Liouvillian solutions of Whittaker-Ince equation. (March 2023)

• Record Type: Journal Article
• Authors: Cheung, Tsz Yung
• Is Part Of: Journal of symbolic computation. Volume 115(2023)
• Page Start: 18
• Page End: 38
• Publication Date: 2023-03
• Subjects: Liouvillian solution -- Whittaker-Ince equation -- Ellipsoidal wave equation -- Picard-Fuchs equation -- Kovacic's algorithm; 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.2022.07.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We apply Kovacic's algorithm, a tool that is developed from differential Galois theory, to discuss the existence of Liouvillian solutions of Whittaker-Ince equation, ellipsoidal wave equation and the Picard-Fuchs equation of a K3 surface. We determine the necessary and sufficient conditions of having Liouvillian solutions for Whittaker-Ince equation when a parameter vanishes; as well as a sufficient condition of having Liouvillian solution when this parameter is non-zero. On the other hand, we prove that ellipsoidal wave equation has no Liouvillian solution. We generalize a Picard-Fuchs equation for certain K3 surface and show that a particular case of this Picard-Fuchs equation cannot have any Liouvillian solutions.