M.N. Rebocho

On the semi-classical character of orthogonal polynomials satisfying structure relations

We prove the semi-classical character of some sequences of orthogonal polynomials, say , , related through relations of the following type: , where denotes the monic polynomial associated with the pth order derivative of , and are complex numbers. The case is studied for a pair of orthogonal polynomials whose corresponding orthogonality measures are coherent. The relation is shown to give a characterization for the semi-classical character of .

Sylvester equations for Laguerre-Hahn orthogonal polynomials on the real line

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.

Discrete Painlevé equations for recurrence coefficients of Laguerre–Hahn orthogonal polynomials of class one

ABSTRACT In this paper we study recurrences for Laguerre–Hahn orthogonal polynomials of class one. It is shown for some families of such Laguerre–Hahn polynomials that the coefficients of the three term recurrence relation satisfy some forms of discrete Painlevé equations, namely and .

Difference and differential equations for deformed Laguerre–Hahn orthogonal polynomials on the unit circle

Sequences of orthogonal polynomials on the unit circle whose Caratheodory 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.

A characterization theorem for semi-classical orthogonal polynomials on non-uniform lattices

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.

Symmetric Semi-classical Orthogonal Polynomials of Class One on q-Quadratic Lattices

In this paper we study discrete semi-classical orthogonal polynomials on non-uniform lattices. In the symmetric class one case we give a closed form expression for the recurrence coefficients of orthogonal polynomials.

On Laguerre–Hahn affine orthogonal polynomials on the unit circle from matrix Sylvester equations

ABSTRACT In this paper are derived recurrences for the reflection coefficients of Laguerre–Hahn affine orthogonal polynomials on the unit circle, including a form of the discrete Painlevé equations . The technique is based on the knowledge of the first-order differential equation for the Carathéodory function, combined with a re-interpretation, in the formalism of matrix Sylvester equations, of compatibility conditions for the differential systems satisfied by the polynomials.

Deformed Laguerre–Hahn orthogonal polynomials on the real line

Abstract One derives discrete dynamical systems related to Laguerre–Hahn orthogonal polynomials. One studies deformations of the recurrence relation coefficients of the orthogonal polynomials under a t -dependence on the coefficients of the Riccati differential equation for the related Stieltjes function.