The aim of this paper is to give linear independence results for the values of Lambert type series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences $$\{n_\ell\}_{\ell\geq1}$$ { n ℓ } ℓ ≥ 1 . For example, the three numbers $$1, \quad \sum_{p:{\rm prime}}\frac{1}{F_{p^2}}, \quad \sum_{p:{\rm prime}}\frac{1}{L_{p^2}} $$ 1 , ∑ p : prime 1 F p 2 , ∑ p : prime 1 L p 2 are linearly independent over $$\mathbb{Q}(\sqrt{5})$$ Q ( 5 ) , where $$\{F_n\}$$ { F n } and $$\{L_n\}$$ { L n } are the Fibonacci and Lucas numbers, respectively.

中文翻译：

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斐波那契数和卢卡斯数的倒数之和的线性独立结果

本文的目的是给出 Lambert 类型级数的线性独立结果。作为一个应用，我们推导出与某些互质序列 $$\{n_\ell\}_{\ell\geq1}$$ { n ℓ } ℓ ≥ 1 相关联的斐波那契数和卢卡斯数的倒数之和的算术性质。例如三个数$$1, \quad \sum_{p:{\rm prime}}\frac{1}{F_{p^2}}, \quad \sum_{p:{\rm prime}}\ frac{1}{L_{p^2}} $$ 1 , ∑ p : prime 1 F p 2 , ∑ p : prime 1 L p 2 在 $$\mathbb{Q}(\sqrt{5} )$$ Q ( 5 ) ，其中 $$\{F_n\}$$ { F n } 和 $$\{L_n\}$$ { L n } 分别是斐波那契数和卢卡斯数。