Taiwanese Journal of Mathematics

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

Devasirvatham John Prabhakaran and Kumar Ranjith Kumar

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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 $m = 17,37$.

Article information

Source
Taiwanese J. Math., Volume 24, Number 3 (2020), 545-551.

Dates
Received: 31 May 2018
Revised: 14 July 2019
Accepted: 28 July 2019
First available in Project Euclid: 19 May 2020

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1589875218

Digital Object Identifier
doi:10.11650/tjm/190706

Subjects
Primary: 33D10
Secondary: 11F20: Dedekind eta function, Dedekind sums 14K25: Theta functions [See also 14H42]

Keywords
modular equation class invariants remarkable product of theta-function

Citation

Prabhakaran, Devasirvatham John; Kumar, Kumar Ranjith. The Explicit Formulae and Evaluations of Ramanujan's Remarkable Product of Theta-functions. Taiwanese J. Math. 24 (2020), no. 3, 545--551. doi:10.11650/tjm/190706. https://projecteuclid.org/euclid.twjm/1589875218


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References

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