Second-order differential equations in the Laguerre–Hahn class

Abstract

Laguerre-Hahn families on the real line are characterized in terms of second order differential equations with matrix coefficients for vectors involving the orthogonal polynomials and their associated polynomials, as well as in terms of second order differential equation for the functions of the second kind. Some characterizations of the classical families are derived.

On linear spectral transformations and the Laguerre–Hahn class

“…Here, A is the same as in (7) and M n , N n are 2 × 2 matrices whose entries are bounded degree polynomials depending on the coefficients of the Riccati equation [16,Theorem 1]. In the semi-classical case, that is, B ≡ 0 in (7), when dealing with weights, there holds the equivalence between the semi-classical character of w, w /w = C/A, and the differential system…”

“…Recall that the semi-classical class is closed under the Christoffel and Geronimus transformations [12]. Whenever dealing with weights, there holds the equivalence between the semi-classical character of w, say w /w = C/A, and the differential system (16) for the corresponding Y n ,…”

“…In what follows we see howB n in (49) and the transfer matrices of {Ỹ n } n≥0 , henceforth denoted byà n , relate to B n in (16) and to the transfer matrices A n of {Y n } n≥0 .…”

“…where A is a polynomial and M n , N n are 2 × 2 matrices with polynomial entries [16]. We study the modifications of systems of type (2) under the general framework of Christoffel and Geroninums transformations of linear functionals [15,17,18].…”

Laguerre–Hahn orthogonal polynomials on the real line

“…Before proceeding, some comments on notations and terminology. Throughout the present work, it will be used the notations contained in some of the author's papers, for instance, [14,15,39], as well as in others references, such as [25,28,90]. In some references, mainly from P. Maroni and co-authors, e.g., [29,34,61], the Stieltjes function is given as S(x) = − n≥0 u n x −n−1 .…”