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On arithmetic properties of an entire function. (Sur les propriétés arithmétiques d’une fonction entière.) (French) Zbl 0942.11035

For \(q\in \mathbb{Q}\), \(|q|>1\), let \(T_q\) be the Tschakaloff’s function defined by \(T_q(z)= \sum_{n=0}^\infty q^{-n(n+1)/2} z^n\). Let \(\mathbb{K}\) be any quadratic extension of \(\mathbb{Q}\).
Under some conditions on \(q\), the author proves that, if \(\alpha\in \mathbb{K}^*\), then \(T_q(\alpha)\not\in \mathbb{K}\). The theorem holds, in particular, if \(q\in \mathbb{Z}\). This beautiful result is obtained by proving that, if \(f(\alpha)\in \mathbb{K}\), then some function \(g\) related to \(T_q\) is a rational function, following the same idea as in his paper [Acta Arith. 55, 233-240 (1990; Zbl 0712.11038)]. However, in the present paper, the author attacks directly the Kronecker determinants of \(g\) and proves, after some spectacular computations, that they vanish.
Reviewer: D.Duverney (Lille)

MSC:

11J72 Irrationality; linear independence over a field
30D15 Special classes of entire functions of one complex variable and growth estimates

Citations:

Zbl 0712.11038
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References:

[1] : Les Nombres p-Adiques, Presses Universitaires de France Collection Sup, Paris, 1975
[2] Bundschuh, Portugaliae Math. 33 pp 1– (1974)
[3] Duverney, Acta Arithm 64 pp 2– (1993) · Zbl 0779.11028
[4] Duverney, Comptes Rendus de 1’Académie des Sciences 320 pp 1041– (1995)
[5] Töpper, Analysis 15 pp 25– (1995)
[6] Tschakaloff, Math Ann 80 pp 62– (1921)
[7] Math Ann 84 pp 100– (1921)
[8] Vänäänen, Math Scand 73 pp 197– (1993) · Zbl 0818.11028
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