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# Talk: Old and new results on Sobolev and semi-classical orthogonal polynomials

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

Date: 2011..08..31
Event: 11Th Orthogonal Polynomials, Special Functions and Applications
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 $$\{n:\gamma_n=0\}\ne \emptyset$$.