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 It can also be given in closed form as where ζ′(z) denotes the derivative of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and Another expression using the polygamma function is[1] Or using the balanced generalization of the polygamma function:[2] where A is the Glaisher constant.
Similar to the Bohr-Mollerup Theorem for the gamma function, the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation
[3] It can be shown that for α > 0: This can be shown by defining a function f such that: Differentiating this identity now with respect to α yields: Applying the logarithm rule we get By the definition of the K-function we write And so Setting α = 0 we have Now one can deduce the identity above.
The K-function is closely related to the gamma function and the Barnes G-function; for natural numbers n, we have More prosaically, one may write The first values are Similar to the multiplication formula for the gamma function: there exists a multiplication formula for the K-Function involving Glaisher's constant:[4]