任意次与任意阶的Ferrers函数及相关函数
经典分析与常微分方程
2022-05-04 v3
摘要
本文推导了任意次与任意阶的第一类Ferrers函数(割线上的关联勒让德函数)的许多新颖积分与级数表示、连接不同阶与不同次的Ferrers函数的各种积分、级数与微分关系,以及一致渐近展开。给出了Ferrers函数四个生成函数的简单证明。基于三类超几何多项式的生成函数,证明了自变量为tanh(a+b)的Ferrers函数的加法定理。作为特例,得到了Gegenbauer多项式与Ferrers关联勒让德函数(关联勒让德多项式)的关系。
引用
@article{arxiv.2110.13564,
title = {Ferrers functions of arbitrary degree and order and related functions},
author = {P. Malits},
journal= {arXiv preprint arXiv:2110.13564},
year = {2022}
}
备注
39 pages. 1) Inessential details are omitted in the proof of Theorem 13. 2) The statement of Theorem 26 is slightly changed. This change has enabled me to write a new generating function for Ferrers functions as a corollary. 3) The statement of Theorem 29 (Theorem 28 in the previous version) is corrected. 4) Typos are corrected