Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 52. no. 3. P. 7 – 23. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-52-3-7-23
Research Article
Full text in English
MSC 68R15; 05C42; 11B05; 11R45; 11B39

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Generalized Natural Density \operatorname{DF}(\mathfrak{F}_k) of Fibonacci Word

D. Abdullah, J. Hamoud^{\ast}

Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudnyi, 141700 Russia

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 nth
root of the absolute value of terms in a random Fibonacci sequence converges to 1.13198824 . . ., with
subsequent refinements by Rittaud yielding a limit of approximately 1.20556943 for the expected value’s
n-th root. Novel definitions, such as the natural density of sets of positive integers and the limiting
density of Fibonacci sequences modulo powers of primes, provide a robust framework for our analysis. We introduce the concept of k-Fibonacci words, extending classical Fibonacci words to higher dimensions, and investigate their patterns alongside sequences like the Thue-Morse and Sturmian words. Our main results include a unique representation theorem for real numbers using Fibonacci numbers, a symmetry identity for sums involving Fibonacci words, \sum_{k=1}^{b} \dfrac{(-1)^k F_a}{F_k F_{k+a}}= \sum_{k=1}^{a} \dfrac{(-1)^k F_b}{F_k F_{k+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.

Key words: density, Fibonacci, word, natural, sequence, balanced.

Received: 30.09.2025; Revised: 09.10.2025; Accepted: 11.10.2025; First online: 10.11.2025

For citation. Abdullah D., Hamoud J. Generalized Natural Density \operatorname{DF}(\mathfrak{F}_k) of Fibonacci Word. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 52: 3, 7-23. EDN: KCDDRW. https://doi.org/10.26117/2079-6641-2025-52-3-7-23.

Funding. Not funding.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: hamoud.math@gmail.com

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Abdullah D., Hamoud J., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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