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Decimal expansion of Gamma(1/4).
40

%I #84 Jan 04 2025 10:32:55

%S 3,6,2,5,6,0,9,9,0,8,2,2,1,9,0,8,3,1,1,9,3,0,6,8,5,1,5,5,8,6,7,6,7,2,

%T 0,0,2,9,9,5,1,6,7,6,8,2,8,8,0,0,6,5,4,6,7,4,3,3,3,7,7,9,9,9,5,6,9,9,

%U 1,9,2,4,3,5,3,8,7,2,9,1,2,1,6,1,8,3,6,0,1,3,6,7,2,3,3,8,4,3,0,0,3,6,1,4,7

%N Decimal expansion of Gamma(1/4).

%C Nesterenko proves that this constant is transcendental (he cites Chudnovsky as the first to show this); in fact it is algebraically independent of Pi and e^Pi over Q. - _Charles R Greathouse IV_, Nov 11 2013

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.

%D Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:13 at page 414.

%H Harry J. Smith, <a href="/A068466/b068466.txt">Table of n, a(n) for n = 1..1000</a>

%H William Duke and Özlem Imamoḡlu, <a href="http://www.numdam.org/item/JTNB_2006__18_1_113_0/">Special values of multiple gamma functions</a>, Journal de théorie des nombres de Bordeaux, Vol. 18. No. 1 (2006), pp. 113-123.

%H Greg J. Fee and Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/gamma14.txt">Gamma(1/4) to 25000 digits</a>.

%H Yu. V. Nesterenko, <a href="https://doi.org/10.1070/SM1996v187n09ABEH000158">Modular functions and transcendence questions</a>, Sbornik: Mathematics, Vol. 187, No. 9 (1996), pp. 1319-1348. (English translation)

%H Simon Plouffe, <a href="https://plouffe.fr/simon/constants/gamma14.txt">Gamma(1/4) to 250000 digits</a>.

%H Dan Romik, <a href="https://arxiv.org/abs/2303.13427">On Viazovska's modular form inequalities</a>, arXiv:2303.13427 [math.NT], 2023.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GammaFunction.html">Gamma Function</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Particular_values_of_the_Gamma_function#General_rational_arguments">Particular values of the Gamma function: General rational arguments</a>.

%H <a href="/index/Ga#gamma_function">Index to sequences related to the Gamma function</a>.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F From _Amiram Eldar_, Jun 12 2021: (Start)

%F Equals sqrt(2*sqrt(2*Pi^3)*G), where G is Gauss's constant (A014549).

%F Equals (2*Pi)^(3/4) * Product_{k>=1} tanh(k*Pi/2) (Duke and Imamoḡlu, 2006). (End)

%F Gamma(1/4) * A068465 = A063448. - _R. J. Mathar_, May 22 2024

%F Equals Product_{n>=1} exp((2*(6*n + 1)*(1 - beta(n)) - (eta(n) - 1))/(4*n)), where eta(n) and beta(n) are the Dirichlet eta and beta functions, respectively. - _Antonio Graciá Llorente_, Sep 05 2024

%e 3.6256099082219083119306851558676720029951676828800654674333...

%p evalf(GAMMA(1/4));

%t RealDigits[Gamma[1/4], 10, 110][[1]] (* _Bruno Berselli_, Dec 13 2012 *)

%o (PARI) default(realprecision, 1080); x=gamma(1/4); for (n=1, 1000, d=floor(x); x=(x-d)*10; write("b068466.txt", n, " ", d)); \\ _Harry J. Smith_, Apr 19 2009

%o (Magma) R:= RealField(100); SetDefaultRealField(R); Gamma(1/4); // _G. C. Greubel_, Mar 10 2018

%Y Cf. A014549, A248557.

%K cons,easy,nonn

%O 1,1

%A _Benoit Cloitre_, Mar 10 2002