An extension of Gauss congruences for Ap\'ery numbers

TL;DR

This paper extends Gauss congruences for a broad class of Apéry-related numbers, connecting them to Bernoulli numbers and deepening the understanding of their congruence properties.

Contribution

It introduces an extended congruence for Apéry-like numbers, linking them to Bernoulli numbers and advancing the theory of their divisibility properties.

Findings

Established an extended Gauss congruence involving Bernoulli numbers.

Connected Apéry-like numbers to deeper congruence structures.

Enhanced understanding of divisibility and congruence properties of special number sequences.

Abstract

Osburn, Sahu and Straub introduced the numbers: \begin{align*} A_n^{(r,s,t)}=\sum_{k=0}^n{n\choose k}^r{n+k\choose k}^s{2k\choose n}^t, \end{align*} for non-negative integers $n,r,s,t$ with $r\ge 2$, which includes two kinds of Ap\'ery numbers and four kinds of Ap\'ery-like numbers as special cases, and showed that the numbers $\{A_n^{(r,s,t)}\}_{n\ge 0}$ satisfy the Gauss congruences of order $3$. We establish an extension of Osburn--Sahu--Straub congruence through Bernoulli numbers, which is one step deep congruence of the Gauss congruence for $A_n^{(r,s,t)}$.