A363081 - OEIS

%I #9 May 20 2023 02:56:42

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

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

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

%N Decimal expansion of Product_{k>=1} (1 - exp(-11*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(-22*Pi))^(5/2) / (phi(exp(-44*Pi)) * theta_3(0, exp(-11*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(11*Pi/24) * Gamma(1/4) * (((-3 + sqrt(11))^(1/4) * sqrt(((2*(47 - 27*sqrt(3))^(1/6) + (47 + 27*sqrt(3))^(1/6)* sqrt(7 + sqrt(33))) * (2 + ((11 + 3*sqrt(11)) * (4 - 3*sqrt(3) + 3*sqrt(11)))^(1/3) + (143 + 33*sqrt(3) + 45*sqrt(11) + 9*sqrt(33))^(1/3))) / (2*(47 - 27*sqrt(3))^(1/6) * (4 + 3*sqrt(3) + sqrt(11)) + (47 + 27*sqrt(3))^(1/6) * (4 - 3*sqrt(3) + sqrt(11)) * sqrt(7 + sqrt(33))))) / (6^(7/8)*(-(11/(-4*22^(1/3) + (1490 + 837*sqrt(3) - 351*sqrt(11) - 306*sqrt(33))^(1/3) + (1490 - 837*sqrt(3) - 351*sqrt(11) + 306*sqrt(33))^(1/3))))^(3/8) * ((1/34012224)*(2 + (11 + 3*sqrt(11))^(1/3) * ((4 - 3*sqrt(3) + 3*sqrt(11))^(1/3) + (4 + 3*sqrt(3) + 3*sqrt(11))^(1/3)))^12 - sqrt(-1 + (2 + (11 + 3*sqrt(11))^(1/3) * ((4 - 3*sqrt(3) + 3*sqrt(11))^(1/3) + (4 + 3*sqrt(3) + 3*sqrt(11))^(1/3)))^24 / 1156831381426176))^(1/8)) / Pi^(3/4)).

%e 0.9999999999999990185682406467667685324890186498523246531748501440722320873182...

%t RealDigits[QPochhammer[E^(-11*Pi)], 10, 120][[1]]

%t RealDigits[QPochhammer[E^(-22*Pi)]^(5/2) / QPochhammer[E^(-44*Pi)] / EllipticTheta[3, 0, Exp[-11*Pi]]^(1/2), 10, 120][[1]]

%t RealDigits[E^(11*Pi/24) * Gamma[1/4] * (((-3 + Sqrt[11])^(1/4) * Sqrt[((2*(47 - 27*Sqrt[3])^(1/6) + (47 + 27*Sqrt[3])^(1/6)* Sqrt[7 + Sqrt[33]]) * (2 + ((11 + 3*Sqrt[11]) * (4 - 3*Sqrt[3] + 3*Sqrt[11]))^(1/3) + (143 + 33*Sqrt[3] + 45*Sqrt[11] + 9*Sqrt[33])^(1/3))) / (2*(47 - 27*Sqrt[3])^(1/6) * (4 + 3*Sqrt[3] + Sqrt[11]) + (47 + 27*Sqrt[3])^(1/6) * (4 - 3*Sqrt[3] + Sqrt[11]) * Sqrt[7 + Sqrt[33]])]) / (6^(7/8)*(-(11/(-4*22^(1/3) + (1490 + 837*Sqrt[3] - 351*Sqrt[11] - 306*Sqrt[33])^(1/3) + (1490 - 837*Sqrt[3] - 351*Sqrt[11] + 306*Sqrt[33])^(1/3))))^(3/8) * ((1/34012224)*(2 + (11 + 3*Sqrt[11])^(1/3) * ((4 - 3*Sqrt[3] + 3*Sqrt[11])^(1/3) + (4 + 3*Sqrt[3] + 3*Sqrt[11])^(1/3)))^12 - Sqrt[-1 + (2 + (11 + 3*Sqrt[11])^(1/3) * ((4 - 3*Sqrt[3] + 3*Sqrt[11])^(1/3) + (4 + 3*Sqrt[3] + 3*Sqrt[11])^(1/3)))^24 / 1156831381426176])^(1/8))/Pi^(3/4)), 10, 120][[1]]

%t RealDigits[E^(11*Pi/24) * Gamma[1/4] * Root[-387420489 + 1578379770*#1^2 - 1299078*#1^6 + 594*#1^10 + 11*#1^12 &, 2] / (Pi^(3/4) * 11^(3/8) * (2*(Root[-5832 - 3888*#1 + 1296*#1^2 + 324*#1^3 - 72*#1^4 - 12*#1^5 + #1^6 &, 2]^12 - Sqrt[-1156831381426176 + Root[-5832 - 3888*#1 + 1296*#1^2 + 324*#1^3 - 72*#1^4 - 12*#1^5 + #1^6 & , 2]^24]))^(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)), A363020 phi(exp(-12*Pi)), A363178 phi(exp(-13*Pi)), A363119 phi(exp(-14*Pi)), A363179 phi(exp(-15*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 18 2023