On the binomial transforms of Apéry-like sequences
Abstract
In the proof of the irrationality of $\zeta(3)$ and $\zeta(2)$, 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 $u_n\equiv \alpha^n \pmod{N_{\alpha}}~(\alpha\in \mathbb{Z},N_{\alpha}\in \mathbb{N}^{+})$ for all of the $15$ Apéry-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 $15$ 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.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2024
- DOI:
- 10.48550/arXiv.2406.18059
- arXiv:
- arXiv:2406.18059
- Bibcode:
- 2024arXiv240618059L
- Keywords:
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- Mathematics - Number Theory;
- Mathematics - Combinatorics;
- 11B50;
- 11B65;
- 05A10
- E-Print:
- 19 pages