On the solutions of holonomic third-order linear irreducible differential equations in terms of hypergeometric functions. (November 2020)

• Record Type: Journal Article
• Authors: Mouafo Wouodjié, Merlin; Koepf, Wolfram
• Is Part Of: Journal of symbolic computation. Volume 101(2020)
• Page Start: 202
• Page End: 241
• Publication Date: 2020-11
• Subjects: 34-XX -- 33C10 -- 33C20 -- 34B30 -- 34Lxx; Bessel functions -- Bessel square functions -- Composition of Bessel square functions with square roots -- Hypergeometric functions -- Operators -- Transformations -- Change of variables -- exp-product -- Gauge transformation -- Singularities -- Generalized exponents -- Exponent differences -- Rational functions -- Zeroes -- Poles; 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.2019.08.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper we present algorithms that combine change of variables, exp-product and gauge transformation to represent solutions of a given irreducible third-order linear differential operator L, with rational function coefficients and without Liouvillian solutions, in terms of functions S ∈ { F 2 0, F 2 1, F 2 2, B ˇ ν 2 } where F q p with p ∈ { 0, 1, 2 }, q = 2, is the generalized hypergeometric function, and B ˇ ν 2 ( x ) = ( B ν ( x ) ) 2 with B ν a Bessel function (see (Abramowitz and Stegun, 1972 )). That means we find rational functions r, r 0, r 1, r 2, f such that the solution of L will be of the form y = exp ⁡ ( ∫ r d x ) ( r 0 S ( f ( x ) ) + r 1 ( S ( f ( x ) ) ) ′ + r 2 ( S ( f ( x ) ) ) ″ ) . An implementation of those algorithms in Maple is available.