Ramanujan type series for 1/𝜋, revisited

Abstract: In this note, we revisit Ramanujan type series for \(\frac{1}{\pi}\) and show how they arise from genus zero subgroups of \({SL}_{2}(\mathbb{Z})\) that are commensurable with \({SL}_{2}(\mathbb{Z})\). 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.