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A355022
Decimal expansion of Im(Li(3, (i+1)/2)), where Li(n, z) is the polylogarithm function and i is the imaginary unit.
0
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, 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, 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
OFFSET
0,1
COMMENTS
This constant is the subject of the paper by Campbell, Levrie and Nimbran (2021).
The real part of Li(3, (i+1)/2) is 35*zeta(3)/64 - 5*Pi^2*log(2)/192 + log(2)^3/48.
LINKS
John M. Campbell, A Wilf-Zeilberger-based solution to the Basel problem with applications, Discrete Math. Lett., Vol. 10 (2022), pp. 21-27.
John M. Campbell, Marco Cantarini, and Jacopo D'Aurizio, Symbolic computations via Fourier-Legendre expansions and fractional operators, Integral Transforms and Special Functions, Vol. 33, No. 2 (2022), pp. 157-175.
John M. Campbell, Jacopo D'Aurizio, and Jonathan Sondow, On the interplay among hypergeometric functions, complete elliptic integrals, and Fourier-Legendre expansions, Journal of Mathematical Analysis and Applications, Vol. 479, No. 1 (2019), pp. 90-121.
John M. Campbell, Paul Levrie, and Amrik Nimbran, A natural companion to Catalan's constant, Journal of Classical Analysis, Vol. 18, No. 2 (2021), pp. 117-135.
John M. Campbell, Paul Levrie, Ce Xu, and Jianqiang Zhao, On a problem involving the squares of odd harmonic numbers, arXiv preprint, arXiv:2206.05026 [math.NT], 2022.
Marco Cantarini and Jacopo D’Aurizio, On the interplay between hypergeometric series, Fourier-Legendre expansions and Euler sums, Bollettino dell'Unione Matematica Italiana, Vol. 12, No. 4 (2019), pp. 623-656.
Mark W. Coffey, Evaluation of a ln tan integral arising in quantum field theory, Journal of Mathematical Physics, Vol. 49, No. 9 (2008), 093508; arXiv preprint, arXiv:0801.0272 [math-ph], 2008.
Amrik Singh Nimbran, Deriving Forsyth-Glaisher type series for 1/Pi and Catalan's constant by an elementary method, Math. Student, Vol. 84, No. 1-2 (2015), pp. 69-86.
Amrik Singh Nimbran, Some New 4F3(1) Hypergeometric Series, preprint, 2021.
Anthony Sofo and Amrik Singh Nimbran, Euler-like sums via powers of log, arctan and arctanh functions, Integral Transforms and Special Functions, Vol. 31, No. 12 (2020), pp. 966-981.
Eric Weisstein's World of Mathematics, Polylogarithm.
Wikipedia, Polylogarithm.
FORMULA
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.
Equals (1/2) * Integral_{x=0..1} log(1-x)^2/(1+x^2) dx.
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).
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.
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.
Equals Sum_{k>=1} sin(Pi*k/4)/(2^(k/2)*k^3).
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.
EXAMPLE
0.57007740708876897819560975759007455106314580991872...
MATHEMATICA
RealDigits[Im[PolyLog[3, (I + 1)/2]], 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jun 25 2022
STATUS
approved