On Schultz's generalization of Borweins' cubic identity

TL;DR

This paper explores Schultz's generalization of Borweins' cubic identity for theta functions, offering new proofs and identities to deepen understanding of elliptic function theory.

Contribution

It presents two novel approaches to derive Schultz's identity, expanding the theoretical framework and generating new Schultz-type identities.

Findings

Provided new proofs of Schultz's identity.

Derived several new Schultz-type identities.

Enhanced understanding of theta function identities.

Abstract

In 1991, the Borweins established a cubic analogue of Jacobi's identity for theta functions, which is used by B.C. Berndt, S. Bhargava, and F.G. Garvan in the development of Ramanujan's cubic theory of elliptic functions. In 2013, D. Schultz discovered an identity for theta series in three variables which generalizes the Borweins' identity. In this article, we revisit Schultz's identity and present two distinct approaches to its derivation. Our investigation not only provides new proofs but also yields several new Schultz-type identities.