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 ${\sum }_{k=0}^{\infty }1∕F_{2^k}=7-\sqrt{5}∕2$, Erdös and Graham asked whether ${\sum }_{k=1}^{\infty }1∕F_{n_k}$ is irrational for any sequence of positive integers $n_1,n_2,\dots$ with $n_{k+1}∕n_k\ge c>1$. We resolve the transcendence counterpart of their question; as a special case of our main theorem, we have that ${\sum }_{k=1}^{\infty }1∕F_{n_k}$ is transcendental when . 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 nontrivial results obtainable by only Mahler’s method for special sequences of the form $n_k=d^k+r$.