Heun functions versus elliptic functions. (English) Zbl 1125.33017
Elaydi, S. (ed.) et al., Difference equations, special functions and orthogonal polynomials. Proceedings of the international conference, Munich, Germany, July 25–30, 2005. Hackensack, NJ: World Scientific (ISBN 978-981-270-643-0/hbk). 664-686 (2007).
Heun functions are defined as a natural generalization of the hypergeometric function, to be the solutions of the Fuchsian differential equation. The author presents some recent progress on Heun functions, gathering results from classical analysis up to elliptic functions. The author describes Picard’s generalization of Floquet’s theory for differential equations with doubly periodic coefficients and gives the detailed forms of the level one Heun functions in terms of Jacobi theta functions. The finite-gap solutions give an interesting alternative integral representation which, at level one, is shown to be equivalent to their elliptic form.
For the entire collection see [Zbl 1117.39001].
For the entire collection see [Zbl 1117.39001].
Reviewer: Stamatis Koumandos (Nicosia)
MSC:
33E30 | Other functions coming from differential, difference and integral equations |
33E05 | Elliptic functions and integrals |
33C75 | Elliptic integrals as hypergeometric functions |
34C25 | Periodic solutions to ordinary differential equations |
34M05 | Entire and meromorphic solutions to ordinary differential equations in the complex domain |