Generalized Natural Density \DF(\mathfrakF_n) of Fibonacci Word

2025/04/14 by Jasem Hamoud, Hamoud, Jasem, Duaa Abdullah +1
Computer Science · Mathematics · Physics and Astronomy · #05C42 #11B05 #11B39 #11R45 #68R15 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Coding theory and cryptography #Combinatorics (math.CO) #F.2.2 #FOS: Mathematics #G.2.0

paper · pdf · doi:10.48550/arxiv.2504.10207

openalex publication_date 2025/04/14 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

This paper explores profound generalizations of the Fibonacci sequence, delving into random Fibonacci sequences, k-Fibonacci words, and their combinatorial properties. We established that the n-th root of the absolute value of terms in a random Fibonacci sequence converges to 1.13198824…, a symmetry identity for sums involving Fibonacci words, ∑_n=1^b \frac(-1)ⁿ F_aF_n F_n+a = ∑_n=1^a \frac(-1)ⁿ F_bF_n F_n+b, and an infinite series identity linking Fibonacci terms to the golden ratio. These findings underscore the intricate interplay between number theory and combinatorics, illuminating the rich structure of Fibonacci-related sequences. We provide, according to this paper, new concepts of density of Fibonacci word.

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