|
%I #15 Jun 10 2019 12:13:55
%S 3,2,9,9,3,3,8,0,1,0,6,0,0,6,4,0,5,9,0,3,9,7,9,0,6,5,2,2,8,6,9,5,2,9,
%T 6,4,6,9,3,6,8,3,0,4,8,0,7,5,8,3,4,2,7,7,3,6,0,2,6,0,3,9,3,6,2,6,0,2,
%U 7,5,7,4,2,5,7,2,6,4,4,0,5,8,4,2,3,3,4,1,5,5,1,7,2,2,6,7,4,9,4,8,8,9,4,3
%N Decimal expansion of p_4(1), a particular radial probability density of a 4-step uniform random walk.
%H Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, <a href="http://dx.doi.org/10.4153/CJM-2011-079-2">Densities of Short Uniform Random Walks</a> p. 974, Canad. J. Math. 64(2012), 961-990.
%H MathOverflow, <a href="http://mathoverflow.net/questions/206352">Integral_{0..infinity} x*[J_0(x)]^5 dx: source and context, if any?
%F p_4(x) = (2*sqrt(16-x^2)*Re(3F2(1/2, 1/2, 1/2; 5/6, 7/6; (16-x^2)^3/(108*x^4))))/(Pi^2*x) where 3F2 is the hypergeometric function.
%F p_4(1) = (2*sqrt(15)*Re(3F2(1/2, 1/2, 1/2; 5/6, 7/6; 125/4)))/Pi^2.
%F p_4(1) = (1/(2*Pi^2))*sqrt((gamma(1/15)*gamma(2/15)*gamma(4/15)*gamma(8/15))/(5*gamma(7/15)*gamma(11/15)*gamma(13/15)*gamma(14/15))).
%F Equals Gamma(1/15) * Gamma(2/15) * Gamma(4/15) * Gamma(8/15) / (8*sqrt(5)*Pi^4). - _Vaclav Kotesovec_, Jun 10 2019
%e 0.329933801060064059039790652286952964693683048075834277360260393626...
%p evalf(GAMMA(1/15)*GAMMA(2/15)*GAMMA(4/15)*GAMMA(8/15) / (8*sqrt(5)*Pi^4), 120); # _Vaclav Kotesovec_, Jun 10 2019
%t RealDigits[(2*Sqrt[15]*Re[HypergeometricPFQ[{1/2, 1/2, 1/2}, {5/6, 7/6}, 125/4]])/Pi^2, 10, 104] // First
%K nonn,cons,walk
%O 0,1
%A _Jean-François Alcover_, Jul 09 2014
|
