Divided-difference operators from the geometric point of view

It is presented a study of general divided-difference operators having the fundamental property of leaving a polynomial of degree n−1 when applied to a polynomial of degree n.info:eu-repo/semantics/updatedVersio

Laguerre-Hahn orthogonal polynomials on the real line

A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coeffcients - the so-called Laguerre-Hahn class. The main goal is to describe analytical aspects, focusing on differential equations for those orthogonal polynomials, difference and differential equations for the recurrence coeffcients, and distributional equations for the corresponding linear functionals.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

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

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

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

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

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: it is established that if ([mu]0,[mu]1) is a coherent pair of measures on the unit circle, then [mu]0 is a semi-classical measure. Moreover, we obtain that the linear functional associated with [mu]1 is a specific rational transformation of the linear functional corresponding to [mu]0. Some examples are given.http://www.sciencedirect.com/science/article/B6WH7-4S2MJ0D-1/1/61050bb3811832f373ff40a48b7461d

Zeros of para-orthogonal polynomials and linear spectral transformations on the unit circle

We study the interlacing properties of zeros of para-orthogonal polynomials associated with a nontrivial probability measure supported on the unit circle d mu and para-orthogonal polynomials associated with a modification of d mu by the addition of a pure mass point, also called Uvarov transformation. Moreover, as a direct consequence of our approach, we present some results related with the Christoffel transformation.The authors thank the referees for their comments and suggestions. This work is partially supported by the CMUC, funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the Fundacao para a Ciência e a Tecnologia (FCT) under the project PEst-C/MAT/UI0324/2013. The research of the first author is supported by the Portuguese Government through the FCT under the grant SFRH/BPD/101139/2014. This author also acknowledges the financial support by the Brazilian Government through the CNPq under the project 470019/2013-1. The research of the first and second author is supported by the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012–36732–C03–01. The second author also acknowledges the financial support by the Brazilian Government through the CAPES under the project 107/2012

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. [...]info:eu-repo/semantics/publishedVersio