Ladder operators and a second--order difference equation for general discrete Sobolev orthogonal polynomials

TL;DR

This paper develops ladder operators and a second-order difference equation for discrete Sobolev orthogonal polynomials involving the Hahn difference operator, generalizing previous results and providing explicit formulas.

Contribution

It introduces a general framework for discrete Sobolev orthogonal polynomials with the Hahn difference operator, deriving ladder operators and a second-order difference equation.

Findings

Constructed ladder operators for the polynomials.

Derived the second-order difference equation.

Provided explicit formulas for all involved functions.

Abstract

We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well--known difference operators $\mathscr{D}_{q}$ and $\Delta$ and, as a limit case, the derivative operator. The objective is twofold. On the one hand, we construct the ladder operators for the corresponding nonstandard orthogonal polynomials and we obtain the second--order difference equation satisfied by these polynomials. On the other hand, we generalise some related results appeared in the literature as we are working in a more general framework. Moreover, we will show that all the functions involved in these constructions can be computed explicitly.