Computing hypergeometric solutions of second order linear differential equations using quotients of formal solutions and integral bases. (November 2017)

• Record Type: Journal Article
• Authors: Imamoglu, Erdal; van Hoeij, Mark
• Is Part Of: Journal of symbolic computation. Volume 83(2017)
• Page Start: 254
• Page End: 271
• Publication Date: 2017-11
• Subjects: Symbolic computation -- Linear differential equations -- Closed form solutions -- Hypergeometric solutions -- Integral bases; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.11.014
• Languages: English
• ISSNs: 0747-7171

Abstract

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.