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

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.