Theory of general theta relations, addition formulas, and theta constants identities

TL;DR

This paper generalizes Jacobi's theta relations to arbitrary products, extending Igusa's derivation method to establish new addition formulas and identities for theta constants, with detailed analysis for the case n=3.

Contribution

It introduces a generalized framework for theta relations of any number of products, expanding the theoretical understanding of theta functions and identities.

Findings

Generalized theta relations for arbitrary n products

Extended Igusa's derivation method for these relations

Explicit analysis of the case n=3

Abstract

Jacobi's theta relations among quartic products of theta functions are generalized to those of arbitrary $n$ products. Igusa's procedure of derivation is extended to prove such general theta relations, from which we obtain general addition formulas and theta constants identities. To complete the proof, the concept of {\it cycle number $\lambda$} is essential. The case of $n=3$ is discussed and examined explicitly.