Linear independence of certain sums of reciprocals of the Lucas numbers

Let h≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h\ge 3$$\end{document} and i be integers with 1≤i≤h-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le i\le h-1$$\end{document}. In this paper, we give linear independence results for the values of the functions gh,i(z):=∑n=1∞zin-z(h-i)n1-zhn,|z|<1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g_{h,i}(z):=\sum _{n=1}^{\infty }\frac{z^{in}-z^{(h-i)n}}{1-z^{hn}} , \quad |z|<1, \end{aligned}$$\end{document}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 1,∑n=1∞1L2n,∑n=1∞1L2n+1,∑n=1∞1L2n-1,∑n=1∞1L4n,∑n=1∞1L4n-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 1,\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}},\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}+1},\quad \sum _{n=1}^{\infty }\frac{1}{L_{2n}-1},\quad \sum _{n=1}^{\infty }\frac{1}{L_{4n}},\quad \sum _{n=1}^{\infty }\frac{1}{L_{4n}-1} \end{aligned}$$\end{document}are linearly independent over the field Q5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Q}}\left( \sqrt{5}\right) $$\end{document}, where {Ln}n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{L_{n}\}_{n\ge 0}$$\end{document} is the classical Lucas sequence.