K-function
Encyclopedia
In mathematics
, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex number
s, 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 polygamma function is
Or using alternative generalization of Polygamma function
:
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
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, similar to the generalization of the factorial
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
to the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
.
Formally, the K-function is defined as
It can also be given in closed form as
where ζ'(z) denotes the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of the Riemann zeta function, ζ(a,z) denotes the Hurwitz zeta function and
Another expression using polygamma function is
Or using alternative generalization of Polygamma function
Generalized polygamma function
In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa and Victor H. Moll...
:
- where A is Glaisher constant.
The K-function is closely related to the Gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
and the Barnes G-function
Barnes G-function
In mathematics, the Barnes G-function G is a function that is an extension of superfactorials to the complex numbers. It is related to the Gamma function, the K-function and the Glaisher-Kinkelin constant, and was named after mathematician Ernest William Barnes...
; for natural numbers n, we have
More prosaically, one may write
The first values are
- 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... .