## The Semiclassical-Sobolev orthogonal polynomials: A general approach

Costas-Santos, R. S. and Moreno Balcazar, J. J.**Journal of Approximation Theory**163, no. 1 (2011), 65 — 83

## Abstract

We say the polynomial sequence, (Qn(x; λ)), is a semiclassical-Sobolev polynomial sequence when it is orthogonal with respect to the inner product (p,r) = ⟨**u**,pr⟩ + λ⟨**u**,*D*p*D*r⟩, where **u** is a semiclassical linear functional, *D* is the differential (or the difference or the *q*-difference, respectively) operator, and λ 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 **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 *D* considered. Finally, we illustrate our results 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.

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## BibTeX

@article {MR2741220, AUTHOR={Costas-Santos, R.S. and Moreno-Balcázar, J.J.}, TITLE={The semiclassical Sobolev orthogonal polynomials: a general approach}, JOURNAL={J. Approx. Theory}, FJOURNAL={Journal of Approximation theory}, VOLUME={163}, NUMBER={1}, PAGES={65--83}, YEAR={2011}, ISSN={0167-8019}, DOI={10.1016/j.jat.2010.03.001}, MRCLASS={33C45 (42C05)}, MRNUMBER={MR2741220}, ZBL={1215.33005}, MRREVIEWER={Miguel A. Pi\~{n}ar}, DOI={10.1016/j.jat.2010.03.001}, URL={https://doi.org/10.1016/j.jat.2010.03.001}, }