Polinomios biortogonales y sus generalizacionesuna perspectiva desde los sistemas integrables

Abstract

The existing connection between the theory of orthogonal polynomials and other branches of mathematics, physics and engineering is truly astonishing. There is no better proof of the usefulness of the theory than the recognition of its constant development and the wide generalizations that the original meaning of orthogonal polynomial has experienced since the dawn of the theory. The original concepts were generalized at the same time as the techniques for their study. Many of these new techniques were suggested by the new connections that kept appearing with di erent branches of mathematics. The approach that this thesis presents towards the study of the orthogonal polynomials is an example of such an interrelationship among disciplines, sharing tools and ideas with the theory of integrable systems. A privileged role throughout this thesis will be played by the notion of semi in nite Gram matrices. These will be associated to a sesquilinear form suited to the kind of orthogonality under study. Additionally, some conditions will be imposed on the Gram matrix with the aim of guaranteeing the existence and uniqueness of the associated biorthogonal sequences. The following step consists of searching for any symmetry that the Gram matrix may have. There are two main reasons why such a task is worth the e ort. In the rst place, each found symmetry can be translated into a property of the biorthogonal sequences, for example: The Hankel structure of the matrix is equivalent to the well known three term recurrence relation satis ed by the standard orthogonal polynomials; the symmetry that the classical (Hermite, Laguerre, Jacobi) matrices possess induces the existence of the second order linear di erential operator of which the classical orthogonal polynomials are solutions; etc. In the second place, the matrices that codify these kind of symmetries also help to surmise possible deformations of the problem, this is, they suggest wise perturbations of the Gram matrix...