mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the -function, typically denoted ''K''(''z''), is a generalization of the

hyperfactorial In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n is the product of the numbers of the form x^x from 1^1 to n^n. Definition The hyperfactorial of a positive integer n is the product of the numbers ...

complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...

s, similar to the generalization of the

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...

to the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

Definition

Formally, the -function is defined as :

K(z)=(2\pi)^ \exp\left binom+\int_0^ \ln \Gamma(t + 1)\,dt\right

It can also be given in closed form as :

K(z)=\exp\bigl zeta'(-1,z)-\zeta'(-1)\bigr /math>

where  denotes the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

of the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...

, denotes the

Hurwitz zeta function In mathematics, the Hurwitz zeta function is one of the many zeta functions. It is formally defined for complex variables with and by :\zeta(s,a) = \sum_^\infty \frac. This series is absolutely convergent for the given values of and and can ...

and :

\zeta'(a,z)\ \stackrel\ \left.\frac\_.

Another expression using the

polygamma function In mathematics, the polygamma function of order is a meromorphic function on the complex numbers \mathbb defined as the th derivative of the logarithm of the gamma function: :\psi^(z) := \frac \psi(z) = \frac \ln\Gamma(z). Thus :\psi^(z) = ...

is :

K(z)=\exp\left psi^(z)+\frac-\frac  \ln 2\pi \right /math>

Or using the balanced generalization of the polygamma function :

: K(z)=A \exp\left psi(-2,z)+\frac\right /math>

where  is the

Glaisher constant Glaisher is a surname, and may refer to: * Cecilia Glaisher (1828–1892), photographer and illustrator *James Glaisher (1809–1903), English meteorologist and astronomer *James Whitbread Lee Glaisher (1848–1928), English mathematician and astro ...

. Similar to the Bohr-Mollerup Theorem for the

Gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

, the log K-function is the unique (up to an additive constant) eventually 2-convex solution to the equation

\Delta f(x)=x\ln(x)

where

\Delta

is the forward difference operator.

Properties

It can be shown that for : :

\int_\alpha^\ln K(x)\,dx-\int_0^1\ln K(x)\,dx=\tfrac\alpha^2\left(\ln\alpha-\tfrac\right)

This can be shown by defining a function such that: :

f(\alpha)=\int_\alpha^\ln K(x)\,dx

Differentiating this identity now with respect to yields: :

f'(\alpha)=\ln K(\alpha+1)-\ln K(\alpha)

Applying the logarithm rule we get :

f'(\alpha)=\ln\frac

By the definition of the -function we write :

f'(\alpha)=\alpha\ln\alpha

And so :

f(\alpha)=\tfrac12\alpha^2\left(\ln\alpha-\tfrac12\right)+C

Setting we have :

\int_0^1 \ln K(x)\,dx=\lim_\left tfrac12 t^2\left(\ln t-\tfrac12\right)\right C \ =C

Now one can deduce the identity above. The -function is closely related to the

and the Barnes -function; for

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...

s , we have :

K(n)=\frac.

More prosaically, one may write :

K(n+1)=1^1 \cdot 2^2 \cdot 3^3 \cdots n^n.

The first values are :1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... .

References

External links

* {{mathworld, title=K-Function, urlname=K-Function Gamma and related functions