Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases. (November 2017) - Imamoglu, Erdal; van Hoeij, Mark
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 (1) exp ⁡ ( ∫ r d x ) ⋅ 2 F 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 (2) exp ⁡ ( ∫ r d x ) ⋅ ( r 0 ⋅ 2 F 1 ( a 1, a 2 ; b 1 ; f ) + r 1 ⋅ 2 F 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 type(1), and then uses the first algorithm.