On the binomial transforms of Apéry-like sequences

Abstract

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