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arXiv · 1606.01576

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

Erdal Imamoglu·Mark van Hoeij
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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.

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Erdal Imamoglu, Mark van Hoeij. 2016-06-05. Computing Hypergeometric Solutions of Second Order Linear Differential Equations using Quotients of Formal Solutions and Integral Bases. https://arxiv.org/abs/1606.01576

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