Orthogonal polynomials with respect to differential operators and matrix orthogonal polynomials

Resumen

This thesis deals with the concept of orthogonal polynomials with respect to a differential operator, the study of the strong asymptotic behavior of eigenpolynomials of exactly solvable operators, and matrix orthogonal polynomials. We consider orthogonal polynomials with respect to either a Jacobi, Laguerre or Hermite operator and a finite positive Borel measure $\mu$ satisfying certain conditions. For a positive integer $m$, we analyze the conditions over the measure $\mu$ in order to guarantee the existence of an infinite sequence of monic polynomials $\{Q_n\}_{n=m+1}^{\infty}$, where each $Q_n$ has degree $n$ and orthogonal with respect to the operator. We consider algebraic and analytic properties of this sequence. A fluid dynamics model for the interpretation of the zeros of these polynomials is also considered. Some of the results obtained for a classical operator are generalized by considering orthogonal polynomials with respect to a wider class of linear differential operators. We analyze the uniqueness and zero location of these polynomials. An interesting phenomena occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator we also find a classification, in terms of a system of difference equations with varying coefficients, of the measures for which it is possible to guarantee the existence of an infinite sequence of orthogonal polynomials. We also obtain a curve which contains the set of accumulation points of the zeros of these polynomials for the case of a first order differential operator giving also the strong asymptotic behavior. We consider as well the study of the strong asymptotic behavior the eigenpolynomials of exactly solvable operators. Under the assumption that the leading coefficient of the operator is a real polynomial, we obtain a formula for the strong asymptotic behavior of the eigenpolynomials on certain compact subsets of the complex plane. As an application, we study the strong asymptotic behavior of a sequence of monic orthogonal polynomials with respect to a Sobolev inner product which are eigenfunctions of a fourth order differential operator. It is also object of study a new class of matrix orthogonal polynomials of arbitrary size satisfying a second order matrix differential equation. For matrix polynomials of size 2, we find an explicit expression of the sequence of orthonormal polynomials with respect to a weight by using a Rodrigues' formula for these polynomials. In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature.