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Supercongruences via Beukers' method (2408.09776v4)

Published 19 Aug 2024 in math.NT, math.CA, and math.CO

Abstract: Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Ap\'ery-like numbers. In this paper, we use Beukers' method to prove some conjectures of the first author concerning the congruences for $$\sum_{k=0}{(p-1)/2}\frac{\binom{2k}k3}{mk}, \ \sum_{k=0}{p-1}\frac{\binom{2k}k2\binom{4k}{2k}}{mk}, \ \sum_{k=0}{p-1}\frac{\binom{2k}k\binom{3k}k\binom{6k}{3k}}{mk}, \ \sum_{n=0}{p-1}\frac{V_n}{mn},\ \sum_{n=0}{p-1}\frac{T_n}{mn},\ \sum_{n=0}{p-1}\frac{D_n}{mn} $$ and $\sum_{n=0}{p-1}(-1)nA_n$ modulo $p3$, where $p$ is an odd prime representable by some suitable binary quadratic form, $m$ is an integer not divisible by $p$, $V_n=\sum_{k=0}n\binom{2k}k2\binom{2n-2k}{n-k}2$, $T_n=\sum_{k=0}n\binom nk2\binom{2k}n2$, $D_n=\sum_{k=0}n\binom nk2\binom{2k}k\binom{2n-2k}{n-k}$ and $A_n$ is the Ap\'ery number given by $A_n=\sum_{k=0}n\binom nk2\binom{n+k}k2$.

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