Elliptic functions to the quintic base

Heng Huat Chan and Zhi-Guo Liu

Vol. 226 (2006), No. 1, 53–64

DOI: 10.2140/pjm.2006.226.53

Abstract

We study two elliptic functions to the quintic base and find two nonlinear second order differential equations satisfied by them. We then derive two recurrence relations involving certain Eisenstein series associated with the group Γ₀(5). These recurrence relations allow us to derive infinite families of identities involving the Eisenstein series and Dedekind η-products. An imaginary transformation for one of the elliptic functions is also derived.

Keywords

Eisenstein series, elliptic functions, Dedekind eta function

Mathematical Subject Classification 2000

Primary: 33E05, 11F11, 11F27

Milestones

Received: 26 September 2004

Revised: 16 January 2005

Accepted: 16 January 2005

Published: 1 July 2006