OFFSET
0,1
LINKS
D. J. Broadhurst, Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity, arXiv:hep-th/9803091, 1998;
Eric Weisstein's MathWorld, Polygamma Function.
Wikipedia, Polygamma Function.
FORMULA
S_2 = (1/27)*(PolyGamma(1, 1/3) - PolyGamma(1, 2/3)).
Also equals 2/3^(3/2) Cl_2(2Pi/3) where Cl_2 is the Clausen function Cl_2(x) = Sum_{n>0} sin(n x)/n^2.
EXAMPLE
0.2604341376321620989557291432080307854550447788484284734073666876556...
MATHEMATICA
S2 = (1/27)*(PolyGamma[1, 1/3] - PolyGamma[1, 2/3]);
RealDigits[S2, 10, 104][[1]]
PROG
(PARI)
polygamma(n, x) = if (n == 0, psi(x), (-1)^(n+1)*n!*zetahurwitz(n+1, x));
(polygamma(1, 1/3) - polygamma(1, 2/3))/27 \\ Gheorghe Coserea, Sep 30 2018
(PARI)
clausen(n, x) = my(z = polylog(n, exp(I*x))); if (n%2, real(z), imag(z));
2/3^(3/2) * clausen(2, 2*Pi/3) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
sumpos(n=0, (2*n+1)/((3*n+1)^2*(3*n+2)^2)) \\ Gheorghe Coserea, Sep 30 2018
(PARI)
4/81*sumalt(n=0, (-1/27)^n*(9/(6*n+1)^2 - 9/(6*n+2)^2 - 12/(6*n+3)^2 - 3/(6*n+4)^2 + 1/(6*n+5)^2)) \\ Gheorghe Coserea, Sep 30 2018
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, Jun 20 2016
STATUS
approved