Hyperfactorial
The hyperfactorial (Sloane and Plouffe 1995) is the function defined by
where K(n) is the K-function.
The hyperfactorial is implemented in the Wolfram Language as Hyperfactorial [n].
For integer values n=1, 2, ... are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (OEIS A002109).
The hyperfactorial can also be generalized to complex numbers, as illustrated above.
The Barnes G-function and hyperfactorial H(z) satisfy the relation
| H(z-1)G(z)=e^((z-1)logGamma(z)) |
(3)
|
for all complex z.
The hyperfactorial is given by the integral
![画像: H(z)=(2pi)^(-z/2)exp[(z+1; 2)+int_0^zln(t!)dt]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Hyperfactorial/NumberedEquation2.svg){kind=link}
and the closed-form expression
| K(z)=exp[zeta^'(-1,z+1)-zeta^'(-1)] |
(5)
|
for R[z]>0, where zeta(z) is the Riemann zeta function, zeta^'(z) its derivative, zeta(a,z) is the Hurwitz zeta function, and
![画像: zeta^'(a,z)=[(dzeta(s,z))/(ds)]_(s=a).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Hyperfactorial/NumberedEquation4.svg){kind=link}
H(z) also has a Stirling-like series
| H(z)∼Ae^(-z^2/4)z^(z(z+1)/2+1/12)×(1+1/(720z^2)-(1433)/(7257600z^4)+...) |
(7)
|
H(-1/2) has the special value
where gamma is the Euler-Mascheroni constant and A is the Glaisher-Kinkelin constant.
The derivative is given by
![画像: (dH(x))/(dx)=H(x){1/2[1-ln(2pi)]+ln(Gamma(x+1))+x}.](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Hyperfactorial/NumberedEquation6.svg){kind=link}
See also
Barnes G-Function, Glaisher-Kinkelin Constant, K-Function, SuperfactorialExplore with Wolfram|Alpha
More things to try:
References
Fletcher, A.; Miller, J. C. P.; Rosenhead, L.; and Comrie, L. J. An Index of Mathematical Tables, Vol. 1, 2nd ed. Reading, MA: Addison-Wesley, p. 50, 1962.Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, p. 477, 1994.Sloane, N. J. A. Sequences A002109/M3706, A143475, and A143476 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
HyperfactorialCite this as:
Weisstein, Eric W. "Hyperfactorial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hyperfactorial.html