The Ramanujan Journal

, Volume 30, Issue 1, pp 39–65

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

• Michael Coons
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

11J82 41A21

## Notes

### Acknowledgements

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

## References

1. 1.
Allouche, J.-P., Peyrière, J., Wen, Z.-X., Wen, Z.-Y.: Hankel determinants of the Thue–Morse sequence. Ann. Inst. Fourier (Grenoble) 48(1), 1–27 (1998)
2. 2.
Adamczewski, B., Rivoal, T.: Irrationality measures for some automatic real numbers. Math. Proc. Camb. Philos. Soc. 147(3), 659–678 (2009)
3. 3.
Brezinski, C.: Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, vol. 50. Birkhäuser, Basel (1980)
4. 4.
Bugeaud, Y.: On the rational approximation of the Thue–Morse–Mahler number. Ann. Inst. Fourier (Grenoble) (to appear) Google Scholar
5. 5.
Coons, M.: Extension of some theorems of W. Schwarz. Can. Math. Bull. (2011). doi: Google Scholar
6. 6.
Duverney, D.: Transcendence of a fast converging series of rational numbers. Math. Proc. Camb. Philos. Soc. 130(2), 193–207 (2001)
7. 7.
Golomb, S.W.: On the sum of the reciprocals of the Fermat numbers and related irrationalities. Can. J. Math. 15, 475–478 (1963)
8. 8.
Liouville, J.: Sur des classes très étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationelles algébriques. C. R. Acad. Sci. Paris 18, 883–885 (1844). 910–911 Google Scholar
9. 9.
Mahler, K.: Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101(1), 342–366 (1929)
10. 10.
Mahler, K.: Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen. Math. Z. 32(1), 545–585 (1930)
11. 11.
Mahler, K.: Uber das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen. Math. Ann. 103(1), 573–587 (1930)
12. 12.
Roth, K.F.: Rational approximations to algebraic numbers. Mathematika 2, 1–20 (1955). Corrigendum 168
13. 13.
Schwarz, W.: Remarks on the irrationality and transcendence of certain series. Math. Scand. 20, 269–274 (1967)