A273087 -id:A273087 - cp's OEIS frontend

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

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, 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, 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

Offset: 1

Vaclav Kotesovec, May 14 2016

			1.0009099218872567629192860041215666718045881467303013308592...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's MathWorld, Jacobi Theta Functions
Wikipedia, Theta function

Cf. A175573, A247217, A273081, A273082, A273083, A273084, A086231, A273087.

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);
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);

RealDigits[EllipticTheta[3, 0, Exp[-Sqrt[6]*Pi]], 10, 105][[1]]
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]]
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]]

th3(x)=1 + 2*suminf(n=1,x^n^2)
th3(exp(-sqrt(6)*Pi)) \\ Charles R Greathouse IV, Jun 06 2016

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)).

Equals (4 - sqrt(2) + sqrt(6))^(1/4) * sqrt(Gamma(1/24)*Gamma(11/24)) / (2^(3/2)*3^(3/8)*Pi^(3/4)).

cp's OEIS Frontend

A247217 Decimal expansion of theta_3(0, exp(-2*Pi)), where theta_3 is the 3rd Jacobi theta function.

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