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On the binomial transforms of Apéry-like sequences

Ji-Cai Liu

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.

On the binomial transforms of Apéry-like sequences

Abstract

In the proof of the irrationality of and , Apéry defined two integer sequences through -term recurrences, which are known as the famous Apéry numbers. Zagier, Almkvist--Zudilin and Cooper successively introduced the other sporadic sequences through variants of Apéry's -term recurrences. All of the sporadic sequences are called Apéry-like sequences. Motivated by Gessel's congruences mod for the Apéry numbers, we investigate the congruences in the form for all of the Apéry-like sequences . Let be the largest positive integer such that for all non-negative integers . We determine the values of for all of the Apéry-like sequences .The binomial transforms of Apéry-like sequences provide us a unified approach to this type of congruences for Apéry-like sequences.

Paper Structure

This paper contains 14 sections, 9 theorems, 99 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Apéry-like sequences
  3. Motivation
  4. Main results
  5. Values of $N_{u_1}$
  6. Structure of the article
  7. Binomial transforms
  8. A key equivalent condition
  9. Gauss congruences
  10. Differential operators
  11. Proof of Theorem \ref{['t-1']}
  12. Proof of Theorem \ref{['t-2']}
  13. An auxiliary result
  14. Proof of Theorem \ref{['t-2']}

Key Result

Theorem 1.1

Let $\{u_n\}_{n\ge 0}$ be one of the $15$ Apéry-like sequences and $\alpha$ be an integer. For all non-negative integers $n$, we have

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Corollary 2.6
  • Lemma 4.1