Relative asymptotics for orthogonal matrix polynomials

In this paper we study sequences of matrix polynomials that satisfy a non-symmetric recurrence relation. To study this kind of sequences we use a vector interpretation of the matrix orthogonality. In the context of these sequences of matrix polynomials we introduce the concept of the generalized matrix Nevai class and we give the ratio asymptotics between two consecutive polynomials belonging to this class. We study the generalized matrix Chebyshev polynomials and we deduce its explicit expression as well as we show some illustrative examples. The concept of a Dirac delta functional is introduced. We show how the vector model that includes a Dirac delta functional is a representation of a discrete Sobolev inner product. It also allows to reinterpret such perturbations in the usual matrix Nevai class. Finally, the relative asymptotics between a polynomial in the generalized matrix Nevai class and a polynomial that is orthogonal to a modification of the corresponding matrix measure by the addition of a Dirac delta functional is deduced

On the semiclassical character of orthogonal polynomials satisfying structure relations

We prove the semiclassical character of some sequences of orthogonal polynomials [...]info:eu-repo/semantics/publishedVersio

Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circle

In this paperwe characterize sequences of orthogonal polynomials on the unit circle whose Carathéodory function satisfies a Riccati differential equation with polynomial coefficients, in terms of matrix Sylvester differential equations. For the particular case of semi-classical orthogonal polynomials on the unit circle, it is derived a characterization in terms of first order linear differential systems.info:eu-repo/semantics/publishedVersio

Complex high order Toda and Volterra lattices

Given a solution of a high order Toda lattice we construct a one parameter family of new solutions. In our method, we use a set of B¨acklund transformations in such a way that each new generalized Toda solution is related to a generalized Volterra solution.Dirección General de Investigación, Ministerio de Educación y Ciencia, MTM2006-13000-C03-02; Universidad Politécnica de Madrid; Comunidad Autónoma de Madrid CCG06-UPM/MTM- 539; CMUC/FC

Structure relations for orthogonal polynomials on the unit circle

Structure relations for orthogonal polynomials on the unit circle are studied. We begin by proving that semi-classical orthogonal polynomials on the unit circle satisfy structure relations of the following type: [...]info:eu-repo/semantics/publishedVersio

Distributional equation for Laguerre- Hahn functionals on the unit circle

Let u be a hermitian linear functional defined in the linear space of Laurent polynomials and F its corresponding Carathéodory function. [...]info:eu-repo/semantics/publishedVersio

Characterizations of Laguerre-Hahn affne orthogonal polynomials on the unit circle

In this work we characterize a monic polynomial sequence, orthogonal with respect to a hermitian linear functional [...]info:eu-repo/semantics/publishedVersio

Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials

High-order non symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems

Coherent pairs of linear functionals on the unit circle

In this paper we extend the concept of coherent pairs of measures from the real line to Jordan arcs and curves. We present a characterization of pairs of coherent measures on the unit circle. Some examples are given