Weighted Sobolev orthogonal polynomials on the unit ball

For the weight function $W_mu(x) = (1-|x|^2)^mu$, $mu > -1$, $lambda > 0$ and $b_mu$ a normalizing constant, a family of mutually orthogonal polynomials on the unit ball with respect to the inner product $$ la f,g ra = {b_mu [int_{BB^d} f(x) g(x) W_mu(x) dx + lambda int_{BB^d} abla f(x) cdot abla g(x) W_mu(x) dx]} $$ are constructed in terms of spherical harmonics and a sequence of Sobolev orthog onal polynomials of one variable. The latter ones, hence, the orthogonal polynomials with respect to $la cdot,cdotra$, can be generated through a recursive formula.

Krall--type Orthogonal Polynomials in several variables

For a bilinear form obtained by adding a Dirac mass to a positive definite moment functional in several variables, explicit formulas of orthogonal polynomials are derived from the orthogonal polynomials associated with the moment functional. Explicit formula for the reproducing kernel is also derived and used to establish certain inequalities for classical orthogonal polynomials.

A generating function for non-standard orthogonal polynomials involving differences: the Meixner case

In this paper we deal with a family of non--standard polynomials orthogonal with respect to an inner product involving differences. This type of inner product is the so--called $Delta$--Sobolev inner product. Concretely, we consider the case in which both measures appearing in the inner product correspond to the Pascal distribution (the orthogonal polynomials associated to this distribution are known as Meixner polynomials). The aim of this work is to obtain a generating function for the $Delta$--Meixner--Sobolev orthogonal polynomials and, by using a limit process, recover a generating function for Laguerre--Sobolev orthogonal polynomials.