On some new general theorems for the explicit evaluations of Ramanujan’s remarkable product of theta-functions

On pages 338 and 339 in his first notebook (Notebooks (2 volumes), 1957]), Ramanujan records eighteen values for a certain product of theta-functions. All these have been proved by B.C. Berndt, H.H. Chan and L.-C. Zhang (Proc. Edinburg Math. Soc. 40:583–612, 1997]). The main purpose of this paper is to establish some new general theorems for the explicit evaluations of Ramanujan’s product of theta-functions and some new explicit evaluations there from.