Generalized Natural Density $\DF(\mathfrak{F}_n)$ of Fibonacci Word
Jasem Hamoud, Duaa Abdullah
TL;DR
The paper advances generalized Fibonacci theory by establishing new density concepts for Fibonacci words, classic and novel representations, and key identities. It proves a symmetry relation $\sum_{n=1}^{b}\frac{(-1)^n F_a}{F_n F_{n+a}}=\sum_{n=1}^{a}\frac{(-1)^n F_b}{F_n F_{n+b}}$, and a telescoping result $\sum_{n\ge1}\frac{1}{F_n F_{n+2k}}=\frac{1}{F_{2k}}\sum_{n=1}^k\frac{1}{F_{2n-1}F_{2n}}$, alongside a unique Fibonacci-base expansion $a=a_0+\sum_{k\ge1}\frac{\bar{\alpha}_k}{F_k}$ with $\bar{\alpha}_k\in\{0,1\}$. It also confirms Viswanath's limit for random Fibonacci sequences, $\lim_{n\to\infty}\sqrt[n]{|t_n|}=1.13198824\ldots$, and discusses zero natural density of Fibonacci numbers. Collectively, these results deepen the link between number theory and combinatorics, with implications for coding theory and symbolic dynamics via $k$-Fibonacci words, Sturmian structures, and non-integer base representations.
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 $1.13198824\ldots$, a symmetry identity for sums involving Fibonacci words, $\sum_{n=1}^{b} \frac{(-1)^n F_a}{F_n F_{n+a}} = \sum_{n=1}^{a} \frac{(-1)^n F_b}{F_n F_{n+b}}$, 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.