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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.

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
Full Text: DOI arXiv
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