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%I #8 Jun 26 2022 10:47:54
%S 5,7,0,0,7,7,4,0,7,0,8,8,7,6,8,9,7,8,1,9,5,6,0,9,7,5,7,5,9,0,0,7,4,5,
%T 5,1,0,6,3,1,4,5,8,0,9,9,1,8,7,2,8,7,3,2,8,6,6,9,7,0,4,7,9,0,1,5,9,2,
%U 9,6,7,3,9,4,3,2,7,5,7,7,1,7,3,6,1,7,0,5,4,8,5,1,4,0,5,1,4,2,1,4,5,3,8,8,5
%N Decimal expansion of Im(Li(3, (i+1)/2)), where Li(n, z) is the polylogarithm function and i is the imaginary unit.
%C This constant is the subject of the paper by Campbell, Levrie and Nimbran (2021).
%C The real part of Li(3, (i+1)/2) is 35*zeta(3)/64 - 5*Pi^2*log(2)/192 + log(2)^3/48.
%H John M. Campbell, <a href="https://doi.org/10.47443/dml.2022.030">A Wilf-Zeilberger-based solution to the Basel problem with applications</a>, Discrete Math. Lett., Vol. 10 (2022), pp. 21-27.
%H John M. Campbell, Marco Cantarini, and Jacopo D'Aurizio, <a href="https://doi.org/10.1080/10652469.2021.1919103">Symbolic computations via Fourier-Legendre expansions and fractional operators</a>, Integral Transforms and Special Functions, Vol. 33, No. 2 (2022), pp. 157-175.
%H John M. Campbell, Jacopo D'Aurizio, and Jonathan Sondow, <a href="https://doi.org/10.1016/j.jmaa.2019.06.017">On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre expansions</a>, Journal of Mathematical Analysis and Applications, Vol. 479, No. 1 (2019), pp. 90-121.
%H John M. Campbell, Paul Levrie, and Amrik Nimbran, <a href="https://lirias.kuleuven.be/retrieve/638220">A natural companion to Catalan's constant</a>, Journal of Classical Analysis, Vol. 18, No. 2 (2021), pp. 117-135.
%H John M. Campbell, Paul Levrie, Ce Xu, and Jianqiang Zhao, <a href="https://arxiv.org/abs/2206.05026">On a problem involving the squares of odd harmonic numbers</a>, arXiv preprint, arXiv:2206.05026 [math.NT], 2022.
%H Marco Cantarini and Jacopo D’Aurizio, <a href="https://doi.org/10.1007/s40574-019-00198-5">On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums</a>, Bollettino dell'Unione Matematica Italiana, Vol. 12, No. 4 (2019), pp. 623-656.
%H Mark W. Coffey, <a href="https://doi.org/10.1063/1.2981311">Evaluation of a ln tan integral arising in quantum field theory</a>, Journal of Mathematical Physics, Vol. 49, No. 9 (2008), 093508; <a href="https://arxiv.org/abs/0801.0272">arXiv preprint</a>, arXiv:0801.0272 [math-ph], 2008.
%H Amrik Singh Nimbran, <a href="https://www.researchgate.net/profile/Amrik-Nimbran/publication/278730648_DERIVING_FORSYTH-GLAISHER_TYPE_SERIES_FOR_AND_CATALAN'S_CONSTANT_BY_AN_ELEMENTARY_METHOD">Deriving Forsyth-Glaisher type series for 1/Pi and Catalan's constant by an elementary method</a>, Math. Student, Vol. 84, No. 1-2 (2015), pp. 69-86.
%H Amrik Singh Nimbran, <a href="http://dx.doi.org/10.13140/RG.2.2.29508.35202">Some New 4F3(1) Hypergeometric Series</a>, preprint, 2021.
%H Anthony Sofo and Amrik Singh Nimbran, <a href="https://doi.org/10.1080/10652469.2020.1765775">Euler-like sums via powers of log, arctan and arctanh functions</a>, Integral Transforms and Special Functions, Vol. 31, No. 12 (2020), pp. 966-981.
%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Polylogarithm">Polylogarithm</a>.
%F Equals (1/16) * Sum_{k>=0} (-1/4)^k/(2*k+1)^3 + (1/2) * Sum_{k>=0} (-1/4)^k/(4*k+1)^3 + (1/4) * Sum_{k>=0} (-1/4)^k/(4*k+3)^3.
%F Equals (1/2) * Integral_{x=0..1} log(1-x)^2/(1+x^2) dx.
%F Equals (1/3) * Integral_{x=0..1} arctan(x)*log(1+x)/x dx - G*log(2)/2 + 3*Pi^3/128 + Pi*log(2)^2/32, where G is Catalan's constant (A006752).
%F Equals Integral_{x=0..1} arctan(x)*log(1-x)/x dx - G*log(2)/2 + 7*Pi^3/128 + Pi*log(2)^2/32.
%F Equals 23*Pi^3/384 + 3*Pi*log(2)^2/32 - sqrt(2) * Sum_{k>=0} binomial(2*k,k)/(2^(3*k)*(2*k+1)^3) = 23*Pi^3/384 + 3*Pi*log(2)^2/32 - sqrt(2) * 4F3({1/2,1/2,1/2,1/2}, {3/2,3/2,3/2}, 1/2), where 4F3 is the generalized hypergeometric function.
%F Equals Sum_{k>=1} sin(Pi*k/4)/(2^(k/2)*k^3).
%F Equals (1/8) * Sum_{k>=0} (-1)^k * H(k)^2/(2*k+1) + 3*Pi^3/128 - 3*Pi*log(2)^2/32, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
%e 0.57007740708876897819560975759007455106314580991872...
%t RealDigits[Im[PolyLog[3, (I + 1)/2]], 10, 100][[1]]
%Y Cf. A001008, A002805, A006752.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Jun 25 2022
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