On higher-order differential equations for Laguerre-Hahn orthogonal polynomials
Abstract
In this work, we present a constructive method to derive homogeneous linear differential equations of arbitrary order for Laguerre--Hahn orthogonal polynomials. The approach is based on a hierarchy of structure relations, built recursively from the fundamental structure relation of the Laguerre--Hahn forms. The fourth-order differential equation, which has been extensively studied in the literature, is naturally recovered as a particular case of this general framework. Special attention is devoted to the semiclassical case. The method is fully algorithmic and has been implemented in {\it MathematicacircledR}. We apply it to two Laguerre--Hahn families of class zero analogous to Hermite, providing explicit structure relations and differential equations of orders five and ten. We also examine the classical Hermite sequence, which is recovered as a limiting case of the first family.
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BibTeX
@misc{khalfallah2026higherorder,
title = {{On higher-order differential equations for Laguerre-Hahn orthogonal polynomials}},
author = {Mohamed Khalfallah and Zélia da Rocha},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22922},
eprint = {2609.22922},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22922},
}
RIS
TY - UNPB
TI - On higher-order differential equations for Laguerre-Hahn orthogonal polynomials
AU - Mohamed Khalfallah
AU - Zélia da Rocha
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22922
UR - https://arxiv.org/abs/2609.22922
AB - In this work, we present a constructive method to derive homogeneous linear differential equations of arbitrary order for Laguerre--Hahn orthogonal polynomials. The approach is based on a hierarchy of structure relations, built recursively from the fundamental structure relation of the Laguerre--Hahn forms. The fourth-order differential equation, which has been extensively studied in the literature, is naturally recovered as a particular case of this general framework. Special attention is devoted to the semiclassical case. The method is fully algorithmic and has been implemented in {\it Mathematica^circledR}. We apply it to two Laguerre--Hahn families of class zero analogous to Hermite, providing explicit structure relations and differential equations of orders five and ten. We also examine the classical Hermite sequence, which is recovered as a limiting case of the first family.
ER -