Hyperfactorial

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

			(1)
			(2)

where is the K-function.

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

For integer values , 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 satisfy the relation

(3)

for all complex .

The hyperfactorial is given by the integral

(4)

and the closed-form expression

(5)

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

(6)

also has a Stirling-like 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.

The derivative is given by

(11)

Wolfram Web Resources