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 |α−p/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.
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Acknowledgements
We wish to thank Yann Bugeaud, Kevin Hare, Cameron Stewart, and Jeffrey Shallit for helpful comments and conversations.
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The research of M. Coons was supported in part by a Fields–Ontario Fellowship and NSERC.
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Coons, M. On the rational approximation of the sum of the reciprocals of the Fermat numbers. Ramanujan J 30, 39–65 (2013). https://doi.org/10.1007/s11139-012-9410-x
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DOI: https://doi.org/10.1007/s11139-012-9410-x