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Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

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Abstract

Irrationality measures are given for the values of the series \(\sum_{n=0}^{\infty} t^{n}/W_{an+b}\), where \(a,b\in\mathbb{Z}^+, 1\le b\le a, (a,b)=1\) and W n is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality of all the numbers

$$ \sum_{n=0}^\infty \frac{1}{f_{an+b}},\quad \sum_{n=0}^\infty \frac{1}{l_{an+b}}, $$

where f n and l n are the Fibonacci and Lucas numbers, respectively.

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Authors and Affiliations

  1. Matemaattisten tieteiden laitos, Linnanmaa, PL 3000, 90014, Oulun Yliopisto, Finland

    Tapani Matala-Aho

  2. Laboratoire de Mathématiques Pures et Appliquées, Université du Littoral, 50 Rue F. Buisson, B.P. 699, 62228, Calais Cédex, France

    Marc Prévost

Authors
  1. Tapani Matala-Aho
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  2. Marc Prévost
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Correspondence to Tapani Matala-Aho.

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2000 Mathematics Subject Classification Primary—11J82, 11B39

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Matala-Aho, T., Prévost, M. Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers. Ramanujan J 11, 249–261 (2006). https://doi.org/10.1007/s11139-006-6511-4

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  • DOI: https://doi.org/10.1007/s11139-006-6511-4

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