Duverney, Daniel; Nishioka, Keiji; Nishioka, Kumiko; Shiokawa, Iekata Transcendence of Jacobi’s theta series. (English) Zbl 0884.11030 Proc. Japan Acad., Ser. A 72, No. 9, 202-203 (1996). 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) Cited in 5 ReviewsCited in 15 Documents MSC: 11J91 Transcendence theory of other special functions 11J85 Algebraic independence; Gel’fond’s method Keywords:transcendence; Jacobi theta series; algebraic independence PDFBibTeX XMLCite \textit{D. Duverney} et al., Proc. Japan Acad., Ser. A 72, No. 9, 202--203 (1996; Zbl 0884.11030) Full Text: DOI References: [1] Tom M. Apsotol: Introduction to Analytic Number Theory. UTM, Springer-Verlag (1976). · Zbl 0335.10001 [2] Tom M. Apsotol: Modular Functions and Dirich-let Series in Number Theory. GTM, Springer-Verlag (1976). [3] C. G. J. Jacobi: Uber die Differentialgleichung, welcher die Reihen 1 \pm 2q + 2qA \pm 2q + etc., 2^ + 2^ + 2 *ff* + etc. Geniige leisten. J. Reine Angew. Math., 36, 97-112 (1847). · ERAM 036.1001cj [4] S. Lang: Introduction to Modular Forms. Springer-Verlag (1776). · Zbl 0344.10011 [5] K. Mahler: On algebraic differential equations satisfied by automorphic functions. J. Austral. Math. Soc, 10, 445-450 (1969). · Zbl 0207.08302 [6] Y. V. Nesterenko: Modular functions and transcendence problems. C. R. Acad. Sci., Paris, ser. 1, 322, 909-914 (1996). · Zbl 0859.11047 [7] Y. V. Nesterenko: Modular functions and transcendence problems. Math. Sb. (to appear). · Zbl 0859.11047 [8] K. Nishioka: A conjecture of Mahler on automorphic functions. Arch. Math., 53, 46-51 (1989). · Zbl 0684.10022 [9] A. Weil: Foundations of Algebraic Geometry. A.M.S. (1962). · Zbl 0168.18701 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.