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A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.

%I A259147 #29 Feb 16 2025 08:33:25
%S A259147 7,4,9,3,1,1,4,7,7,8,0,0,0,0,2,7,8,7,4,2,9,6,2,5,6,5,8,7,8,3,3,8,0,3,
%T A259147 1,1,9,0,4,0,9,2,5,2,7,9,0,1,1,7,3,9,2,8,3,1,2,0,6,7,3,1,0,1,3,1,3,5,
%U A259147 8,8,5,3,7,5,5,1,7,4,7,2,5,8,6,1,3,4,7,5,6,3,5,7,6,5,5,8,5,8,4,0,4,6,3,7,9
%N A259147 Decimal expansion of phi(exp(-Pi/2)), where phi(q) = Product_{n>=1} (1-q^n) is the Euler modular function.
%H A259147 Istvan Mezo, <a href="https://arxiv.org/abs/1106.2703">Several special values of Jacobi theta functions</a>, arXiv:1106.2703 [math.CA], 2011-2013.
%H A259147 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>
%H A259147 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%H A259147 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%H A259147 Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler_function">Euler function</a>
%F A259147 phi(q) = QPochhammer(q,q) = (q;q)_infinity.
%F A259147 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 A259147 phi(exp(-Pi/2)) = ((sqrt(2) - 1)^(1/3)*(4 + 3*sqrt(2))^(1/24) * exp(Pi/48) * Gamma(1/4))/(2^(5/6)*Pi^(3/4)).
%F A259147 phi(exp(-Pi/2)) = (sqrt(2)-1)^(1/4) * exp(Pi/48) * Gamma(1/4)/(2^(13/16)*Pi^(3/4)). - _Vaclav Kotesovec_, Jul 03 2017
%e A259147 0.74931147780000278742962565878338031190409252790117392831206731...
%t A259147 phi[q_] := QPochhammer[q, q]; RealDigits[phi[Exp[-Pi/2]], 10, 105] // First
%Y A259147 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)), A259148 phi(exp(-Pi)), A259149 phi(exp(-2*Pi)), A292888 phi(exp(-3*Pi)), A259150 phi(exp(-4*Pi)), A292905 phi(exp(-5*Pi)), A363018 phi(exp(-6*Pi)), A259151 phi(exp(-8*Pi)), A363019 phi(exp(-10*Pi)), A363020 phi(exp(-12*Pi)), A292864 phi(exp(-16*Pi)), A363021 phi(exp(-20*Pi)).
%Y A259147 Cf. A292819, A292823, A292827.
%K A259147 nonn,cons,easy
%O A259147 0,1
%A A259147 _Jean-François Alcover_, Jun 19 2015