A theta function identity and its implications

Abstract

In this paper we prove a general theta function identity with four parameters by employing the complex variable theory of elliptic functions. This identity plays a central role for the cubic theta function identities. We use this identity to re-derive some important identities of Hirschhorn, Garvan and Borwein about cubic theta functions. We also prove some other cubic theta function identities. A new representation for ∏ n = 1 ∞ ( 1 − q n ) 10 \prod _{n=1}^\infty (1-q^n)^{10} is given. The proofs are self-contained and elementary.