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Decimal expansion of Pi^3/48 + Pi*log(2)^2/4.
1

%I #30 Aug 28 2024 15:46:29

%S 1,0,2,3,3,1,1,0,1,2,2,3,6,3,7,0,3,2,3,0,8,4,8,2,0,5,0,4,0,8,8,4,8,6,

%T 7,3,8,3,1,8,7,2,0,9,7,6,7,4,7,3,2,8,1,3,0,3,0,5,1,3,4,2,7,6,3,6,2,9,

%U 5,3,3,4,3,9,7,5,6,0,8,6,6,8,2,9,2,3,4

%N Decimal expansion of Pi^3/48 + Pi*log(2)^2/4.

%C The first part of Ramanujan's question 308 in the Journal of the Indian Mathematical Society (III, 168) asked "Show that Integral_{t=0..Pi/2} t * cotan(t) * log(sin(t)) dt = -Pi^3/48 - Pi*log(2)^2/4".

%H V. S. Adamchik <a href="https://doi.org/10.1016/S0377-0427(96)00167-7">On Stirling numbers and Euler sums</a>, J. Comput. Appl. Math. 79 (1997) 119-130. Equals [1/2,3] apart from factor 8/sqrt Pi.

%H B. C. Berndt, Y. S. Choi and S. Y. Kang, <a href="https://faculty.math.illinois.edu/~berndt/jims.ps">The problems submitted by Ramanujan to the Journal of Indian Math. Soc.</a>, in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q308, JIMS III).

%H B. C. Berndt, Y. S. Choi and S. Y. Kang, <a href="https://citeseerx.ist.psu.edu/pdf/ae75da0be9fb455e2c55daa5fca46ae63e6a60bd">The problems submitted by Ramanujan to the Journal of Indian Math. Soc.</a>, in: Continued fractions, Contemporary Math., 236 (1999), 15-56 (see Q308, JIMS III).

%H R. J. Mathar, <a href="https://arxiv.org/abs/2408.15212">Chebyshev expansion of x^m*(-log x)^l in the interval 0<=x<=1</a>, arXiv:2408.15212 (2024)

%H Pedro Ribeiro, <a href="https://doi.org/10.1080/00029890.2018.1460990">Problem 12051</a>, The American Mathematical Monthly, Vol. 125, No. 6 (2018), p. 563; <a href="https://doi.org/10.1080/00029890.2020.1691901">A Series Involving Central Binominal [sic] Coefficients</a>, Solution to Problem 12051 by Hongwei Chen, ibid., Vol. 127, No. 2 (2020), pp. 182-183.

%F Equals Sum_{k>=0} binomial(2*k,k)/(4^k*(2*k+1)^3) (Ribeiro, 2018). - _Amiram Eldar_, Oct 04 2021

%F Equals 4F3(1/2,1/2,1/2,1/2 ; 3/2,3/2,3/2 ; 1) [Adamchik]. - _R. J. Mathar_, Aug 19 2024

%F Equals A196877/2. - _R. J. Mathar_, Aug 23 2024

%e 1.0233110122363703230848205040884867383187209767473281303051342763...

%t RealDigits[Pi^3/48 + Pi*Log[2]^2/4, 10, 100][[1]] (* _Amiram Eldar_, Oct 04 2021 *)

%o (PARI) Pi^3/48+Pi*log(2)^2/4

%o (PARI) -intnum(x=0,Pi/2,x*cotan(x)*log(sin(x)))

%K nonn,cons

%O 1,3

%A _Hugo Pfoertner_, Sep 17 2018