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Communications in Mathematics

Editor-in-Chief: Rossi, Olga

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Mathematical Citation Quotient (MCQ) 2016: 0.28

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2336-1298
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Diophantine Approximations of Infinite Series and Products

Ondřej Kolouch
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  • Department of Mathematics, Faculty of Science, University of Ostrava, 30. Dubna 22, Ostrava, Czech Republic
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/ Lukáš Novotný
Published Online: 2016-08-20 | DOI: https://doi.org/10.1515/cm-2016-0006

Abstract

This survey paper presents some old and new results in Diophantine approximations. Some of these results improve Erdos’ results on irrationality. The results in irrationality, transcendence and linear independence of infinite series and infinite products are put together with idea of irrational sequences and expressible sets.

Keywords: Infinite products; irrationality; linear independence; expressible set

References

  • [1] C. Badea: The irrationality of certain Infinite products. Studia Univ. Babeş-Bolyai Math. 31 (3) (1986) 3-8.Google Scholar

  • [2] C. Badea: The irrationality of certain Infinite series. Glasgow Mathematical Journal 29 (2) (1987) 221-228.CrossrefGoogle Scholar

  • [3] D. Duverney: Sur les séries de nombres rationnels à convergence rapide. Comptes Rendus de l'Académie des Sciences, Series I, Mathematics 328 (7) (1999) 553-556.Google Scholar

  • [4] P. Erdős: Problem 4321. The American Mathematical Monthly 64 (7) (1950).Google Scholar

  • [5] P. Erdős: Some Problems and Results on the Irrationality of the Sum of Infinite Series. Journal of Mathematical Sciences 10 (1975) 1-7.Google Scholar

  • [6] P. Erdős: Erdős problem no. 6. 1995 Prague Midsummer Combinatorial Workshop, KAM Series (95-309) (ed. M. Klazar) (KAM MPP UK, Prague, 1995)(1995).Google Scholar

  • [7] P. Erdős, E. G. Straus: On the irrationality of certain Ahmes series. Journal of Indian Mathematical Society 27 (1964) 129-133.Google Scholar

  • [8] J. Hančl: Expression of Real Numbers with the Help of Infinite Series. Acta Arithmetica LIX (2) (1991) 97-104.Google Scholar

  • [9] J. Hančl: Criterion for Irrational Sequences. Journal of Number Theory 43 (1) (1993) 88-92.Google Scholar

  • [10] J. Hančl, F. Filip: Irrationality Measure of Sequences. Hiroshima Math. J. 35 (2) (2005) 183-195.Google Scholar

  • [11] J. Hančl, O. Kolouch: Erdős’ method for determining the irrationality of products. Bull. Aust. Math. Soc. 84 (3) (2011) 414-424.CrossrefGoogle Scholar

  • [12] J. Hančl, O. Kolouch: Irrationality of Infinite products. Publ. Math. Debrecen 83 (4) (2013) 667-681.Web of ScienceCrossrefGoogle Scholar

  • [13] J. Hančl, O. Kolouch, L. Novotný: A Criterion for linear independence of Infinite products. An. St. Univ. Ovidius Constanta 23 (2) (2015) 107-120.Web of ScienceGoogle Scholar

  • [14] J. Hančl, O. Kolouch, S. Pulcerová, J. Štìpnička: A note on the transcendence of Infinite products. Czechoslovak Math. J. 62 (137) (2012) 613-623.Web of ScienceGoogle Scholar

  • [15] J. Hančl, K. Korčeková, L. Novotný: Productly linearly independent sequences. Stud. Sci. Math. Hung. 52 (2015) 350-370.Web of ScienceGoogle Scholar

  • [16] J. Hančl, R. Nair, L. Novotný: On expressible sets of products. Period. Math. Hung. 69 (2) (2014) 199-206.CrossrefGoogle Scholar

  • [17] J. Hančl, R. Nair, L. Novotný, J. Šustek: On the Hausdorff dimension of the expressible set of certain sequences. Acta Arithmetica 155 (1) (2012) 85-90.Web of ScienceGoogle Scholar

  • [18] J. Hančl, R. Nair, J. Šustek: On the Lebesgue measure of the expressible set of certain sequences. Indag. Mathem. 17 (4) (2006) 567-581.CrossrefGoogle Scholar

  • [19] J. Hančl, A. Schinzel, J. Šustek: On Expressible Sets of Geometric Sequences. Funct. Approx. Comment. Math. 38 (2008) 341-357.Google Scholar

  • [20] T. Kurosawa, Y. Tachiya, T. Tanaka: Algebraic independence of the values of certain infinite products and their derivatives related to Fibonacci and Lucas numbers (Analytic Number Theory: Number Theory through Approximation and Asymptotics). In: Proceedings of Institute for Mathematical Sciences, Kyoto University. Research Institute for Mathematical Sciences (2014) 81-93.Google Scholar

  • [21] F. Luca, Y. Tachiya: Algebraic independence of Infinite products generated by Fibonacci and Lucas numbers. Hokkaido Math. J. 43 (2014) 1-20.Google Scholar

  • [22] M. A. Nyblom: On the construction of a family of transcendental valued Infinite products. Fibonacci Quart. 42 (4) (2004) 353-358.Google Scholar

  • [23] J. Sándor: Some classes of irrational numbers. Studia Universitatis Babeş-Bolyai Mathematica 29 (1984) 3-12.Google Scholar

  • [24] K. Väänänen: On the approximation of certain infinite products. Math. Scand. 73 (2) (1993) 197-208. Google Scholar

About the article

Received: 2016-06-02

Accepted: 2016-07-29

Published Online: 2016-08-20

Published in Print: 2016-08-01


Citation Information: Communications in Mathematics, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2016-0006.

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© 2016. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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