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)

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."