Ramanujan's type of product of theta functions and its applications

Abstract

In this paper, we initiate a generous amount of new-found general theorems for explicit evaluations of product of the theta functions $b_{m, n}$ using Kronecker's limit formula and other various novel explicit evaluations that were introduced thereupon. Also, we have obtained a few novel values of Ramanujan's class invariant $g_n$ using new-found values of $b_{m, n}$. Eventually, we have shown that $b_{m, n}$ is the unit in certain algebraic number fields.