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Decimal expansion of Product_{k>=1} (1 - exp(-5*Pi*k)).
17

%I #7 May 13 2023 05:05:07

%S 9,9,9,9,9,9,8,4,9,2,9,8,2,4,9,7,4,9,9,8,2,8,5,5,6,8,4,2,4,9,9,5,1,3,

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

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

%N Decimal expansion of Product_{k>=1} (1 - exp(-5*Pi*k)).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dedekind_eta_function">Dedekind eta function</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler_function">Euler function</a>

%F Equals exp(5*Pi/8) * Gamma(1/4) * (9 + 4*sqrt(5))^(1/4) * (-exp(5*Pi/2) + sqrt(exp(5*Pi) + 64*r^24))^(1/4) / (2^(7/4) * sqrt(5) * Pi^(3/4) * r^5), where r = A292904 = 1.00000015070175025002398949386987146797376100643050740569...

%F Equals exp(5*Pi/24) * Gamma(1/4) * (7 + 3*sqrt(5) + 12*sqrt(14*sqrt(5) - 30))^(1/8) / (2*sqrt(5)*Pi^(3/4)). - _Vaclav Kotesovec_, May 13 2023

%e 0.999999849298249749982855684249951337192226280495972174466518680326272...

%t RealDigits[QPochhammer[E^(-5*Pi)], 10, 120][[1]]

%t RealDigits[E^(5*Pi/8) * Gamma[1/4] * (9 + 4*Sqrt[5])^(1/4) * (-E^(5*Pi/2) + Sqrt[E^(5*Pi) + 64*r^24])^(1/4) / (2^(7/4) * Sqrt[5] * Pi^(3/4) * r^5)/.r -> (r/.FindRoot[2^(3/4)*r^6 + 2^(17/8)*E^(5*Pi/24)*r^5 + 2^(5/8)*E^(25*Pi/24)*r - E^(5*Pi/4) == 0, {r, 1}, WorkingPrecision -> 130]), 10, 120][[1]]

%t RealDigits[E^(5*Pi/24) * Gamma[1/4]*(7 + 3*Sqrt[5] + 12*Sqrt[14*Sqrt[5] - 30])^(1/8) / (2*Sqrt[5]*Pi^(3/4)), 10, 120][[1]] (* _Vaclav Kotesovec_, May 13 2023 *)

%Y Cf. A292862, A292863, A259147, A259148, A259149, A292888, A259150, A259151, A292864.

%Y Cf. A292904.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Sep 26 2017