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- Classification of Laguerre-Hahn orthogonal polynomials of class onePublication . Rebocho, M. N.; Filipuk, GalinaWe study orthogonal polynomials related to Stieltjes functions satisfying Riccati type differential equations with polynomial coefficients, AS0 = BS2 + CS + D, with max {deg(A); deg(B)} <= 3; deg(C) <= 2. We derive recurrences for the three-term recurrence relation coefficients of the orthogonal polynomials, including connections with some forms of discrete Painlevé equations.
- Divided-difference operators from the geometric point of viewPublication . Rebocho, M. N.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.
- 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.
- Distributional equation for Laguerre- Hahn functionals on the unit circlePublication . Branquinho, A.; Rebocho, M. N.Let u be a hermitian linear functional defined in the linear space of Laurent polynomials and F its corresponding Carathéodory function. [...]
- On the second order holonomic equation for Sobolev-type orthogonal polynomialsPublication . Rebocho, Maria das NevesIt is presented a general approach to the study of orthogonal polynomials related to Sobolev inner products which are defined in terms of divided-difference operators having the fundamental property of leaving a polynomial of degree $n-1$ when applied to a polynomial of degree $n$. This paper gives analytic properties for the orthogonal polynomials, including the second order holonomic difference equation satisfied by them.
- 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.
- Discrete Painlevé Equations for Recurrence Coefficients of Laguerre-Hahn Orthogonal Polynomials of Class OnePublication . Rebocho, M. N.; Filipuk, GalinaIn 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, dPI and dPIV .
- Discrete semi-classical orthogonal polynomials of class one on quadratic latticesPublication . Rebocho, M. N.; Filipuk, GalinaWe study orthogonal polynomials on quadratic lattices with respect to a Stieltjes function, S, that satisfies a difference equation ADS = CMS+D; where A is a polynomial of degree less or equal than 3 and C is a polynomial of degree greater or equal than 1 and less or equal than 2. We show systems of difference equations for the orthogonal polynomials that arise from the so-called compatibility conditions. Some closed formulae for the recurrence relation coefficients are obtained.
- 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 [...]
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