Laguerre–Hahn orthogonal polynomials on the real line

Abstract

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.

Integral Representations of a Generalized Linear Hermite Functional

“…Of course, this method can be applied not only to other (symmetric) classical orthogonal polynomials but to any other symmetric orthogonal polynomial sequence for which a hypergeometric representation is known. This is something we should do in order to obtain novel integral representations for other Special functions; for example we could consider some other generalization for the Hermite linear functional, as well as some Laguerre-Hahn or semi-classical, orthogonal polynomials (see, e.g., [8,9] and the references therein).…”

Integrable differential systems for deformed Laguerre–Hahn orthogonal polynomials

“…where I denotes the 2 × 2 identity matrix. In the semi-classical case, taking B ≡ 0 in (39) and combining with the spectral differential equation for the weight, ∂ x ln w = C/A, then a second solution is obtained,…”

“…Another direction of study proceeds with the analysis of differential equations of the Riccati type equation ( 4) focusing on transformations of orthogonal polynomials corresponding to rational Stieltjes spectral transformations [30,55] (see also [27,36]). Further details on these topics may be found in the survey paper [39].…”