Francisco Marcellán

Classical orthogonal polynomials: A functional approach

We characterize the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) using the distributional differential equation D(φu)=ψu. This result is naturally not new. However, other characterizations of classical orthogonal polynomials can be obtained more easily from this approach. Moreover, three new properties are obtained.

Orthogonal polynomials on Sobolev spaces: old and new directions

Abstract During the last years, orthogonal polynomials on Sobolev spaces have attracted considerable attention. Algebraic properties, distribution of their zeros and Fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case. In this paper we present an introductory survey about the subject. The origin of the problems and their development show the interest and the promising future of this field.

RELATIVE ASYMPTOTICS FOR POLYNOMIALS ORTHOGONAL WITH RESPECT TO A DISCRETE SOBOLEV INNER PRODUCT

AbstractWe investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products

On a class of polynomials orthogonal with respect to a discrete Sobolev inner product

\left\langle {h,{\text{ }}g} \right\rangle = \int h g d\mu + \sum {_{j = 1}^m } \sum {_{i = 0}^{N_j } M_{j,i} h^{(i)} (c_j )} g^{(i)} (c_j )

A distributional study of discrete classical orthogonal polynomials

, where μ is a certain type of complex measure on the real line, andcj are complex numbers in the complement of supp(μ). The Sobolev orthogonal polynomials are compared with the orthogonal polynomials corresponding to the measure μ.

Orthogonal polynomials and coherent pairs: the classical case

Abstract Orthogonal polynomials may be fully characterized by the following recurrence relation: Pn(x) = (x − βn-1)Pn-1(x)-γn-1Pn-2(x), with P0(x)=1, P1(x) = x - β0 and γn ≠ 0. Here we study how the structure and the spectrum of these polynomials get modified by a local perturbation in the β and γ parameters of a co-recursive (βk → βk + μ), co-dilated (γk → λγk and co-modified (βk → βk + μ; γk → λγk) nature for an arbitrary (but fixed) kth element (1 ⩽ k). Specifically, Stieltjes functions, differential equations and distributions of zeros as well as representations of the new perturbed polynomials in terms of the old unperturbed ones are given. This type of problems is strongly related to the boundary value problems of finite-difference equations and to the quantum mechanical study of physical many-body systems (atoms, molecules, nuclei and solid state systems).

On a Class of Matrix Orthogonal Polynomials on the Real Line

This paper analyzes polynomials orthogonal with respect to the Sobolev inner product @(Lg) = I f(x)g(x)e(x)dx+~-‘f”‘(c)g”‘(c) iF with I E IR+, c E [R, and p(x) is a weight function. We study this family of orthogonal polynomials, as linked to the polynomials orthogonal with respect to Q(X) and we find the recurrence relation verified by such a family. If the weight Q is semiclassical we obtain a second order differential equation for these polynomials. Finally, an illustrative example is shown.

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

This paper discusses recurrence relations for sequences of polynomials which are orthogonal with respect to the Sobolev inner product defined on the set of polynomials