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A242168
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Decimal expansion of the integral of the q-Pochhammer symbol (reciprocal of the partition function) over the real interval -1 to 1.
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5
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1, 2, 8, 8, 3, 0, 0, 8, 8, 8, 6, 7, 3, 9, 2, 1, 2, 3, 0, 1, 8, 0, 9, 0, 1, 4, 9, 3, 9, 3, 0, 9, 6, 3, 4, 4, 4, 2, 2, 5, 8, 7, 3, 8, 0, 7, 1, 3, 8, 7, 9, 6, 1, 9, 5, 0, 3, 2, 0, 1, 4, 9, 4, 2, 6, 9, 8, 6, 4, 4, 2, 4, 1, 8, 5, 2, 0, 4, 9, 7, 8, 8, 7, 6, 8, 2, 0, 9, 3, 4, 4, 4, 4, 1, 1, 1, 3, 3, 9, 8, 1, 3, 6, 3, 3
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OFFSET
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1,2
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COMMENTS
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As a function, the q-Pochhammer symbol is an irregularly left-skewed bell curve. It has limiting value 0 at -1 and 1, and its maximum is at -0.411248... (decimal value given by A143441).
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LINKS
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FORMULA
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Equals 4*sqrt(3/23)*Pi * (2*sinh(sqrt(23)*Pi/6) + sqrt(2)*sinh(sqrt(23)*Pi/4)) / (2*cosh(sqrt(23)*Pi/3)-1). - Vaclav Kotesovec, Jun 02 2015
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EXAMPLE
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1.2883008886739212301809014939309634442258738...
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MAPLE
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evalf(4*sqrt(3/23)*Pi * (2*sinh(sqrt(23)*Pi/6) + sqrt(2)*sinh(sqrt(23)*Pi/4)) / (2*cosh(sqrt(23)*Pi/3)-1), 120); # Vaclav Kotesovec, Jun 02 2015
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MATHEMATICA
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NIntegrate[QPochhammer[q, q], {q, -1, 1}, WorkingPrecision -> 45]
RealDigits[4*Sqrt[3/23]*Pi*(2*Sinh[Sqrt[23]*Pi/6] + Sqrt[2]*Sinh[Sqrt[23]*Pi/4]) / (2*Cosh[Sqrt[23]*Pi/3]-1), 10, 105][[1]] (* Vaclav Kotesovec, Jun 02 2015 *)
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PROG
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(PARI) eta2(q)=if(q==0, 1, my(p=log(10^-38)/log(abs(q)), N=floor(sqrt(2*p/3))); sum(n=-N, N, (-1)^n*q^((3*n^2-n)/2), 0.))
intnum(q=-.99999, .99999, eta2(q)) \\ Bill Allombert, May 06 2014
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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