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

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.