OFFSET
-2,2
COMMENTS
In general, for 0 <= x < 1, QPochhammer(x) = (-8*x*QPochhammer(x^4)^8 + sqrt(QPochhammer(x^2)^24/QPochhammer(x^4)^8 + 64*x^2*QPochhammer(x^4)^16))^(1/8).
LINKS
Eric Weisstein's World of Mathematics, Dedekind Eta Function
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol
Wikipedia, Dedekind eta function
Wikipedia, Euler function
FORMULA
Equals exp(Pi/384) * Gamma(1/4) / (((-16 + 6*sqrt(-24 + 22*sqrt(2)))/ (-8*(8 - 3*sqrt(-24 + 22*sqrt(2)))^2 + sqrt((-16 + 6*sqrt(-24 + 22*sqrt(2)))* (32*(-8 + 3*sqrt(-24 + 22*sqrt(2)))^3 + sqrt(2)*(99 + 70*sqrt(2))* (-136 + 96*sqrt(2) + 3*sqrt(2792 - 1984*sqrt(2) + sqrt(-1201560 + 849766*sqrt(2))))^3))))^(1/8) * (2*Pi)^(3/4)).
EXAMPLE
0.001311568896679258886414738411968760891094304411218465289631108459577842...
MATHEMATICA
RealDigits[QPochhammer[E^(-Pi/16)], 10, 120][[1]]
RealDigits[E^(Pi/384) * Gamma[1/4] / (((-16 + 6*Sqrt[-24 + 22*Sqrt[2]])/ (-8*(8 - 3*Sqrt[-24 + 22*Sqrt[2]])^2 + Sqrt[(-16 + 6*Sqrt[-24 + 22*Sqrt[2]])* (32*(-8 + 3*Sqrt[-24 + 22*Sqrt[2]])^3 + Sqrt[2]*(99 + 70*Sqrt[2])* (-136 + 96*Sqrt[2] + 3*Sqrt[2792 - 1984*Sqrt[2] + Sqrt[-1201560 + 849766*Sqrt[2]]])^3)]))^(1/8) * (2*Pi)^(3/4)), 10, 120][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Dec 17 2023
STATUS
approved