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The Ramanujan Journal

, Volume 30, Issue 1, pp 39–65 | Cite as

On the rational approximation of the sum of the reciprocals of the Fermat numbers

  • Michael CoonsEmail author
Article

Abstract

Let \(\mathcal{G}(z):=\sum_{n\geqslant0} z^{2^{n}}(1-z^{2^{n}})^{-1}\) denote the generating function of the ruler function, and \(\mathcal {F}(z):=\sum_{n\geqslant} z^{2^{n}}(1+z^{2^{n}})^{-1}\); note that the special value \(\mathcal{F}(1/2)\) is the sum of the reciprocals of the Fermat numbers \(F_{n}:=2^{2^{n}}+1\). The functions \(\mathcal{F}(z)\) and \(\mathcal{G}(z)\) as well as their special values have been studied by Mahler, Golomb, Schwarz, and Duverney; it is known that the numbers \(\mathcal {F}(\alpha)\) and \(\mathcal{G}(\alpha)\) are transcendental for all algebraic numbers α which satisfy 0<α<1.

For a sequence u, denote the Hankel matrix \(H_{n}^{p}(\mathbf {u}):=(u({p+i+j-2}))_{1\leqslant i,j\leqslant n}\). Let α be a real number. The irrationality exponent μ(α) is defined as the supremum of the set of real numbers μ such that the inequality |αp/q|<q μ has infinitely many solutions (p,q)∈ℤ×ℕ.

In this paper, we first prove that the determinants of \(H_{n}^{1}(\mathbf {g})\) and \(H_{n}^{1}(\mathbf{f})\) are nonzero for every n⩾1. We then use this result to prove that for b⩾2 the irrationality exponents \(\mu(\mathcal{F}(1/b))\) and \(\mu(\mathcal{G}(1/b))\) are equal to 2; in particular, the irrationality exponent of the sum of the reciprocals of the Fermat numbers is 2.

Keywords

Irrationality exponents Padé approximants Hankel determinants Fermat numbers 

Mathematics Subject Classification (2010)

11J82 41A21 

Notes

Acknowledgements

We wish to thank Yann Bugeaud, Kevin Hare, Cameron Stewart, and Jeffrey Shallit for helpful comments and conversations.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia

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