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

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.