Transcendental Series of Reciprocals of Fibonacci and Lucas Numbers

Abstract

Let $F_1=1,F_2=1,\dots$ be the Fibonacci sequence. Motivated by the identity $\sum _{k=0}^{\infty }\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erdös and Graham asked whether $\sum _{k=1}^{\infty }\frac{1}{F_{n_k}}$ is irrational for any sequence of positive integers $n_1,n_2,\dots$ with $\frac{n_{k+1}}{n_k}\ge c>1$. We resolve the transcendence counterpart of their question: as a special case of our main theorem, we have that $\sum _{k=1}^{\infty }\frac{1}{F_{n_k}}$ is transcendental when $\frac{n_{k+1}}{n_k}\ge c>2$. The bound $c>2$ is best possible thanks to the identity at the beginning. This paper provides a new way to apply the Subspace Theorem to obtain transcendence results and extends previous non-trivial results obtainable by only Mahler's method for special sequences of the form $n_k=d^k+r$.