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Transcendence of special values of log double sine function. (English) Zbl 1310.11076

Summary: In [J. Number Theory 129, No. 9, 2154–2165 (2009; Zbl 1175.11039); ibid. 130, No. 5, 1251 (2010; Zbl 1216.11070)], S. Gun et al. studied transcendental values of the logarithm of the gamma function. They showed that for any rational number \(x\) with \(0 < x < \frac{1}{2}\), the number \(\log \Gamma(x) + \log \Gamma(1-x)\) is transcendental with at most one possible exception. In this paper, we study transcendental values of log double sine function using their method.

MSC:

11J81 Transcendence (general theory)
11J86 Linear forms in logarithms; Baker’s method
11J91 Transcendence theory of other special functions
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References:

[1] A. Baker, Transcendental number theory , Cambridge Univ. Press, London, 1975. · Zbl 0297.10013
[2] E. W. Barnes, On the theory of the multiple gamma function, Trans. Cambridge Philos. Soc. 19 (1904), 374-425.
[3] C. Deninger, Local \(L\)-factors of motives and regularized determinants, Invent. Math. 107 (1992), no. 1, 135-150. · Zbl 0762.14015
[4] S. Gun, M. R. Murty and P. Rath, Transcendence of the log gamma function and some discrete periods, J. Number Theory 129 (2009), no. 9, 2154-2165. · Zbl 1175.11039
[5] S. Gun, M. R. Murty and P. Rath, Corrigendum to “Transcendence of the log gamma function and some discrete periods” [J. Number Theory 129 (9) (2009) 2154-2165], J. Number Theory 130 (2010), 1251. · Zbl 1175.11039
[6] S. Koyama and N. Kurokawa, Values of the double sine function, J. Number Theory 123 (2007), no. 1, 204-223. · Zbl 1160.11044
[7] N. Kurokawa and S. Koyama, Multiple sine functions, Forum Math. 15 (2003), no. 6, 839-876. · Zbl 1065.11065
[8] N. Kurokawa and M. Wakayama, Extremal values of double and triple trigonometric functions, Kyushu J. Math. 58 (2004), no. 1, 141-166. · Zbl 1058.33001
[9] M. Lerch, Další studie v oboru Malmsténovských řad, Rozpravy České Akad. 3 (1894), no. 28, 1-61.
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