On Schultz's generalization of Borweins' cubic identity

TL;DR

This work revisits Schultz's two-variable cubic theta identity and provides two independent proofs, then extends the framework to two-variable generalizations of Jacobi's identity. It develops a web of analogues based on theta-function transformations, eta-products, and Macdonald-type identities, linking Borwein's cubic identity to Chan–Chan–Sole and Chan–Cooper–Toh results. The paper also establishes two-variable generalizations and cubic analogues of Borweins, including identities of Borwein-type and X^2+Y^2=Z^2+W^2 structures, with several new A,B,C series satisfying A^3+B^3=C^3. Collectively, these results enrich Ramanujan's elliptic theory to the cubic base and illuminate connections among modular forms, theta identities, and lattice-sum transformations.

Abstract

Around 1991, J.M. and P.B. Borwein established a cubic analogue of Jacobi's fundamental identity for theta functions. Their identity serves as the foundation for the subsequent development by B.C. Berndt, S. Bhargava, and F.G. Garvan of Ramanujan's theory of elliptic functions to the cubic base. 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 identities of a similar type. Furthermore, this study enables us to construct new two-variable generalizations of Jacobi's original classical identity.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Theorem 2.1
• proof : Proof of \ref{['Borweins-xy']}
• Remark 2.1
• Theorem 3.1
• Remark 3.1
• Theorem 3.2
• proof : Second proof of \ref{['key-to-Schultz']}
• Theorem 5.1