Series solutions of linear ODEs by Newton-Raphson method on quotient $D$-modules

TL;DR

This paper introduces a $D$-module framework for solving linear differential equations using long division and Newton-Raphson methods, revealing connections between special function series and eigenvalue problems.

Contribution

It presents a novel $D$-module approach to differential equations, linking series solutions, eigenvalue problems, and factorizations through a valuation-theoretic analogue of Hensel's lemma.

Findings

Unified explanation for classical special function series

Generation of eigenvalue problems from remainder maps

Facilitation of factorizations of hypergeometric operators

Abstract

We develop a $D-$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.