Supercongruences via Beukers' method
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 $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k^3}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k\binom{6k}{3k}}{m^k}, \ \sum_{n=0}^{p-1}\frac{V_n}{m^n},\ \sum_{n=0}^{p-1}\frac{T_n}{m^n},\ \sum_{n=0}^{p-1}\frac{D_n}{m^n} $$ and $\sum_{n=0}^{p-1}(-1)^nA_n$ modulo $p^3$, where $p$ is an odd prime representable by some suitable binary quadratic form, $m$ is an integer not divisible by $p$, $V_n=\sum_{k=0}^n\binom{2k}k^2\binom{2n-2k}{n-k}^2$, $T_n=\sum_{k=0}^n\binom nk^2\binom{2k}n^2$, $D_n=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2n-2k}{n-k}$ and $A_n$ is the Apéry number given by $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.09776
- arXiv:
- arXiv:2408.09776
- Bibcode:
- 2024arXiv240809776S
- Keywords:
-
- Mathematics - Number Theory;
- Mathematics - Classical Analysis and ODEs;
- Mathematics - Combinatorics;
- 11A07;
- 11B65;
- 11E25;
- 11F03;
- 11F20
- E-Print:
- 59 pages