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Transcendental series of reciprocals of Fibonacci and Lucas numbers

### Khoa Dang Nguyen

Vol. 16 (2022), No. 7, 1627–1654
##### 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 ${n}_{k+1}∕{n}_{k}\ge c>2$. 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 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
Primary: 11J87
Secondary: 11B39