K-function

The function is in mathematics a special function, which is commonly referred to. It generalizes the hyper Faculty on the complex numbers.

The Hyper factorial of a natural number is defined by

For the function now is to apply

And it should be extended to the number area of the complex numbers.

Definitions

One possible definition of the function is:

Where is the complex generalization of the binomial coefficients and Γ the gamma function.

Another possibility is

Where for the Riemann zeta function and the Hurwitz zeta function are (there are the derivatives of each hand. )

The relationship of function to the gamma function and the barnesschen function is defined by the formula

Expressed.

For natural values of the K- function agree by definition the same as the value of the hyper factorial function. The first of these values are

The value is explicitly given by

Where is the constant of Glaisher - Kinkelin.

Applies with the barnesschen G function

For all

Benoit Cloitre 2003 showed the following formula:

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