Supercongruences via Beukers' method

Zhi-Hong Sun, Dongxi Ye

Published 2024-08-19Version 1

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 $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k^3}{m^k}, \quad\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k}, \quad \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k\binom{6k}{3k}}{m^k}, \quad \sum_{k=0}^{p-1}\frac{V_k}{m^k} \quad\hbox{and}\quad \sum_{k=0}^{p-1}\frac{T_k}{m^k} $$ modulo $p^3$, where $p>3$ is a prime, $m$ is an integer not divisible by $p$, $V_n=\sum_{k=0}^n\binom{2k}k^2\binom{2n-2k}{n-k}^2$ and $T_n=\sum_{k=0}^n\binom nk^2\binom{2k}n^2$.