K function

In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of the factorial to the Gamma function.

Formally, the K-function is defined as

K ( z ) = ( 2 π ) ( − z + 1 ) / 2 exp ⁡ [ ( z 2 ) + ∫ 0 z − 1 ln ⁡ ( Γ ( t + 1 ) ) d t ] .

It can also be given in closed form as

K ( z ) = exp ⁡ [ ζ ′ ( − 1 , z ) − ζ ′ ( − 1 ) ]

where ζ'(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and

ζ ′ ( a , z )   = d e f   [ ∂ ζ ( s , z ) ∂ s ] s = a .

Another expression using polygamma function is

K ( z ) = exp ⁡ ( ψ ( − 2 ) ( z ) + z 2 − z 2 − z 2 ln ⁡ ( 2 π ) )

Or using balanced generalization of Polygamma function:

K ( z ) = A e ψ ( − 2 , z ) + z 2 − z 2 where A is Glaisher constant.

The K-function is closely related to the Gamma function and the Barnes G-function; for natural numbers n, we have

K ( n ) = ( Γ ( n ) ) n − 1 G ( n ) .

More prosaically, one may write

K ( n + 1 ) = 1 1 2 2 3 3 ⋯ n n .

The first values are

1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... ((sequence A002109 in the OEIS)).