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Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases. (English) Zbl 1375.65097

Summary: 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 \[ \exp(\int r d x) \cdot_2 F_1(a_1, a_2; b_1; f) \eqno{(1)} \] 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 \[ \begin{aligned} \exp(\int r d x) \cdot(r_0 \cdot_2 F_1(a_1, a_2; b_1; f) \\ + r_1 \cdot_2 F_1^\prime(a_1, a_2; b_1; f))\end{aligned} \eqno{(2)} \] 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 type (1), and then uses the first algorithm.

MSC:

65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
68W30 Symbolic computation and algebraic computation
34A30 Linear ordinary differential equations and systems
33C05 Classical hypergeometric functions, \({}_2F_1\)
65D20 Computation of special functions and constants, construction of tables
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References:

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