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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.
- 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.
- 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.
- Orthogonal polynomials on systems of non-uniform lattices from compatibility conditionsPublication . Rebocho, M. N.; Filipuk, GalinaWe deduce difference equations in the matrix form for Laguerre-Hahn orthogonal polynomials on systems of non-uniform lattices, the so-called compatibility conditions, involving the transfer matrices. As a consequence, we obtain closed form expressions for the recurrence relation coefficients of the Laguerre-Hahn polynomials of class zero.
- Symmetric semi-classical orthogonal polynomials of class one on q-quadratic latticesPublication . Rebocho, M. N.; Filipuk, GalinaIn 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.
- The Symmetric Semi-classical Orthogonal Polynomials of Class Two and Some of Their ExtensionsPublication . Rebocho, M. N.; Filipuk, GalinaWe study a large class of orthogonal polynomials, containing the semi-classical symmetric orthogonal polynomials of class two. We show difference equations for the recurrence coefficients of the orthogonal polynomials as well as for related quantities. Some of these recurrences are identified with discrete Painlevé equations.
- Nonlinear difference equations for a modified Laguerre weight: Laguerre-Freud equations and asymptoticsPublication . Rebocho, M. N.; Filipuk, Galina; Chen, YangIn this paper we derive second and third order nonlinear difference equations for one of the recurrence coefficients in the three term recurrence relation of polynomials orthogonal with respect to a modified Laguerre weight. We show how these equations can be obtained from the Backlund transformations of the third Painlevé equation. We also show how to use nonlinear difference equations to derive a few terms in the formal asymptotic expansions in n of the recurrence coefficients.
- Differential equations for families of semi- classical orthogonal polynomials within class onePublication . Rebocho, M. N.; Filipuk, GalinaIn 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.