Journal of Integer Sequences, Vol. 23 (2020), Article 20.2.4

On a Curious Identity of Ramanujan


Hùng Việt Chu
Department of Mathematics
University of Illinois at Urbana-Champaign
Champaign, IL 61820
USA

Abstract:

Ramanujan discovered the following identity:
\begin{align*}\sqrt{2 \left(1 - \frac{1}{3^2}\right) \left(1 - \frac{1}{7^2}\rig... ...ight) \left(1 + \frac{1}{11}\right) \left(1 + \frac{1}{19}\right) . \end{align*}
We find necessary and sufficient conditions for the integers appearing in analogous identities, prove that there are only finitely many such identities, and provide a method to generate many interesting variations.

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Received September 27 2019; revised version received September 29 2019; January 3 2020. Published in Journal of Integer Sequences, February 17 2020.

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