Supercongruences involving Apéry-like numbers and Bernoulli numbers

Ji-Cai Liu

Published 2024-03-28Version 1

For non-negative integers $n,r,s,t$, let $A_n^{(r,s,t)}=\sum_{k=0}^n{n\choose k}^r{n+k\choose k}^s{2k\choose n}^t$, which includes six Ap\'ery-like numbers as special cases. We establish one step deep congruences of the Gauss congruences for $\{A_n^{(r,s,t)}\}_{n\ge 0}$ in the form: \begin{align*} A_{np}^{(r,s,t)}\equiv A_n^{(r,s,t)}+p^3B_{p-3}\mathcal{A}^{(r,s,t)}_n\pmod{p^4}, \end{align*} for all primes $p\ge 5$ and all positive integers $n,r$ with $r\ge 2$, where $\mathcal{A}^{(r,s,t)}_n$ is independent of $p$ and $B_{p-3}$ is the $(p-3)$th Bernoulli numbers. This type of supercongruences for another two Ap\'ery-like numbers are also established.