Second--order difference equation for Sobolev--type orthogonal polynomials. Part I: theoretical results

Abstract

We consider a general Sobolev--type 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 to construct the ladder operators for the corresponding nonstandard orthogonal polynomials and deduce the second--order differential--difference equation satisfied by these polynomials. Moreover, we will show that all the functions involved in these constructions can be computed explicitly.