Addition formulas for Jacobi theta functions, Dedekind’s eta function, and Ramanujan’s congruences

Zhi-Guo Liu

Vol. 240 (2009), No. 1, 135–150

DOI: 10.2140/pjm.2009.240.135

Abstract

Previously, we proved an addition formula for the Jacobi theta function, which allows us to recover many important classical theta function identities. Here, we use this addition formula to derive a curious theta function identity, which includes Jacobi’s quartic identity and some other important theta function identities as special cases. We give new series expansions for η²(τ), η⁶(τ), η⁸(τ), and η¹⁰(τ), where η(τ) is Dedekind’s eta function. The series expansions for η⁶(τ) and η¹⁰(τ) lead to simple proofs of Ramanujan’s congruences p(7n + 5) ≡ 0 (mod 7) and p(11n + 6) ≡ 0 (mod 11), respectively.

Keywords

elliptic function, theta function, Dedekind’s eta function, Ramanujan’s congruence

Mathematical Subject Classification 2000

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

Milestones

Received: 27 March 2008

Revised: 28 November 2008

Accepted: 1 December 2008

Published: 2 March 2009