Laguerre Hahn Orthogonal Polynomials Research Articles

Integrable differential systems for deformed Laguerre–Hahn orthogonal polynomials

Our work studies sequences of orthogonal polynomials of the Laguerre–Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, which are subject to a deformation parameter t. We derive systems of differential equations and give Lax pairs, yielding nonlinear differential equations in t for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a Möbius transformation of a Stieltjes function related to a deformed Jacobi weight is studied in detail, showing this system is governed by a differential equation of the Painlevé type P. The particular case of P arising here has the same four parameters as the solution found by Magnus (1995 J. Comput. Appl. Math. 57 215–37) but differs in the boundary conditions.

Classification of Laguerre–Hahn orthogonal polynomials of class one

AbstractWe study orthogonal polynomials related to Stieltjes functions satisfying Riccati type differential equations with polynomial coefficients, , with . We derive recurrences for the three‐term recurrence relation coefficients of the orthogonal polynomials, including connections with some forms of discrete Painlevé equations.

Laguerre–Hahn orthogonal polynomials on the real line

A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coefficients — 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 coefficients, and distributional equations for the corresponding linear functionals.

Some Laguerre-Hahn Orthogonal Polynomials of Class One

Some Laguerre-Hahn Orthogonal Polynomials of Class One

On linear spectral transformations and the Laguerre–Hahn class

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

Orthogonal polynomials on systems of non-uniform lattices from compatibility conditions

We 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.

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

ABSTRACTIn 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 .

Characterization theorem for Laguerre–Hahn orthogonal polynomials on non-uniform lattices

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.

Deformed Laguerre–Hahn orthogonal polynomials on the real line

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.

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.

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

Sequences of orthogonal polynomials on the unit circle whose Carathéodory 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.

On difference equations for orthogonal polynomials on nonuniform lattices1

By the study of various properties of some divided-difference equations, we simplify the definition of classical orthogonal polynomials given by Atakishiyev et al., 1995, On classical orthogonal polynomials, Constructive Approximation, 11, 181–226, then prove that orthogonal polynomials obtained by some modifications of the classical orthogonal polynomials on nonuniform lattices satisfy a single fourth-order linear homogeneous divided-difference equation with polynomial coefficients. Moreover, we factorize and solve explicitly these divided-difference equations. Also, we prove that the product of two functions, each of which satisfying a second-order linear homogeneous divided-difference equation is a solution of a fourth-order linear homogeneous divided-difference equation. This result holds in particular when the divided-difference operator is carefully replaced by the Askey–Wilson operator , following pioneering work by Magnus 1988, Associated Askey–Wilson polynomials as Laguerre–Hahn orthogonal polynomials, Lecture Notes in Mathematics (Berlin: Springer), vol. 1329, pp. 261–278, connecting and divided-difference operators. Finally, we propose a method to look for polynomial solutions of linear divided-difference equations with polynomial coefficients.

Factorization of fourth-order differential equations for perturbed classical orthogonal polynomials

We factorize the fourth-order differential equations satisfied by the Laguerre–Hahn orthogonal polynomials obtained from some perturbations of classical orthogonal polynomials such as: the rth associated (for generic r), the general co-recursive, the general co-recursive associated, the general co-dilated and the general co-modified classical orthogonal polynomials. Moreover, we find four linearly independent solutions of the fourth-order differential equations, and show that the factorization obtained for modifications of classical orthogonal polynomials is still valid, with some minor changes when the polynomial family modified is semi-classical. Finally, we extend the validity of the results obtained for the associated classical orthogonal polynomials with integer order of association from integers to reals.

Computational Aspects of Three-Term Recurrence Relations

Computational Aspects of Three-Term Recurrence Relations