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Hyperfactorial

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Hyperfactorial

The hyperfactorial (Sloane and Plouffe 1995) is the function defined by

H(n)=K(n+1)
(1)
=product_(k=1)^(n)k^k,
(2)

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).

HyperfactorialReIm
HyperfactorialContours

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

H(z)=(2pi)^(-z/2)exp[(z+1; 2)+int_0^zln(t!)dt]
(4)

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

zeta^'(a,z)=[(dzeta(s,z))/(ds)]_(s=a).
(6)

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)

(OEIS A143475 and A143476).

H(-1/2) has the special value

H(-1/2)=e^(-[(ln2)/3+12zeta^'(-1)]/8)
(8)
=2^(1/12)pi^(1/8)e^([gamma-1-zeta^'(2)/zeta(2)]/8)
(9)
=(A^(3/2))/(2^(1/24)e^(1/8)),
(10)

where gamma is the Euler-Mascheroni constant and A is the Glaisher-Kinkelin constant.

The derivative is given by

(dH(x))/(dx)=H(x){1/2[1-ln(2pi)]+ln(Gamma(x+1))+x}.
(11)

See also

Barnes G-Function, Glaisher-Kinkelin Constant, K-Function, Superfactorial

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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

Hyperfactorial

Cite this as:

Weisstein, Eric W. "Hyperfactorial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hyperfactorial.html

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