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

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.