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A \(q\)-analogue of the Fukazawa-Gel’fond-Gramain theorem. (Un \(q\)-analogue du théorème de Fukazawa-Gel’fond-Gramain.) (French. English summary) Zbl 1294.30056

Summary: Let \(q\in \mathbb Z\) such that \(|q|\geqslant 2\). In this note, we show that if \(f\) is an entire function such that \(f(q^{ n }+iq^{ m })\in \mathbb Z[i]\) for \(n,m\in \mathbb N\), and if \(f\) is of sufficiently slow growth, then \(f\) is a polynomial.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
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References:

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