On the binomial transforms of Ap\'ery-like sequences

TL;DR

This paper investigates congruences of Apéry-like sequences, focusing on binomial transforms to unify understanding of their properties and determine maximal moduli for which these sequences satisfy specific exponential congruences.

Contribution

It introduces a unified approach using binomial transforms to analyze congruences of all 15 Apéry-like sequences and determines the largest moduli for these congruences.

Findings

Determined the maximal moduli for congruences of Apéry-like sequences.

Unified the analysis of these sequences through binomial transforms.

Extended Gessel's congruences to all Apéry-like sequences.

Abstract

In the proof of the irrationality of $\zeta(3)$ and $\zeta(2)$, Ap\'ery defined two integer sequences through $3$-term recurrences, which are known as the famous Ap\'ery numbers. Zagier, Almkvist--Zudilin and Cooper successively introduced the other $13$ sporadic sequences through variants of Ap\'ery's $3$-term recurrences. All of the $15$ sporadic sequences are called Ap\'ery-like sequences. Motivated by Gessel's congruences mod $24$ for the Ap\'ery numbers, we investigate the congruences in the form $u_n\equiv \alpha^n \pmod{N_{\alpha}}~(\alpha\in \mathbb{Z},N_{\alpha}\in \mathbb{N}^{+})$ for all of the $15$ Ap\'ery-like sequences $\{u_n\}_{n\ge 0}$. Let $N_{\alpha}$ be the largest positive integer such that $u_n\equiv \alpha^n \pmod{N_{\alpha}}$ for all non-negative integers $n$. We determine the values of $\max\{N_{\alpha}|\alpha \in \mathbb{Z}\}$ for all of the $15$ Ap\'ery-like sequences $\{u_n\}_{n\ge 0}$.The binomial transforms of Ap\'ery-like sequences provide us a unified approach to this type of congruences for Ap\'ery-like sequences.