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

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

Milestones

Received: 27 March 2008
Revised: 28 November 2008
Accepted: 1 December 2008
Published: 2 March 2009

Authors