**Old and new results on Sobolev and semi-classical orthogonal polynomials**

**Date: **
2011..08..31

**Event: **
11Th Orthogonal Polynomials, Special Functions and Applications

**Venue: **
Universidad Carlos III de Madrid. Leganes, Spain.

## Abstract

We present on the one hand algebraic and differential/difference properties for semiclassical-Sobolev polynomial, which are orthogonal with respect to the inner product \[ \left\langle p, r \right\rangle _S=\left\langle {{\bf u}} ,{p\, r}\right \rangle +\lambda \left \langle {{\bf u}}, {{\mathscr D}p \,{\mathscr D}r}\right\rangle, \] where \(\bf u\) is a semiclassical linear functional, \(\mathscr D\) is the differential (or the difference or the \(q\)-difference, respectively) operator, and \(\lambda\) is a positive constant; as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional \(\bf u\). And on the other hand we present a degenerate version of the Favard's Theorem which is valid for all sequences of polynomials satisfying a three-term recurrence relation \[ xp_n=\alpha_n p_{n+1}+\beta p_n+\gamma_n p_{n-1}, \] even when some coefficient \(\gamma_n\) vanishes, i.e., the set \(\{n:\gamma_n=0\}\ne \emptyset\).## Download

Link | Size | Description |
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T10_OPSFA11.pdf | 660 KB | Slides (PDF, 34 pages, 24 slides) |