## 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].

Show Summary Details# A Note on Transcendental Power Series Mapping the Set of Rational Numbers into Itself

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More options …# Communications in Mathematics

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Editor-in-Chief: Rossi, Olga

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Mathematical Citation Quotient (MCQ) 2016: 0.28

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

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[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

**Received**: 2015-11-27

**Accepted**: 2016-07-14

**Published Online**: 2017-06-28

**Published in Print**: 2017-06-27

**Citation Information: **Communications in Mathematics, Volume 25, Issue 1, Pages 1–4, ISSN (Online) 2336-1298, DOI: https://doi.org/10.1515/cm-2017-0001.

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

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