Zeckendorf expansion, Dirichlet series and infinite series involving the infinite Fibonacci word

TL;DR

This paper explores the properties of Dirichlet series and infinite series related to the Zeckendorf expansion and the infinite Fibonacci word, revealing new connections between number representations and series analysis.

Contribution

It introduces novel Dirichlet series based on Zeckendorf expansions and the Fibonacci word, establishing new relations and computing specific infinite series values.

Findings

Derived new Dirichlet series involving Fibonacci-based expansions

Established relations between series and Fibonacci word structures

Computed explicit values of certain infinite series

Abstract

Let $\beta=\frac{1+\sqrt{5}}{2}$, $(a_n)_{n \in \mathbb{N}^+}$ be a non-uniform morphic sequence involving the infinite Fibonacci word and $(\delta(n))_{n \in \mathbb{N}^+}$ be a positive sequence such that for all positive integers $n$, $\delta(n)=\frac{1}{\sqrt{5}}\sum_{j \geq 0}\epsilon_j\beta^{j+2}$ if the unique Zeckendorf expansion of $n$ is $n=\sum_{j \geq 0}\epsilon_jF_{j+2}$ with Fibonacci numbers $F_0,F_1,F_2...$. We define and study some Dirichlet series in the form of $\sum_{n\geq 1}\frac{a_n}{(\delta(n))^s}$ and relations between them. Moreover, we compute the values of some infinite series involving the infinite Fibonacci word.