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Differential equations for families of semi- classical orthogonal polynomials within class one

Publication . Rebocho, M. N.; Filipuk, Galina

In this paper we study families of semi-classical orthogonal polynomials within class one. We derive general second or third order ordinary differential equations (with respect to certain parameters) for the recurrence coefficients of the three-term recurrence relation of these polynomials and show that in particular well-known cases, e.g. related to the modified Airy and Laguerre weights, these equations can be reduced to the second and the fourth Painlevé equations.

2018Preprint

Open access

Laguerre-Hahn orthogonal polynomials on the real line

Publication . Rebocho, M. N.

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.

2020Preprint

Open access

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

Publication . 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.

2018Preprint

Open access

Symmetric semi-classical orthogonal polynomials of class one on q-quadratic lattices

Publication . Rebocho, M. N.; Filipuk, Galina

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.

2018Preprint

Open access

On linear spectral transformations and the Laguerre-Hahn class

Publication . Rebocho, M. N.; Castillo, Kenier

We study the Christoffel, Geronimus, and Uvarov transformations for Laguerre-Hahn orthogonal polynomials on the real line. It is analysed the modification of the corresponding difference-differential equations that characterize the systems of orthogonal polynomials and the consequences for the three-term recurrence relation coefficients.

2017Preprint

Open access