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Ramanujan-type series for
$\frac {1}{\pi }$, revisited
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 67 / Issue 2 / June 2024
- Published online by Cambridge University Press:
- 05 October 2023, pp. 350-368
- Print publication:
- June 2024
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- Article
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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.
