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Decimal expansion of phi(exp(-4*Pi)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.
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%I #29 May 19 2023 14:30:41

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

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

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

%N Decimal expansion of phi(exp(-4*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(-4*Pi)) = exp(Pi/6)*Gamma(1/4)/(2^(11/8)*Pi^(3/4)).

%F A259150 = A259148 * exp(Pi/8)/sqrt(2). - _Vaclav Kotesovec_, Jul 03 2017

%e 0.99999651264548223429509891685211924765750978932634584844773269100472...

%t phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-4*Pi]], 10, 103] // 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), A292862 phi(exp(-Pi/8)), A292863 phi(exp(-Pi/4)), A259147 phi(exp(-Pi/2)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*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. A000706, A292822, A292826.

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Jun 19 2015