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

Abstract

We consider a general discrete Sobolev inner product involving the Hahn difference operator, so this includes the well--known difference operators ${\mathcal{D}}_q$ 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.

Preprint of the version published in Journal of Difference Equations and Applications with the title "Second--order difference equation for Sobolev--type orthogonal polynomials: Part I: theoretical results"