Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases

TL;DR

This paper introduces two algorithms for computing hypergeometric solutions of second order linear differential equations with rational coefficients, utilizing modular reduction, Hensel lifting, and transformations to identify solutions involving hypergeometric functions.

Contribution

It presents novel algorithms that systematically find hypergeometric solutions by combining formal solution quotients, integral bases, and equation transformations.

Findings

Algorithms successfully compute hypergeometric solutions for a class of differential equations.

Use of modular reduction and Hensel lifting enhances solution efficiency.

Transformative approach broadens the applicability of hypergeometric solution methods.

Abstract

We present two algorithms for computing hypergeometric solutions of second order linear differential operators with rational function coefficients. Our first algorithm searches for solutions of the form \[ \exp(\int r \, dx)\cdot{_{2}F_1}(a_1,a_2;b_1;f) \] where $r,f \in \overline{\mathbb{Q}(x)}$, and $a_1,a_2,b_1 \in \mathbb{Q}$. It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form \[ \exp(\int r \, dx)\cdot \left( r_0 \cdot{_{2}F_1}(a_1,a_2;b_1;f) + r_1 \cdot{_{2}F_1}'(a_1,a_2;b_1;f) \right) \] where $r_0, r_1 \in \overline{\mathbb{Q}(x)}$, as follows: It tries to transform the input equation to another equation with solutions of the first type, and then uses the first algorithm.