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On integrality and asymptotic behavior of the $(k, l)$-Göbel sequences

Hibiki Gima, Toshiki Matsusaka, Taichi Miyazaki, Shunta Yara

Abstract

Recently, Matsuhira, Matsusaka, and Tsuchida revisited old studies on the integrality of $k$-Göbel sequences and showed that the first 19 terms are always integers for any integer $k\ge 2$. In this article, we further explore two topics: Ibstedt's $(k,l)$-Göbel sequences and Zagier's asymptotic formula for the $2$-Göbel sequence, and extend their results.

On integrality and asymptotic behavior of the $(k, l)$-Göbel sequences

Abstract

Recently, Matsuhira, Matsusaka, and Tsuchida revisited old studies on the integrality of -Göbel sequences and showed that the first 19 terms are always integers for any integer . In this article, we further explore two topics: Ibstedt's -Göbel sequences and Zagier's asymptotic formula for the -Göbel sequence, and extend their results.
Paper Structure (10 sections, 13 theorems, 53 equations, 1 figure, 1 table)

This paper contains 10 sections, 13 theorems, 53 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. How long can $(k,l)$-Göbel sequences remain integers?
  3. Notations and key properties
  4. Proof of \ref{['Main1']}
  5. Zagier's asymptotic formula and its generalization
  6. The constant $C_{k,l}$
  7. Asymptotic behavior
  8. The asymptotic coefficients $a_{k,r}$
  9. Further observations
  10. Methods for Computing $N_{k,l}$

Key Result

Theorem 1.2

We have $\min_{k, l \ge 2} N_{k,l} = 7$, which implies that $g_{k,l}(n) \in \mathbb{Z}$ for any $k, l \ge 2$ and $1 \le n \le 6$. Moreover, we have $N_{k,l} = 7$ if and only if $k \equiv 2 \pmod{6}$ and $l \equiv 3 \pmod{7}$.

Figures (1)

  • Figure 1: The plots of $C_{k,l}(n)$ for $1\leq n\leq 15$.

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • proof : Proof of \ref{['Main1']}
  • Lemma 3.1
  • ...and 22 more