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Decimal expansion of theta_3(5*i/sqrt(5)), an explicit particular value of the cubic theta function theta_3.
3

%I #8 Jun 29 2015 11:33:22

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

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

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

%N Decimal expansion of theta_3(5*i/sqrt(5)), an explicit particular value of the cubic theta function theta_3.

%H Daniel Schultz, <a href="http://dx.doi.org/10.1016/j.aim.2013.08.021">Cubic theta functions.</a> Adv. Math. 248, 618-697 (2013). p. 72.

%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/DedekindEtaFunction.html">Dedekind Eta Function</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dedekind_eta_function">Dedekind eta function</a>

%F theta_3(tau) = eta(tau/3)^3 + 3*eta(3*tau)^3)/eta(tau), where 'eta' is the Dedekind eta modular elliptic function.

%F theta_3(5*i/sqrt(5)) = sqrt((1/30)*(1 + 3*sqrt(3) + 3*(sqrt(5) + sqrt(15)))*Gamma(1/20)*Gamma(3/20)*Gamma(7/20)*Gamma(9/20))/(4*Pi^(3/2)).

%e 1.00000474760635187188637766879472447144246277056540611713536529453...

%t Sqrt[(1/30)*(1 + 3*Sqrt[3] + 3*(Sqrt[5] + Sqrt[15])) * Gamma[1/20] * Gamma[3/20] * Gamma[7/20] * Gamma[9/20]]/(4*Pi^(3/2)) // RealDigits[#, 10, 104]& // First

%o (PARI) sqrt((1/30)*(1 + 3*sqrt(3) + 3*(sqrt(5) + sqrt(15)))*gamma(1/20)*gamma(3/20)*gamma(7/20)*gamma(9/20))/(4*Pi^(3/2)) \\ _Michel Marcus_, Jun 29 2015

%Y Cf. A259498, A259500, A259501.

%K nonn,cons,easy

%O 1,7

%A _Jean-François Alcover_, Jun 29 2015