Linear independence of certain sums of reciprocals of the Lucas numbers

Abstract

Let $h\ge 3$ and i be integers with $1\le i\le h-1$. In this paper, we give linear independence results for the values of the functions

$$\begin{aligned} g_{h,i}(z):=\sum _{n=1}^{\infty }\frac{z^{in}-z^{(h-i)n}}{1-z^{hn}} , \quad |z|<1, \end{aligned}$$

at suitable algebraic points. As an application, we deduce arithmetical properties of certain sums of reciprocals of linear recurrence sequences. For example, the six numbers

$$\begin{aligned} 1,\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}},\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}+1},\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}-1},\quad \sum _{n=1}^{\infty }\frac{1}{L_{4n}},\quad \sum _{n=1}^{\infty }\frac{1}{L_{4n}-1} \end{aligned}$$

are linearly independent over the field ${\mathbb {Q}}\left( \sqrt{5}\right)$, where $\{L_{n}\}_{n\ge 0}$ is the classical Lucas sequence.