The semiclassical--Sobolev orthogonal polynomials: a general approach

TL;DR

This paper develops a unified algebraic and differential/difference framework for semiclassical Sobolev orthogonal polynomials, encompassing various operators and introducing new polynomial families.

Contribution

It provides a general approach to study semiclassical Sobolev orthogonal polynomials with different operators, including new polynomial families and their properties.

Findings

Derived algebraic and differential/difference properties of these polynomials

Established relations with classical semiclassical orthogonal polynomials

Introduced new families of Sobolev orthogonal polynomials

Abstract

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ <p, r>_S=<{{\bf u}} ,{p\, r}> +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr D}r}>, $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time.