Linear independence results for certain sums of reciprocals of Fibonacci and Lucas numbers

TL;DR

This paper establishes linear independence of certain sums involving reciprocals of Fibonacci and Lucas numbers, revealing new arithmetical properties and independence results over quadratic fields.

Contribution

It provides novel linear independence results for sums of reciprocals of Fibonacci and Lucas numbers, extending understanding of their algebraic independence.

Findings

The sums of reciprocals over prime squares are linearly independent over d5(\u221a5).

New arithmetical properties of Fibonacci and Lucas sums are derived.

Linear independence results apply to specific coprime sequences.

Abstract

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