Some Eisenstein Series Identities Associated with the Borwein Functions

Abstract

In this note, we show how certain Eisenstein series identities associated with the Borweins’ theta-functions can be derived from a well-known identity involving theta-functions and one identity of Ramanujan. We employ the theory of elliptic functions to derive some related theta-function identities. These theta-function identities give a different approach to the Eisenstein series identities. By using some Eisenstein series identities of this note, we provide completely new proofs of the Borweins’ cubic theta function identity and the well-known Jacobi identity in the theory of modular forms.