Series solutions of linear ODEs by Newton-Raphson method on quotient $D$-modules
Classical Analysis and ODEs
2021-11-18 v2 Mathematical Physics
math.MP
Abstract
We develop a module approach to various kinds of solutions to several classes of important differential equations by long divisions of different differential operators. The zeros of remainder maps of such long divisions are handled by an analogue of Hensel's lemma established recently from valuation theory. In particular, this explains the common origin of some classically known special function series solutions of Heun equations and usual Frobenius series solutions. Moreover, these remainder maps also generate eigenvalue problems that lead to non-trivial factorizations of certain generalized hypergeometric operators.
Cite
@article{arxiv.2111.05103,
title = {Series solutions of linear ODEs by Newton-Raphson method on quotient $D$-modules},
author = {Yik Man Chiang and Avery Ching and Chiu Yin Tsang},
journal= {arXiv preprint arXiv:2111.05103},
year = {2021}
}
Comments
Two more references are added; typos of minor natures are corrected on pages 7, 25 and 38