A244995 Decimal expansion of p_4(1), a particular radial probability density of a 4-step uniform random walk.
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, 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, 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
Offset: 0
Examples
0.329933801060064059039790652286952964693683048075834277360260393626...
Links
- Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin, Densities of Short Uniform Random Walks p. 974, Canad. J. Math. 64(2012), 961-990.
- MathOverflow, Integral_{0..infinity} x*[J_0(x)]^5 dx: source and context, if any?
Programs
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Maple
evalf(GAMMA(1/15)*GAMMA(2/15)*GAMMA(4/15)*GAMMA(8/15) / (8*sqrt(5)*Pi^4), 120); # Vaclav Kotesovec, Jun 10 2019
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Mathematica
RealDigits[(2*Sqrt[15]*Re[HypergeometricPFQ[{1/2, 1/2, 1/2}, {5/6, 7/6}, 125/4]])/Pi^2, 10, 104] // First
Formula
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.
p_4(1) = (2*sqrt(15)*Re(3F2(1/2, 1/2, 1/2; 5/6, 7/6; 125/4)))/Pi^2.
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))).
Equals Gamma(1/15) * Gamma(2/15) * Gamma(4/15) * Gamma(8/15) / (8*sqrt(5)*Pi^4). - Vaclav Kotesovec, Jun 10 2019