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

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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].

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

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

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

[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.

[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.

[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.

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

Journal Information

CiteScore 2017: 0.33

SCImago Journal Rank (SJR) 2017: 0.128
Source Normalized Impact per Paper (SNIP) 2017: 0.476

Mathematical Citation Quotient (MCQ) 2017: 0.43

Target Group

researchers in the fields of: algebraic structures, calculus of variations, combinatorics, control and optimization, cryptography, differential equations, differential geometry, fuzzy logic and fuzzy set theory, global analysis, mathematical physics and number theory

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