A universal identity for theta functions of degree eight and applications

TL;DR

This paper presents a universal identity for degree-eight theta functions, enabling the derivation of numerous classical and new elliptic modular function results, and unifying various historical identities under a common framework.

Contribution

It introduces a general theta function identity that acts as a generating tool for elliptic modular function identities, unifying many previous results and providing new insights.

Findings

Derived hundreds of results about elliptic modular functions

Unified classical identities from Jacobi, Kiepert, Ramanujan, and Weierstrass

Proposed a new conjecture related to theta functions

Abstract

Previously, we proved an identity for theta functions of degree eight, and several applications of it were also discussed. This identity is a natural extension of the addition formula for the Weierstrass sigma-function. In this paper we will use this identity to reexamine our work in theta function identities in the past two decades. Hundreds of results about elliptic modular functions, both classical and new, are derived from this identity with ease. Essentially, this general theta function identity is a theta identities generating machine. Our investigation shows that many well-known results about elliptic modular functions with different appearances due to Jacobi, Kiepert, Ramanujan and Weierstrass among others, actually share a common source. This paper can also be seen as a summary of my past work on theta function identities. A conjecture is also proposed.