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Transcendence of Jacobi’s theta series. (English) Zbl 0884.11030

The main result of this paper is the following theorem. Let \[ y(q)\in \Biggl\{1+ 2\sum^\infty_{n= 1} q^{n^2},\;1+\sum^\infty_{n= 1}(-1)^n q^{n^2},\;2q^{1/4}\sum^\infty_{n= 1} q^{n(n- 1)}\Biggr\} \] and \(\alpha\) be an algebraic number such that \(0<|\alpha|< 1\). Then the numbers \(y(\alpha)\), \(y'(\alpha)\) and \(y''(\alpha)\) are algebraically independent. The proof uses Nesterenko’s theorem, Nishioka’s result and simple calculations.
Reviewer: J.Hančl (Ostrava)

MSC:

11J91 Transcendence theory of other special functions
11J85 Algebraic independence; Gel’fond’s method
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References:

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