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The semiclassical Sobolev orthogonal polynomials: a general approach. (English) Zbl 1215.33005
A linear functional $${\mathbf u}$$ (defined on the space of polynomials $$\mathbb P)$$ is called a semiclassical functional if it satisfies a distributional equation with polynomial coefficients:
${\mathcal D}(\varphi{\mathbf u})=\psi{\mathbf u},$ where $${\mathcal D}$$ is the differential, or the difference or the $$q$$-difference operator. The polynomial sequence $$(Q_n^{(\lambda)})$$ is called a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product
$\langle p,r\rangle _S=\langle{\mathbf u},pr\rangle+\lambda\langle{\mathbf u},{\mathcal D} p{\mathcal D} r\rangle.$ The authors 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 $${\mathbf u}$$. In particular, a linear operator $${\mathcal J}$$ such that
$\langle{\mathcal J}p, r\rangle_S=\langle p,{\mathcal J}r\rangle_S,\quad p,r\in\mathbb P,$ is constructed.
The main goal of the 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 $${\mathcal D}$$ considered. Finally, the results are illustrated by applying them to some known families of Sobolev orthogonal polynomials (Jacobi-Sobolev and $$\Delta$$-Meixner-Sobolev ones) as well as to some new ones introduced for the first time, such as $$q$$-Freud-Sobolev ones and another family related to a 1-singular semiclassical functional considered by Medem.

##### MSC:
 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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