×

zbMATH — the first resource for mathematics

On the rational approximation of the sum of the reciprocals of the Fermat numbers. (English) Zbl 1271.11075
Ramanujan J. 30, No. 1, 39-65 (2013); addendum ibid. 37, No. 1, 109–111 (2015).
The author proves that if \(a\geq 2\) is a positive integer then the irrationality exponent of the numbers \[ \sum_{n=1}^\infty \frac{a^{2^n}}{1+a^{2^n}}\quad \text{and}\quad \sum_{n=1}^\infty \frac{a^{2^n}}{1-a^{2^n}} \] is equal to \(2\). The proof is based on the Padé approximation and uses the Hankel determinants. As a special case we obtain that the sum of reciprocal Fermat numbers has an irrationality exponent equal to \(2\).

MSC:
11J82 Measures of irrationality and of transcendence
41A21 Padé approximation
PDF BibTeX XML Cite
Full Text: DOI
References:
[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) · Zbl 0974.11010
[2] Adamczewski, B., Rivoal, T.: Irrationality measures for some automatic real numbers. Math. Proc. Camb. Philos. Soc. 147(3), 659–678 (2009) · Zbl 1205.11080
[3] Brezinski, C.: Padé-type Approximation and General Orthogonal Polynomials. International Series of Numerical Mathematics, vol. 50. Birkhäuser, Basel (1980) · Zbl 0418.41012
[4] Bugeaud, Y.: On the rational approximation of the Thue–Morse–Mahler number. Ann. Inst. Fourier (Grenoble) (to appear) · Zbl 1285.11100
[5] Coons, M.: Extension of some theorems of W. Schwarz. Can. Math. Bull. (2011). doi: 10.4153/CMB-2011-037-9 · Zbl 1275.11026
[6] Duverney, D.: Transcendence of a fast converging series of rational numbers. Math. Proc. Camb. Philos. Soc. 130(2), 193–207 (2001) · Zbl 0999.11037
[7] Golomb, S.W.: On the sum of the reciprocals of the Fermat numbers and related irrationalities. Can. J. Math. 15, 475–478 (1963) · Zbl 0115.04501
[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
[9] Mahler, K.: Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101(1), 342–366 (1929) · JFM 55.0115.01
[10] Mahler, K.: Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen. Math. Z. 32(1), 545–585 (1930) · JFM 56.0186.01
[11] Mahler, K.: Uber das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen. Math. Ann. 103(1), 573–587 (1930) · JFM 56.0185.03
[12] Roth, K.F.: Rational approximations to algebraic numbers. Mathematika 2, 1–20 (1955). Corrigendum 168 · Zbl 0064.28501
[13] Schwarz, W.: Remarks on the irrationality and transcendence of certain series. Math. Scand. 20, 269–274 (1967) · Zbl 0164.05701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.