A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$

Ji-Cai Liu

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arXiv · 2008.06675

A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$

Ji-Cai Liu

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Abstract

We prove three supercongruences for sums of Almkvist-Zudilin numbers, which confirm some conjectures of Zudilin and Z.-H. Sun. A typical example is the Ramanujan-type supercongruence: \begin{align*} \sum_{k=0}^{p-1} \frac{4k+1}{81^k}\gamma_k \equiv \left(\frac{-3}{p}\right) p\pmod{p^3}, \end{align*} which is corresponding to Chan and Verrill's formula for $1/\pi$: \begin{align*} \sum_{k=0}^\infty \frac{4k+1}{81^k}\gamma_k = \frac{3\sqrt{3}}{2\pi}. \end{align*} Here $\gamma_n$ are the Almkvist-Zudilin numbers.

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Ji-Cai Liu. 2020-08-15. A $p$-adic analogue of Chan and Verrill's formula for $1/\pi$. https://arxiv.org/abs/2008.06675

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