关于Laguerre-Hahn正交多项式的高阶微分方程

中文摘要 AI 辅助

本文提出了一种构造性方法，用于推导任意阶的Laguerre-Hahn正交多项式所满足的齐次线性微分方程。该方法基于结构关系的层级体系，该体系由Laguerre-Hahn形式的基本结构关系递归构建。文献中广泛研究的四阶微分方程，自然地被作为该一般框架的一个特例而恢复。特别关注半经典情形。该方法完全算法化，并已在{\it Mathematica$^{\circledR}$}中实现。我们将其应用于两类类似于Hermite的零类Laguerre-Hahn族，给出了显式的结构关系以及五阶和十阶的微分方程。我们还考察了经典的Hermite序列，它作为第一族的极限情形被恢复。

英文摘要

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.

发表机构

• University of Monastir（莫纳斯提尔大学）
• Universidade do Porto（波尔图大学）

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