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 $n$, $\delta (n)=\frac{1}{\sqrt{5}}{\sum }_{j\ge 0}{\epsilon }_j{\beta }^{j+2}$ if the unique Zeckendorf expansion of $n$ is $n={\sum }_{j\ge 0}{\epsilon }_j{F}_{j+2}$ with Fibonacci numbers $F_0,F_1,F_2...$. We define and study some Dirichlet series in the form of ${\sum }_{n\ge 1}\frac{a_n}{(\delta (n){)}^s}$ and relations between them. Moreover, we compute the values of some infinite series involving the infinite Fibonacci word.