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June, 2020 The Explicit Formulae and Evaluations of Ramanujan's Remarkable Product of Theta-functions
Devasirvatham John Prabhakaran, Kumar Ranjith Kumar
Taiwanese J. Math. 24(3): 545-551 (June, 2020). DOI: 10.11650/tjm/190706

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 $m = 17,37$.

Citation

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Devasirvatham John Prabhakaran. Kumar Ranjith Kumar. "The Explicit Formulae and Evaluations of Ramanujan's Remarkable Product of Theta-functions." Taiwanese J. Math. 24 (3) 545 - 551, June, 2020. https://doi.org/10.11650/tjm/190706

Information

Received: 31 May 2018; Revised: 14 July 2019; Accepted: 28 July 2019; Published: June, 2020
First available in Project Euclid: 19 May 2020

zbMATH: 07251186
MathSciNet: MR4100708
Digital Object Identifier: 10.11650/tjm/190706

Subjects:
Primary: 33D10
Secondary: 11F20, 14K25

Rights: Copyright © 2020 The Mathematical Society of the Republic of China

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Vol.24 • No. 3 • June, 2020
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