# The semiclassical-Sobolev orthogonal polynomials: a general approach - Mathematics > Classical Analysis and ODEs

The semiclassical-Sobolev orthogonal polynomials: a general approach - Mathematics > Classical Analysis and ODEs - Descarga este documento en PDF. Documentación en PDF para descargar gratis. Disponible también para leer online.

Abstract: We say that the polynomial sequence $Q^{\lambda} n$ is a semiclassicalSobolev polynomial sequence when it is orthogonal with respect to the innerproduct $$S=<{{\bf u}} ,{p\, r}> +\lambda <{{\bf u}}, {{\mathscr D}p\,{\mathscr D}r}>,$$ where ${\bf u}$ is a semiclassical linear functional,${\mathscr D}$ is the differential, the difference or the $q$-differenceoperator, and $\lambda$ is a positive constant. In this paper we get algebraicand differential-difference properties for such polynomials as well asalgebraic relations between them and the polynomial sequence orthogonal withrespect to the semiclassical functional $\bf u$. The main goal of this articleis to give a general approach to the study of the polynomials orthogonal withrespect to the above nonstandard inner product regardless of the type ofoperator ${\mathscr D}$ considered. Finally, we illustrate our results byapplying them to some known families of Sobolev orthogonal polynomials as wellas to some new ones introduced in this paper for the first time.

Autor: R.S. Costas-Santos, J.J. Moreno-Balcázar

Fuente: https://arxiv.org/