Supercongruences via Beukers' method

2024/08/19 by Sun, Zhi-Hong, Ye, Dongxi
#11A07 #11B65 #11E25 #11F03 #11F20 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2408.09776

Abstract

Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Apéry-like numbers. In this paper, we use Beukers' method to prove some conjectures of the first author concerning the congruences for ∑_k=0^(p-1)/2\frac\binom2kk³m^k, ∑_k=0^p-1\frac\binom2kk²\binom4k2km^k, ∑_k=0^p-1\frac\binom2kk\binom3kk\binom6k3km^k, ∑_n=0^p-1(V_n)/(mⁿ), ∑_n=0^p-1(T_n)/(mⁿ), ∑_n=0^p-1(D_n)/(mⁿ) and ∑_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=∑_k=0ⁿ\binom2kk²\binom2n-2kn-k², T_n=∑_k=0ⁿ\binom nk²\binom2kn², D_n=∑_k=0ⁿ\binom nk²\binom2kk\binom2n-2kn-k and A_n is the Apéry number given by A_n=∑_k=0ⁿ\binom nk²\binomn+kk².

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