A universal identity for theta functions of degree eight and applications

zbMATH Open1487.33015arXiv2104.14705MaRDI QIDQ5011710

Author name not available (Why is that?)

Publication date: 27 August 2021

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.

Full work available at URL: https://arxiv.org/abs/2104.14705

zbMATH Keywords

theta function elliptic function addition formula Ramanujan's modular equations

Mathematics Subject Classification ID

Theta series; Weil representation; theta correspondences (11F27) Holomorphic modular forms of integral weight (11F11) Dedekind eta function, Dedekind sums (11F20) Elliptic functions and integrals (33E05)