On classical orthogonal polynomials related to Hahn's operator

TL;DR

This paper characterizes when certain linear functionals related to Hahn's operator produce orthogonal polynomial sequences, using polynomial coefficients and a Rodrigues-type formula without assuming regularity.

Contribution

It provides necessary and sufficient conditions for the regularity of functionals associated with Hahn's operator, based solely on polynomial coefficients.

Findings

Conditions involving polynomial coefficients determine regularity.

A Rodrigues-type formula holds without assuming regularity.

Characterization of orthogonal polynomials related to Hahn's operator.

Abstract

Let ${\bf u}$ be a nonzero linear functional acting on the space of polynomials. Let $\mathbf{D}_{q,\omega}$ be a Hahn operator acting on the dual space of polynomials. Suppose that there exist polynomials $\phi$ and $\psi$, with $\mathrm{deg}\,\phi\leq2$ and $\mathrm{deg}\,\psi\leq1$, so that the functional equation $$ \mathbf{D}_{q,\omega}(\phi {\bf u})=\psi{\bf u} $$ holds, where the involved operations are defined in a distributional sense. In this note we state necessary and sufficient conditions, involving only the coefficients of $\phi$ and $\psi$, such that ${\bf u}$ is regular, that is, there exists a sequence of orthogonal polynomials with respect to ${\bf u}$. A key step in the proof relies upon the fact that a distributional Rodrigues-type formula holds without assuming that ${\bf u}$ is regular.