Orthogonal Polynomials on the Unit Ball and Fourth-Order Partial Differential Equations

The purpose of this work is to analyse a family of mutually orthogonal polynomials on the unit ball with respect to an inner product which includes an additional term on the sphere. First, we will get connection formulas relating classical multivariate orthogonal polynomials on the ball with our family of orthogonal polynomials. Then, using the representation of these polynomials in terms of spherical harmonics, algebraic and differential properties will be deduced

Multivariate Orthogonal Polynomials and Modified Moment Functionals

Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so-called Uvarov and Christoffel modifications obtained by adding to the moment functional a finite set of mass points, or by multiplying it times a polynomial of total degree 2, respectively. Orthogonal polynomials associated with modified moment functionals will be studied, as well as the impact of the modification in useful properties of the orthogonal polynomials. Finally, some illustrative examples will be given

Perturbations in the Nevai matrix class of orthogonal matrix polynomials

24 pages, no figures.-- MSC2000 codes: 15A54, 15A21, 42C05.MR#: MR1855403 (2002i:42037)Zbl#: Zbl 0992.15022In this paper we study a Jacobi block matrix and the behavior of the limit of its entries when a perturbation of its spectral matrix measure by the addition of a Dirac delta matrix measure is introduced.The work of the second author was supported by Dirección General de Enseñanza Superior (DGES) of Spain under grant PB96-0120-CO3-01 and INTAS Project INTAS-93-0219 Ext, and the work of the third author was supported by DGES under grant PB 95-1205, INTAS-93-0219-ext and Junta de Andalucía, Grupo de Investigación FQM 229.Publicad

What is beyond coherent pairs of orthogonal polynomials?

AbstractUsually, coherent pairs of orthogonal polynomials have been considered in the wider context of Sobolev orthogonality. In this paper, we focus our attention on the problem of coherence between two orthogonal polynomial sequences in terms of the corresponding linear functionals. We deduce some conditions about the linear functionals in order that the corresponding orthogonal polynomial sequences constitute a coherent pair

An asymptotic result for Laguerre-Sobolev orthogonal polynomials

AbstractLet {Sn} denote the sequence of polynomials orthogonal with respect to the Sobolev inner product (f,g)s = ∫0+∞ f(x)g(x)xαe−xdx+λ∫0+∞ f′(x)g′(x)xαe−xdx where α > − 1, λ > 0 and the leading coefficient of the Sn is equal to the leading coefficient of the Laguerre polynomial Ln(α). Then, if x∈Cß[0,+∞), limn→∞Sn(x)Ln(α−1)(x) is a constant depending on λ

Lax-type pairs in the theory of bivariate orthogonal polynomials

Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyse a Lax-type pair system for the coefficients of the three term relations. We also deduce several characterizations relating the Lax-type pair, the shape of the weight, Stieltjes function, moments, a differential equation for the weight, and the bidimensional Toda-type systems