On transcendental entire functions mapping \(\mathbb{Q}\) into itself (Q2326487)

The authors prove that there is no transcendental entire function \(f(z)\in\mathbb C[[z]]\) such that \(f(\mathbb Q)\subset\mathbb Q\) and denominator of \(f(\frac pq)\) is \(O(q)\), for all rational numbers \(\frac pq\), where \(q\) is sufficiently large. The proof is based on the fact that every holomorphic function can be written as an infinite linear combination of special polynomials with bounded coefficients.