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A273086 Decimal expansion of theta_3(0, exp(-sqrt(6)*Pi)), where theta_3 is the 3rd Jacobi theta function.

%I A273086 #18 Feb 16 2025 08:33:34
%S A273086 1,0,0,0,9,0,9,9,2,1,8,8,7,2,5,6,7,6,2,9,1,9,2,8,6,0,0,4,1,2,1,5,6,6,
%T A273086 6,7,1,8,0,4,5,8,8,1,4,6,7,3,0,3,0,1,3,3,0,8,5,9,2,4,1,7,9,6,8,1,3,9,
%U A273086 5,8,5,4,2,0,8,7,9,5,0,0,5,6,3,3,2,7,5,4,2,2,0,2,2,1,8,2,9,1,1,4,7,4,2,1,8
%N A273086 Decimal expansion of theta_3(0, exp(-sqrt(6)*Pi)), where theta_3 is the 3rd Jacobi theta function.
%H A273086 G. C. Greubel, <a href="/A273086/b273086.txt">Table of n, a(n) for n = 1..10000</a>
%H A273086 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A273086 Wikipedia, <a href="http://en.wikipedia.org/wiki/Theta_function#Explicit_values">Theta function</a>
%F A273086 Equals ((6 + sqrt(6*(3 + 2*sqrt(2)))) * Gamma(1/24) * Gamma(5/24) * Gamma(7/24) * Gamma(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)).
%F A273086 Equals (4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(Gamma(1/24)*Gamma(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)).
%e A273086 1.0009099218872567629192860041215666718045881467303013308592...
%p A273086 evalf(((6 + sqrt(6*(3 + 2*sqrt(2)))) * GAMMA(1/24) * GAMMA(5/24) * GAMMA(7/24) * GAMMA(11/24))^(1/4) / (2*6^(3/8)*Pi^(3/4)), 120);
%p A273086 evalf((4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(GAMMA(1/24)*GAMMA(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)), 120);
%t A273086 RealDigits[EllipticTheta[3, 0, Exp[-Sqrt[6]*Pi]], 10, 105][[1]]
%t A273086 RealDigits[((6 + Sqrt[6*(3 + 2*Sqrt[2])]) * Gamma[1/24] * Gamma[5/24] * Gamma[7/24] * Gamma[11/24])^(1/4) / (2*6^(3/8)*Pi^(3/4)), 10, 105][[1]]
%t A273086 RealDigits[(4 - Sqrt[2] + Sqrt[6])^(1/4) * Sqrt[Gamma[1/24]*Gamma[11/24]] / (2^(3/2)*3^(3/8)*Pi^(3/4)), 10, 105][[1]]
%o A273086 (PARI) th3(x)=1 + 2*suminf(n=1,x^n^2)
%o A273086 th3(exp(-sqrt(6)*Pi)) \\ _Charles R Greathouse IV_, Jun 06 2016
%Y A273086 Cf. A175573, A247217, A273081, A273082, A273083, A273084, A086231, A273087.
%K A273086 nonn,cons
%O A273086 1,5
%A A273086 _Vaclav Kotesovec_, May 14 2016