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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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A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself

Diego Marques / Elaine Silva
Published Online: 2017-06-28 | DOI: https://doi.org/10.1515/cm-2017-0001

Abstract

In this note, we prove that there is no transcendental entire function f(z) ∈ ℚ[[z]] such that f(ℚ) ⊆ ℚ and den f(p/q) = F(q), for all sufficiently large q, where F(z) ∈ ℤ[z].

Keywords: Liouville numbers; Mahler’s question; power series

References

  • [1] K. Mahler: Arithmetic properties of lacunary power series with integral coefficients. J. Austral. Math. Soc. 5 (1965) 56{64.Google Scholar

  • [2] K. Mahler: Some suggestions for further research. Bull. Austral. Math. Soc. 29 (1984) 101{108.CrossrefGoogle Scholar

  • [3] E. Maillet: Introduction ¸ la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions. Gauthier-Villars, Paris (1906).Google Scholar

  • [4] D. Marques, C.G. Moreira: A variant of a question proposed by K. Mahler concerning Liouville numbers. Bull. Austral. Math. Soc. 91 (2015) 29{33.CrossrefGoogle Scholar

  • [5] D. Marques, J. Ramirez: On transcendental analytic functions mapping an uncountable class of U-numbers into Liouville numbers. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015) 25{28.CrossrefGoogle Scholar

  • [6] D. Marques, J. Ramirez, E. Silva: A note on lacunary power series with rational coe_cients. Bull. Austral. Math. Soc. 93 (2015) 1{3.Google Scholar

  • [7] D. Marques, J. Schleischitz: On a problem posed by Mahler. J. Austral. Math. Soc. 100 (2016) 86{107.Google Scholar

About the article

Received: 2015-11-27

Accepted: 2016-07-14

Published Online: 2017-06-28

Published in Print: 2017-06-27


Citation Information: Communications in Mathematics, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2017-0001.

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© 2017. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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