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

06/05/2016

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by   Erdal Imamoglu, et al.

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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 (∫ r dx)·_2F_1(a_1,a_2;b_1;f) where r,f ∈Q(x), and a_1,a_2,b_1 ∈Q. It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form (∫ r dx)·( r_0 ·_2F_1(a_1,a_2;b_1;f) + r_1 ·_2F_1'(a_1,a_2;b_1;f) ) where r_0, r_1 ∈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.