Kronecker theta function and a decomposition theorem for theta functions I

TL;DR

This paper introduces a decomposition theorem for theta functions based on the Kronecker theta function, enabling easy derivation of numerous classical and new identities, including a new addition formula.

Contribution

It presents a novel decomposition theorem for theta functions using the Kronecker theta function, unifying and simplifying the derivation of many identities.

Findings

Derived many classical theta identities

Established a new addition formula for theta functions

Revisited and unified results by Ramanujan, Weierstrass, and others

Abstract

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's $_1\psi_1$ 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.