An addition formula for the Jacobian theta function with applications

TL;DR

This paper presents a new addition formula for the Jacobian theta function, enabling derivation of known and novel theta function identities and series expansions, expanding the analytical tools available in elliptic function theory.

Contribution

The paper introduces an addition formula equivalent to Liu's, providing a new approach to derive existing and new theta function identities.

Findings

Derived new series expansions for theta functions

Reproduced known theta identities using the new formula

Established several novel theta function identities

Abstract

Liu established an addition formula for the Jacobian theta function by using the theory of elliptic functions. From this addition formula he obtained the Ramanujan cubic theta function identity, Winquist's identity, a theta function identities with five parameters, and many other interesting theta function identities. In this paper we will give an addition formula for the Jacobian theta function which is equivalent to Liu's addition formula. Based on this formula we deduce some known theta function identities as well as new identities. From these identities we shall establish certain new series expansions for