On the binomial transforms of Apéry-like sequences

Abstract

In the proof of the irrationality of $\hat{I}\P (3)$ and $\hat{I}\P (2)$, 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 $u_n\equiv \hat{I}{\pm }^n(\mathrm{mod}{N}_{\hat{I}}\pm )(\hat{I}\pm \in \mathbb{Z},N_{\hat{I}}\pm \in {\mathbb{N}}^{+})$ for all of the $15$ Apéry-like sequences $\{u_n{\}}_{n\ge 0}$. Let $N_{\hat{I}}\pm$ be the largest positive integer such that $u_n\equiv \hat{I}{\pm }^n(\mathrm{mod}{N}_{\hat{I}}\pm )$ for all non-negative integers $n$. We determine the values of $\max \{N_{\hat{I}}\pm |\hat{I}\pm \in \mathbb{Z}\}$ 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.

19 pages