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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.



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

Source: https://arxiv.org/



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