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Decimal expansion of Product_{k>=1} (1 - exp(-Pi*k/16)).
5

%I #6 Dec 17 2023 09:42:04

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

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

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

%N Decimal expansion of Product_{k>=1} (1 - exp(-Pi*k/16)).

%C In general, for 0 <= x < 1, QPochhammer(x) = (-8*x*QPochhammer(x^4)^8 + sqrt(QPochhammer(x^2)^24/QPochhammer(x^4)^8 + 64*x^2*QPochhammer(x^4)^16))^(1/8).

%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="http://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(Pi/384) * Gamma(1/4) / (((-16 + 6*sqrt(-24 + 22*sqrt(2)))/ (-8*(8 - 3*sqrt(-24 + 22*sqrt(2)))^2 + sqrt((-16 + 6*sqrt(-24 + 22*sqrt(2)))* (32*(-8 + 3*sqrt(-24 + 22*sqrt(2)))^3 + sqrt(2)*(99 + 70*sqrt(2))* (-136 + 96*sqrt(2) + 3*sqrt(2792 - 1984*sqrt(2) + sqrt(-1201560 + 849766*sqrt(2))))^3))))^(1/8) * (2*Pi)^(3/4)).

%F Equals (-8*exp(-Pi/16)*A292863^8 + sqrt(A292862^24 / A292863^8 + 64*exp(-Pi/8) * A292863^16))^(1/8).

%e 0.001311568896679258886414738411968760891094304411218465289631108459577842...

%t RealDigits[QPochhammer[E^(-Pi/16)], 10, 120][[1]]

%t RealDigits[E^(Pi/384) * Gamma[1/4] / (((-16 + 6*Sqrt[-24 + 22*Sqrt[2]])/ (-8*(8 - 3*Sqrt[-24 + 22*Sqrt[2]])^2 + Sqrt[(-16 + 6*Sqrt[-24 + 22*Sqrt[2]])* (32*(-8 + 3*Sqrt[-24 + 22*Sqrt[2]])^3 + Sqrt[2]*(99 + 70*Sqrt[2])* (-136 + 96*Sqrt[2] + 3*Sqrt[2792 - 1984*Sqrt[2] + Sqrt[-1201560 + 849766*Sqrt[2]]])^3)]))^(1/8) * (2*Pi)^(3/4)), 10, 120][[1]]

%Y Cf. A292862, A292863, A259147, A259148, A259149.

%K nonn,cons

%O -2,2

%A _Vaclav Kotesovec_, Dec 17 2023