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

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}γ_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}γ_k = \frac{3\sqrt{3}}{2π}. \end{align*} Here ${\gamma }_n$ are the Almkvist-Zudilin numbers.

12 pages

References in corpus (1)

• Super congruences concerning binomial coefficients and Apéry-like numbers