
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 , , , 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 and .
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