A363179 - OEIS

%I #4 May 19 2023 14:22:00

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

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

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

%N Decimal expansion of Product_{k>=1} (1 - exp(-15*Pi*k)).

%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/questions/3101157/additional-values-of-dedekinds-eta-function-in-radical-form">Additional values of Dedekind's eta function in radical form</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Theta_function#Explicit_values">Theta function</a>.

%F Equals phi(exp(-30*Pi))^(5/2) / (phi(exp(-60*Pi)) * theta_3(0, exp(-15*Pi))^(1/2)), where phi(q) = Product_{k>=1} (1 - q^k) is the Euler modular function and theta_3 is the 3rd Jacobi theta function.

%F Equals exp(5*Pi/8) * Gamma(1/4) * (2 - sqrt(3))^(55/24) * (2 + sqrt(3))^(13/12) * (sqrt(5) - 2)^(5/4) * (3 + sqrt(5)) * (2 + sqrt(2)*3^(3/4)*5^(1/4) + sqrt(2)*15^(1/4))^(3/2) * (-15^(1/4) + sqrt(4 + sqrt(15)))^5 * ((15^(1/4) + sqrt(4 + sqrt(15)))^(5/2) / (Pi^(3/4) * 2048 * 3^(3/8) * sqrt(5) * (2*(7 + 3*sqrt(3) + sqrt(5) + sqrt(2)*3^(1/4)*5^(3/4) + sqrt(2)*15^(1/4) + sqrt(15)))^(1/4) * (((2 + sqrt(3))^4 * (1 + sqrt(5))^12 * (15^(1/4) + sqrt(4 + sqrt(15)))^12) / 16777216 - sqrt(-1 + ((2 + sqrt(3))^8 * (1 + sqrt(5))^24 * (15^(1/4) + sqrt(4 + sqrt(15)))^24) / 281474976710656))^(1/8))).

%e 0.99999999999999999999657741145587875913219208544734891061914001399856284...

%t RealDigits[QPochhammer[E^(-15*Pi)], 10, 120][[1]]

%t RealDigits[QPochhammer[E^(-30*Pi)]^(5/2) / QPochhammer[E^(-60*Pi)] / EllipticTheta[3, 0, Exp[-15*Pi]]^(1/2), 10, 120][[1]]

%t RealDigits[E^(5*Pi/8) * Gamma[1/4] * (2 - Sqrt[3])^(55/24) * (2 + Sqrt[3])^(13/12) * (Sqrt[5] - 2)^(5/4) * (3 + Sqrt[5]) * (2 + Sqrt[2]*3^(3/4)*5^(1/4) + Sqrt[2]*15^(1/4))^(3/2) * (-15^(1/4) + Sqrt[4 + Sqrt[15]])^5 * ((15^(1/4) + Sqrt[4 + Sqrt[15]])^(5/2) / (Pi^(3/4) * 2048 * 3^(3/8) * Sqrt[5] * (2*(7 + 3*Sqrt[3] + Sqrt[5] + Sqrt[2]*3^(1/4)*5^(3/4) + Sqrt[2]*15^(1/4) + Sqrt[15]))^(1/4) * (((2 + Sqrt[3])^4 * (1 + Sqrt[5])^12 * (15^(1/4) + Sqrt[4 + Sqrt[15]])^12) / 16777216 - Sqrt[-1 + ((2 + Sqrt[3])^8 * (1 + Sqrt[5])^24 * (15^(1/4) + Sqrt[4 + Sqrt[15]])^24) / 281474976710656])^(1/8))), 10, 120][[1]]

%Y Cf. A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A363117 phi(exp(-7*Pi)), A259151 phi(exp(-8*Pi)), A363118 phi(exp(-9*Pi)), A363019 phi(exp(-10*Pi)), A363081 phi(exp(-11*Pi)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A292864 phi(exp(-16*Pi)), A363120 phi(exp(-18*Pi)), A363021 phi(exp(-20*Pi)).

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, May 19 2023