The explicit evaluations of Ramanujan's remarkable product of theta-functions a_{m, n}

Abstract

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 $m$.