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

2021/06/10 by Li, Shuo
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2106.05672

Abstract

Let β=(1+√(5))/(2), (a_n)_{n ∈ ℕ⁺} be a non-uniform morphic sequence involving the infinite Fibonacci word and (δ(n))_{n ∈ ℕ⁺} be a positive sequence such that for all positive integers n, δ(n)=(1)/(√(5))∑_{j ≥ 0}ε_jβ^j+2 if the unique Zeckendorf expansion of n is n=∑_{j ≥ 0}ε_jF_j+2 with Fibonacci numbers F₀,F₁,F₂.... We define and study some Dirichlet series in the form of ∑_{n≥ 1}(a_n)/((δ(n))^s) and relations between them. Moreover, we compute the values of some infinite series involving the infinite Fibonacci word.

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