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A386714 Decimal expansion of Integral_{x=0..1} {1/x}^3 * {1/(1-x)}^3 dx, where {} denotes fractional part.

%I A386714 #7 Jul 31 2025 08:41:27
%S A386714 0,1,4,6,1,1,7,0,2,6,1,6,5,3,2,6,9,5,6,8,4,2,7,2,0,3,5,4,7,3,8,7,3,5,
%T A386714 6,5,0,7,6,0,6,8,1,1,5,0,2,6,8,3,5,6,1,6,8,7,0,7,2,8,0,1,8,3,5,6,3,5,
%U A386714 6,5,4,6,7,9,9,4,4,6,5,8,5,9,8,3,1,9,6,3,1,7,5,9,4,3,4,6,3,7,1,1,5,7,3,9,8
%N A386714 Decimal expansion of Integral_{x=0..1} {1/x}^3 * {1/(1-x)}^3 dx, where {} denotes fractional part.
%D A386714 Ovidiu Furdui, Limits, Series, and Fractional Part Integrals: Problems in Mathematical Analysis, New York: Springer, 2013. See section 2.11, page 101.
%H A386714 Ovidiu Furdui, <a href="https://doi.org/10.1080/10652469.2012.708869">A class of fractional part integrals and zeta function values</a>, Integral Transforms and Special Functions, Vol. 24, No. 6 (2013), pp. 485-490.
%F A386714 Equals -zeta(2) + 3*gamma + 36*log(A) - 6*log(2*Pi) + 2, where gamma is Euler's constant and A is the Glaisher-Kinkelin constant.
%F A386714 In general, for m >= 2, Integral_{x=0..1} {1/x}^m * {1/(1-x)}^m dx = 2 * (Sum_{j=2..m-1} (-1)^(m+j-1) * (zeta(j)-1)) + (-1)^m - (2*m) * Sum_{k>=0} (zeta(2*k+m) - zeta(2*k+m+1))/(k+m) (note that the first sum vanishes when m = 2).
%e A386714 0.01461170261653269568427203547387356507606811502683...
%t A386714 RealDigits[-Zeta[2] + 3*EulerGamma + 36*Log[Glaisher] - 6*Log[2*Pi] + 2, 10, 120, -1][[1]]
%o A386714 (PARI) -zeta(2) + 3*Euler + 36*(1/12-zeta'(-1)) - 6*log(2*Pi) + 2
%Y A386714 Cf. A001620 (gamma), A013661, A061444, A074962 (A), A225746.
%Y A386714 Cf. A147533 (m=1), A386713 (m=2), this constant (m=3).
%K A386714 nonn,cons
%O A386714 0,3
%A A386714 _Amiram Eldar_, Jul 31 2025