well as the structure relation.…”

“…and the fact that these polynomials fulfill (15) suggests the study of the action of the companion operators on the function:…”

Characterizations of classical orthogonal polynomials on quadratic lattices

“…Probably the first results in this direction go back to Bochner [3], Favard [5] and Hahn [9]. Moreover, some recent characterizations can be found in [2,6,7], by using either differential operators as Bochner or linear functionals as introduced by Maroni [16,15]. Recently a new characterization of classical continuous, discrete and their q-analogues was given by Verde-Star [21,22] by using a matrix approach.…”

“…In a recent paper [6] the authors gave a characterization theorem for classical orthogonal polynomials on a lattice as described above by using the Pearson-type equation. Moreover, in [7] and by using the functional approach, the authors stated and proved a characterization theorem for classical orthogonal polynomials on non-uniform lattices including the Askey-Wilson polynomials.…”