## 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

Zbl 0712.11038
Full Text:

### 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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