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

D. J. Prabhakaran; K. Ranjith kumar

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arXiv · 2004.14631

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

D. J. Prabhakaran·K. Ranjith kumar

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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 $m$.

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D. J. Prabhakaran, K. Ranjith kumar. 2020-04-30. The explicit evaluations of Ramanujan's remarkable product of theta-functions a_{m, n}. https://arxiv.org/abs/2004.14631

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