The Ramanujan Journal

, Volume 11, Issue 2, pp 249–261 | Cite as

Quantitative irrationality for sums of reciprocals of Fibonacci and Lucas numbers

Article

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 Wn 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 fn and ln are the Fibonacci and Lucas numbers, respectively.

Keywords

Irrationality measure Padé approximation Cyclotomic polynomial q-series 

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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Matemaattisten tieteiden laitosOulun YliopistoFinland
  2. 2.Laboratoire de Mathématiques Pures et AppliquéesUniversité du LittoralCalais CédexFrance

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