Kronecker theta function and a decomposition theorem for theta functions I
Abstract
The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identities is proved.
Cite
@article{arxiv.2012.01670,
title = {Kronecker theta function and a decomposition theorem for theta functions I},
author = {Zhi-Guo Liu},
journal= {arXiv preprint arXiv:2012.01670},
year = {2020}
}
Comments
23 pages. Accepted by the Ramanujan Journal