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A259500
Decimal expansion of theta_3(i/sqrt(7)), an explicit particular value of the cubic theta function theta_3.
3
1, 5, 6, 3, 4, 6, 8, 0, 9, 4, 3, 2, 7, 9, 0, 0, 5, 5, 2, 0, 4, 9, 9, 5, 8, 9, 2, 4, 0, 8, 9, 3, 0, 0, 9, 4, 0, 6, 7, 5, 1, 2, 6, 0, 0, 1, 8, 7, 6, 7, 4, 1, 2, 6, 9, 7, 2, 4, 1, 6, 5, 3, 3, 5, 1, 9, 1, 6, 9, 3, 3, 4, 3, 7, 1, 5, 9, 0, 9, 7, 0, 0, 2, 2, 5, 5, 3, 0, 2, 1, 0, 4, 3, 0, 2, 6, 2, 5, 6, 2, 8, 3, 7, 6, 1
OFFSET
1,2
LINKS
Daniel Schultz, Cubic theta functions. Adv. Math. 248, 618-697 (2013). p. 72.
Eric Weisstein's MathWorld, Dedekind Eta Function
FORMULA
theta_3(tau) = eta(tau/3)^3 + 3*eta(3*tau)^3)/eta(tau), where 'eta' is the Dedekind eta modular elliptic function.
theta_3(i/sqrt(7)) = (sqrt((1/2)*(5 + sqrt(21))*sqrt((1/2)*(sqrt(3) + sqrt(7))) - 3^(7/4)/2)*Gamma(1/7)*Gamma(2/7)*Gamma(4/7))/((2^(5/2)*3^(1/8)*7^(-1/4)*Pi^2)).
EXAMPLE
1.56346809432790055204995892408930094067512600187674126972416533519...
MATHEMATICA
(Sqrt[(1/2)*(5 + Sqrt[21])*Sqrt[(1/2)*(Sqrt[3] + Sqrt[7])] - 3^(7/4)/2] * Gamma[1/7] * Gamma[2/7] * Gamma[4/7])/((2^(5/2)*3^(1/8)*7^(-1/4)*Pi^2)) // RealDigits[#, 10, 105]& // First
CROSSREFS
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved