A355022 Decimal expansion of Im(Li(3, (i+1)/2)), where Li(n, z) is the polylogarithm function and i is the imaginary unit.

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

Table of n, a(n) for n=0..104.

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

Cf. A001008, A002805, A006752.

Sequence in context: A048658 A001111 A374961 * A133079 A116916 A080332

Adjacent sequences: A355019 A355020 A355021 * A355023 A355024 A355025

Keyword

nonn,cons

Author

Amiram Eldar, Jun 25 2022

Status

approved