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)


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: DOI


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