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Decimal expansion of G(2/3) where G is the Barnes G-function.
8

%I #15 Jun 11 2024 02:07:22

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

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

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

%N Decimal expansion of G(2/3) where G is the Barnes G-function.

%H V. S. Adamchik, <a href="http://arxiv.org/abs/math/0308086">Contributions to the Theory of the Barnes function</a>, arXiv:math/0308086 [math.CA], 2003.

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Barnes_G-function">Barnes G-function</a>.

%F (3^(1/72)*e^(1/9 + (-2*Pi^2 + 3*PolyGamma(1, 1/3))/(36*sqrt(3)*Pi)))/(A^(4/3)*Gamma(2/3)^(1/3)), where PolyGamma(1, .) is the derivative of the digamma function and A the Glaisher-Kinkelin constant (A074962).

%F G(1/3) * G(2/3) = A252798 * A252799 = 3^(7/36) * exp(2/9) / (A^(8/3) * 2^(1/3) * Pi^(1/3) * Gamma(1/3)^(1/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Mar 01 2015

%e 0.7768493857761814773011834392215499808040471363453813...

%t RealDigits[BarnesG[2/3], 10, 105] // First

%Y Cf. A074962, A087013, A087014, A087015, A087016, A087017, A252798.

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Dec 22 2014