Article contents
Ramanujan-type series for
$\frac {1}{\pi }$, revisited
Published online by Cambridge University Press: 05 October 2023
Abstract
In this note, we revisit Ramanujan-type series for $\frac {1}{\pi }$ and show how they arise from genus zero subgroups of
$\mathrm {SL}_{2}(\mathbb {R})$ that are commensurable with
$\mathrm {SL}_{2}(\mathbb {Z})$. As illustrations, we reproduce a striking formula of Ramanujan for
$\frac {1}{\pi }$ and a recent result of Cooper et al., as well as derive a new rational Ramanujan-type series for
$\frac {1}{\pi }$. As a byproduct, we obtain a Clausen-type formula in some general sense and reproduce a Clausen-type quadratic transformation formula closely related to the aforementioned formula of Ramanujan.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society
Footnotes
This work was supported by the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2023A1515010298).
References
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