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%I #26 Dec 17 2023 08:57:11
%S 9,9,8,1,2,9,0,6,9,9,2,5,9,5,8,5,1,3,2,7,9,9,6,2,3,2,2,2,4,5,2,7,3,8,
%T 7,8,1,3,0,7,3,8,4,3,5,3,6,5,8,1,6,4,6,1,7,5,4,0,7,8,1,4,0,2,8,2,9,9,
%U 8,5,8,0,4,6,6,0,1,9,2,8,0,7,3,5,7,1,8,2,4,4,7,3,8,7,7,7,3,7,9,3,7,7,1,9
%N Decimal expansion of phi(exp(-2*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.
%H Istvan Mezo, <a href="http://arxiv.org/pdf/1106.2703.pdf">Several special values of Jacobi theta functions</a> arXiv:1106.2703v3 [math.CA] 24 Sep 2013
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H Eric Weisstein's MathWorld, <a href="http://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler_function">Euler function</a>
%F phi(q) = QPochhammer(q,q) = (q;q)_infinity.
%F phi(q) also equals theta'(1, 0, sqrt(q))^(1/3)/(2^(1/3)*q^(1/24)), where theta' is the derivative of the elliptic theta function theta(a,u,q) w.r.t. u.
%F phi(exp(-2*Pi)) = exp(Pi/12)*Gamma(1/4)/(2*Pi^(3/4)).
%e 0.99812906992595851327996232224527387813073843536581646175407814028299858...
%t phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-2Pi]], 10, 104] // First
%Y Cf. A048651 phi(1/2), A100220 phi(1/3), A100221 phi(1/4), A100222 phi(1/5), A132034 phi(1/6), A132035 phi(1/7), A132036 phi(1/8), A132037 phi(1/9), A132038 phi(1/10), A368211 phi(exp(-Pi/16)), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-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)), A363081 phi(exp(-11*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)).
%Y Cf. A292821, A292825, A292829.
%K nonn,cons,easy
%O 0,1
%A _Jean-François Alcover_, Jun 19 2015
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