  On pages 338 and 339 in his first notebook, Ramanujan defined the remarkable
product of theta-functions $a_{m, n}$. Also he recorded eighteen explicit
values depending on two parameters, namely, $m$, and $n$, where these are odd
integers. All these values were established by Berndt et al. In this paper, we
initiate to study explicit evaluations of $a_{m, n}$ for even values of $m$. We
establish a new general theorems for the explicit evaluations of $a_{m, n}$
involving class invariant of Ramanujan. For this purpose, we establish several
new P-Q mixed modular equations. Using these theorems, we calculate several new
explicit values of $a_{m, n}$ for $n = 3, 5, 7, 13$ and even values of $m$.
