The Explicit Formulae and Evaluations of Ramanujan's Remarkable Product of Theta-functions

Abstract

On pages 338 and 339 in his first notebook, Ramanujan defined remarkable product of theta-functions $a_{m,n}$ and also recorded eighteen explicit values depending on two parameters $m$ and $n$. All these values have been established by Berndt et al. In this paper, we establish a new general formulae for the explicit evaluations of $a_{3m,3}$ and $a_{m,9}$ by using $P$-$Q$ mixed modular equation and values for certain class invariant of Ramanujan. Using these formulae, we calculate some new explicit values of $a_{3m,3}$ for $m = 2,7,13,17,25,37$ and $a_{m,9}$ for $m = 17,37$.