An extension of the quintuple product identity and its applications

Zhi-Guo Liu

Vol. 246 (2010), No. 2, 345–390

DOI: 10.2140/pjm.2010.246.345

Abstract

Using the theory of elliptic theta functions, we establish a theta function identity that may be regarded as an extension of the quintuple identity, with many other results, both classical and new, included as special cases. It allows us to give a new derivation of the Ramanujan–Watson modular equation of the seventh order. We give new proofs of some Eisenstein series identities of Ramanujan related to modular equations of degree 7.

Keywords

elliptic function, theta function, modular equation, Eisenstein series, Jacobi’s quartic identity, sum of squares, Ramanujan’s identities, quintuple product identity

Mathematical Subject Classification 2000

Primary: 11F11, 11F20, 11F27, 33E05

Milestones

Received: 2 April 2009

Accepted: 29 July 2009

Published: 1 June 2010