Linear independence results for sums of reciprocals of Fibonacci and Lucas numbers

Abstract

The aim of this paper is to give linear independence results for the values of Lambert type series. As an application, we derive arithmetical properties of the sums of reciprocals of Fibonacci and Lucas numbers associated with certain coprime sequences \(\{n_\ell\}_{\ell\geq1}\). For example, the three numbers

$$1, \quad \sum_{p:{\rm prime}}\frac{1}{F_{p^2}}, \quad \sum_{p:{\rm prime}}\frac{1}{L_{p^2}} $$

are linearly independent over \(\mathbb{Q}(\sqrt{5})\), where \(\{F_n\}\) and \(\{L_n\}\) are the Fibonacci and Lucas numbers, respectively.

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Acknowledgements

The authors would like to deeply thank Professor Hajime Kaneko for pointing out the reference [8]. They express their sincere gratitude to Professor Joseph Vandehey for his comments on Erdős’s paper [7]. The authors also would like to gratefully acknowledge Professor Masataka Ono and Professor Wadim Zudilin for pointing out several misprints in the original manuscript. The authors are greatly indebted to the referee for giving helpful suggestions to brush up the original manuscript.

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Correspondence to Y. Suzuki.

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This work was supported by JSPS KAKENHI Grant Numbers JP16J00906 and JP18K03201.

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Duverney, D., Suzuki, Y. & Tachiya, Y. Linear independence results for sums of reciprocals of Fibonacci and Lucas numbers. Acta Math. Hungar. (2020). https://doi.org/10.1007/s10474-020-01060-3

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Key words and phrases

  • linear independence
  • Lambert series
  • Fibonacci number
  • Lucas number

Mathematics Subject Classification

  • primary 11J72
  • secondary 11A41