Theta vocabulary I

TL;DR

This paper provides a comprehensive, annotated list of transformation properties and identities of the four theta functions, revealing underlying patterns and deriving higher-term identities from fundamental bilinear relations.

Contribution

It introduces a unified presentation of theta function identities and shows how complex identities emerge from basic bilinear relations.

Findings

All 3, 4, and 5-term identities of degree four derive from six fundamental bilinear identities.

The paper uncovers patterns behind various theta-function identities.

It offers a ready-to-use reference for transformation properties of theta functions.

Abstract

This paper is an annotated list of transformation properties and identities satisfied by the four theta functions $\theta _1$, $\theta _2$, $\theta _3$, $\theta _4$ of one complex variable, presented in a ready-to-use form. An attempt is made to reveal a pattern behind various identities for the theta-functions. It is shown that all possible 3, 4 and 5-term identities of degree four emerge as algebraic consequences of the six fundamental bilinear 3-term identities connecting the theta-functions with modular parameters $\tau$ and $2\tau$.