Hyperfactorial

The hyperfactorial (Sloane and Plouffe 1995) is the function defined for a nonnegative integer by

			(1)
			(2)

where is the K-function.

The hyperfactorial is implemented in the Wolfram Language as Hyperfactorial[n].

For positive integers , 2, ..., the hyperfactorial values are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (OEIS A002109).

The hyperfactorial can also be generalized to complex numbers, as illustrated above. This continuation has a branch cut along the real ray .

The Barnes G-function and hyperfactorial satisfy the relation

(3)

for . Here denotes the analytic logarithm of the gamma function that is real for positive real arguments, as implemented by LogGamma[z], rather than the principal value of the natural logarithm of .

For , the hyperfactorial is given by the integral

(4)

using the same analytic logarithm along the straight segment from 0 to . It also has the closed-form expression

(5)

for , where is the Riemann zeta function, its derivative, is the Hurwitz zeta function, and

(6)

As on the real axis, has the asymptotic series

(7)

(OEIS A143475 and A143476).

has the special value

			(8)
			(9)
			(10)

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

For real , the derivative is given by

(11)