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A note on transcendental power series mapping the set of rational numbers into itself. (English) Zbl 1425.11138

Summary: In this note, we prove that there is no transcendental entire function \(f(z)\in\mathbb{Q}[[z]]\) such that \(f(\mathbb{Q})\subseteq\mathbb{Q}\) and \(\mathrm{den}\,f(p/q)=F(q)\), for all sufficiently large \(q\), where \(F(z)\in\mathbb{Z}[z]\).

MSC:

11J81 Transcendence (general theory)
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