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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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