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.

在本文中，我们重新审视$\frac {1}{\pi }$的拉马努金型级数，并展示它们如何从$\mathrm {SL}_{2}(\mathbb {R})$的属零子群中产生与$\mathrm {SL}_{2}(\mathbb {Z})$相当。作为说明，我们重现了$\frac {1}{\pi }$的拉马努扬惊人公式和 Cooper 等人的最新结果，并导出了$\frac {1}的新有理拉马努金型级数{\pi}$。作为副产品，我们获得了某种一般意义上的克劳森型公式，并重现了与上述拉马努金公式密切相关的克劳森型二次变换公式。