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A new factorization of the generalized period-doubling sequences through kernel words and gaps sequences

K. Ernest Bognini, Hamdi Ammar

TL;DR

The paper addresses factorizing generalized period-doubling sequences $\mathbf{P}_k$ over the alphabet $\mathcal{A}_k$ by introducing kernel words and gap sequences. It develops a kernel-word framework, derives recurrence-based definitions for kernel numbers $r_i$ and kernel words $R_i$, and defines gap sequences $G_n$ between occurrences of kernel words. For the binary case $\mathbf{P}_2$, the authors establish a unique factorization $\mathbf{P}_2=\prod_{i\ge1}R_i$, and extend the approach to $k\ge3$ by formulating recursive constructions of $R_n$ and $G_n$ and obtaining explicit gap-length growth, yielding a complete kernel–gap factorization of $\mathbf{P}_k$. These results generalize prior tribonacci/k-bonacci related kernel concepts and provide a structured decomposition framework for generalized period-doubling words, with potential applications to expansive symbolic dynamics and combinatorics on words.

Abstract

In this paper, we study some new factorizations of period-doubling sequences over a $k$-letter alphabet, where $k\geq 2$. First, we define the combinatorial and arithmetic properties of these sequences. Then, we define the kernel words of period-doubling sequences and demonstrate how to factorize a binary sequence using its kernel words. Next, we define gap sequences for period-doubling sequences and explore their relationship with kernel words. Lastly, we present a factorization of period-doubling sequences for $k\geq 3$ based on kernel words and gap sequences.

A new factorization of the generalized period-doubling sequences through kernel words and gaps sequences

TL;DR

The paper addresses factorizing generalized period-doubling sequences over the alphabet by introducing kernel words and gap sequences. It develops a kernel-word framework, derives recurrence-based definitions for kernel numbers and kernel words , and defines gap sequences between occurrences of kernel words. For the binary case , the authors establish a unique factorization , and extend the approach to by formulating recursive constructions of and and obtaining explicit gap-length growth, yielding a complete kernel–gap factorization of . These results generalize prior tribonacci/k-bonacci related kernel concepts and provide a structured decomposition framework for generalized period-doubling words, with potential applications to expansive symbolic dynamics and combinatorics on words.

Abstract

In this paper, we study some new factorizations of period-doubling sequences over a -letter alphabet, where . First, we define the combinatorial and arithmetic properties of these sequences. Then, we define the kernel words of period-doubling sequences and demonstrate how to factorize a binary sequence using its kernel words. Next, we define gap sequences for period-doubling sequences and explore their relationship with kernel words. Lastly, we present a factorization of period-doubling sequences for based on kernel words and gap sequences.

Paper Structure

This paper contains 9 sections, 14 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Combinatorial and Arithmetical Properties of $\textbf{P}_{k}$
  4. New factorization of $\textbf{P}_k$ using kernel words and gap sequences
  5. Kernel words and some of its related properties
  6. Factorization of $\textbf{P}_{k},\ k\geq 2$ with respect to gap sequences and kernel words
  7. Factorization of $\textbf{P}_{2}$ into kernel words
  8. Factorization of $\textbf{P}_{k},\ k\geq 3$ using kernel words and gap sequences
  9. Conclusion

Key Result

Lemma 1

Let $k\geq 2$ be a fixed natural number. Then, for all $n\geq 2$, we have:

Theorems & Definitions (38)

  • Remark 1
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • Remark 2
  • Lemma 2
  • Theorem 2
  • Definition 1
  • ...and 28 more