Transcendental Series of Reciprocals of Fibonacci and Lucas Numbers

TL;DR

This paper proves that certain infinite series of reciprocals of Fibonacci numbers are transcendental when the sequence of indices grows sufficiently fast, extending previous methods and resolving a question related to irrationality and transcendence.

Contribution

It establishes the transcendence of sums of reciprocals of Fibonacci numbers for sequences with ratio growth greater than 2, using a novel application of the Subspace Theorem.

Findings

Proves transcendence of specific Fibonacci reciprocal series.

Extends transcendence results beyond Mahler's method.

Provides new techniques for applying the Subspace Theorem.

Abstract

Let $F_1=1,F_2=1,\ldots$ be the Fibonacci sequence. Motivated by the identity $\displaystyle\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}=\frac{7-\sqrt{5}}{2}$, Erd\"os and Graham asked whether $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is irrational for any sequence of positive integers $n_1,n_2,\ldots$ with $\frac{n_{k+1}}{n_k}\geq c>1$. We resolve the transcendence counterpart of their question: as a special case of our main theorem, we have that $\displaystyle\sum_{k=1}^{\infty}\frac{1}{F_{n_k}}$ is transcendental when $\frac{n_{k+1}}{n_k}\geq c>2$. 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 $n_k=d^k+r$.