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

2016/06/05 by Erdal Imamoglu, Mark van Hoeij, Imamoglu, Erdal +1
Computer Science · Engineering · Mathematics · #68W30 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Computer and information sciences #FOS: Mathematics #I.1.2 #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #acm:68W30 #cs.SC #math.CA #msc:68W30

paper · pdf · doi:10.48550/arxiv.1606.01576

arxiv created 2016/06/05 · openalex publication_date 2016/06/05 · arxiv updated 2016/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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(∫ r dx)⋅₂F₁(a₁,a₂;b₁;f) where r,f ∈ ℚ(x), and a₁,a₂,b₁ ∈ ℚ. It uses modular reduction and Hensel lifting. Our second algorithm tries to find solutions in the form exp(∫ r dx)⋅ ( r₀ ⋅₂F₁(a₁,a₂;b₁;f) + r₁ ⋅₂F₁'(a₁,a₂;b₁;f) ) where r₀, r₁ ∈ 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.

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