Some Eisenstein series identities related to modular equations of the seventh order

Zhi-Guo Liu

Vol. 209 (2003), No. 1, 103–130

DOI: 10.2140/pjm.2003.209.103

Abstract

In this paper we will use one well-known modular equation of seventh order, one theta function identity of S. McCullough and L.-C. Shen, 1994, and the complex variable theory of elliptic functions to prove some new septic identities for theta functions. Then we use these identities to provide new proofs of some Eisenstein series identities in Ramanujan’s notebooks or “lost” notebook. We also derive a new identity for Eisenstein series and some curious trigonometric identities.

Milestones

Received: 28 August 2001

Revised: 22 October 2001

Published: 1 March 2003