Transcendental series of reciprocals of Fibonacci and Lucas numbers

Nguyen, Khoa Dang

doi:10.2140/ant.2022.16.1627

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Transcendental series of reciprocals of Fibonacci and Lucas numbers

Khoa Dang Nguyen

Vol. 16 (2022), No. 7, 1627–1654

DOI: 10.2140/ant.2022.16.1627

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 $n_{k + 1} ∕ n_{k} \geq c > 1$ . 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 $n_{k + 1} ∕ n_{k} \geq c > 2$ . 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 nontrivial results obtainable by only Mahler’s method for special sequences of the form $n_{k} = d^{k} + r$ .

Keywords

algebraic numbers, transcendental numbers, subspace theorem

Mathematical Subject Classification

Primary: 11J87

Secondary: 11B39

Milestones

Received: 27 November 2020

Revised: 20 September 2021

Accepted: 23 October 2021

Published: 16 October 2022

Authors

Khoa Dang Nguyen
Department of Mathematics and Statistics The University of Calgary Calgary AB Canada

Vol. 16, No. 7, 2022