Zhi-Guo Liu - A universal identity for theta functions of degree eight and applications

hrj:7432 - Hardy-Ramanujan Journal, May 6, 2021, Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan - https://doi.org/10.46298/hrj.2021.7432
A universal identity for theta functions of degree eight and applications

Authors: Zhi-Guo Liu

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.


Volume: Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan
Published on: May 6, 2021
Submitted on: April 30, 2021
Keywords: theta function,addition formula,Ramanujan's modular equations,Elliptic function,2010 Mathematics Subject Classification. 33E05, 11F11, 11F20, 11F27,[MATH]Mathematics [math]


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