On classical orthogonal polynomials related to Hahn's operator

Abstract

Let $\mathbf{u}$ 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 $\varphi$ and $\psi$, with $\mathrm{deg}\varphi \le 2$ and $\mathrm{deg}\psi \le 1$, so that the functional equation $${\mathbf{D}}_{q,\omega }(\varphi \mathbf{u})=\psi \mathbf{u}$$ 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 $\varphi$ and $\psi$, such that $\mathbf{u}$ is regular, that is, there exists a sequence of orthogonal polynomials with respect to $\mathbf{u}$. A key step in the proof relies upon the fact that a distributional Rodrigues-type formula holds without assuming that $\mathbf{u}$ is regular.