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- Characterization theorem for Laguerre- Hahn orthogonal polynomials on non-uniform latticesPublication . Rebocho, M. N.; Branquinho, A.A characterization theorem for Laguerre–Hahn orthogonal polynomials on non-uniform lattices is stated and proved.This theorem proves the equivalence between the Riccati equation for the formal Stieltjes function, linear first-order difference relations for the orthogonal polynomials as well as for the associated polynomials of the first kind, and linear first-order difference relations for the functions of the second kind.
- A characterization theorem for semi-classical orthogonal polynomials on non-uniform latticesPublication . Rebocho, M. N.; Filipuk, Galina; Chen, Yang; Branquinho, A.It is proved a characterization theorem for semi-classical orthogonal polynomials on non- uniform lattices that states the equivalence between the Pearson equation for the weight and some systems involving the orthogonal polynomials as well as the functions of the second kind. As a consequence, it is deduced the analogue of the so-called compatibility conditions in the ladder operator scheme. The classical orthogonal polynomials on non- uniform lattices are then recovered under such compatibility conditions, through a closed formula for the recurrence relation coefficients.
- Deformed Laguerre-Hahn orthogonal polynomials on the real linePublication . Branquinho, A.; Rebocho, M. N.We study families of orthogonal polynomials on the real line whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients. We derive discrete dynamical systems, obtained as a result of deformations of the recurrence relation coefficients of the orthogonal polynomials related to the above referred Stieltjes functions.
- On the semiclassical character of orthogonal polynomials satisfying structure relationsPublication . Branquinho, A.; Rebocho, M. N.We prove the semiclassical character of some sequences of orthogonal polynomials [...]
- Second- order differential equations in the Laguerre-Hahn classPublication . Rebocho, Maria das Neves; Branquinho, A.; Paiva, Anabela; Foulquie-Moreno, AnaLaguerre-Hahn families on the real line are characterized in terms of second order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.
- Structure relations for orthogonal polynomials on the unit circlePublication . Branquinho, A.; Rebocho, M. N.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: [...]
- Matrix Sylvester equations in the theory of orthogonal polynomials on the unit circlePublication . Branquinho, A.; Rebocho, M. N.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.
- Sylvester equations for Laguerre-Hahn orthogonal polynomials on the real linePublication . Branquinho, A.; Paiva, Anabela; Rebocho, M. N.Matrix Sylvester differential equations are introduced in the study of Laguerre-Hahn orthogonal polynomials. Matrix Sylvester differential systems are shown to yield representations for the Laguerre-Hahn orthogonal polynomials. Lax pairs are given, formed from the differential system and the recurrence relation, that yield discrete non-linear equations for the three term recurrence relation coefficients of the Laguerre-Hahn orthogonal polynomials.
- Difference and differential equations for deformed Laguerre-Hahn orthogonal polynomials on the unit circlePublication . Branquinho, A.; Rebocho, M. N.Sequences of orthogonal polynomials on the unit circle whose Carath´eodory function satisfies a Riccati-type differential equation with polynomial coefficients are studied. We deduce discrete Lax equations which lead to difference equations for the corresponding sequences of reflection parameters, and we analyze the continuous differential equations that arise when deformations through a dependence on a parameter t occur.