拉马努金型系列，重访

Pub Date : 2023-10-05 DOI:10.4153/s0008439523000772

Dongxi Ye

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引用次数: 0

摘要

在本文中，我们重新讨论$\frac {1}{\pi }$的ramanujan型级数，并说明它们是如何从与$\mathrm {SL}_{2}(\mathbb {Z})$可通约的$\mathrm {SL}_{2}(\mathbb {R})$的属零子群中产生的。作为说明，我们重现了拉马努金对于$\frac {1}{\pi }$的惊人公式和Cooper等人最近的结果，并为$\frac {1}{\pi }$导出了一个新的有理拉马努金型级数。作为副产品，我们得到了一个一般意义上的clausen型公式，并得到了一个与上述拉马努金公式密切相关的clausen型二次变换公式。

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Ramanujan type series for , revisited

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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