## K-Function

An extension of the -function

 (1)

defined by
 (2)

Here, is the G-Function defined by
 (3)

The -function is given by the integral

 (4)

and the closed-form expression
 (5)

where is the Riemann Zeta Function, its Derivative, is the Hurwitz Zeta Function, and
 (6)

also has a Stirling-like series

 (7)

where
 (8) (9) (10)

and is the Euler-Mascheroni Constant (Gosper).

The first few values of for , 3, ... are 1, 4, 108, 27648, 86400000, 4031078400000, ... (Sloane's A002109). These numbers are called Hyperfactorials by Sloane and Plouffe (1995).

Sloane, N. J. A. Sequence A002109/M3706 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.