Geodesic
Ramanujan-type series for $\frac {1}{\pi }$, revisited
Canadian mathematical bulletin, Tome 67 (2024) no. 2, pp. 350-368

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DOI

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.
DOI : 10.4153/S0008439523000772
Mots-clés : Orbifold uniformizations, 1⁄π, Clausen-type transformations 
Ye, Dongxi. Ramanujan-type series for $\frac {1}{\pi }$, revisited. Canadian mathematical bulletin, Tome 67 (2024) no. 2, pp. 350-368. doi: 10.4153/S0008439523000772
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     title = {Ramanujan-type series for $\frac {1}{\pi }$, revisited},
     journal = {Canadian mathematical bulletin},
     pages = {350--368},
     year = {2024},
     volume = {67},
     number = {2},
     doi = {10.4153/S0008439523000772},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439523000772/}
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