K-function

Last updated

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.

Contents

Definition

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 where is the forward difference operator. [3]

Properties

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

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

Related Research Articles

<span class="mw-page-title-main">Gamma function</span> Extension of the factorial function

In mathematics, the gamma function is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except the non-positive integers. For every positive integer n,

<span class="mw-page-title-main">Riemann zeta function</span> Analytic function in mathematics

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined as

<span class="mw-page-title-main">Exponential distribution</span> Probability distribution

In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.

In mathematics, Catalan's constantG, is defined by

<span class="mw-page-title-main">Euler's constant</span> Relates logarithm and harmonic series

Euler's constant is a mathematical constant, usually denoted by the lowercase Greek letter gamma, defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by log:

<span class="mw-page-title-main">Harmonic number</span> Sum of the first n whole number reciprocals; 1/1 + 1/2 + 1/3 + ... + 1/n

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:

<span class="mw-page-title-main">Polygamma function</span> Meromorphic function

In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers defined as the (m + 1)th derivative of the logarithm of the gamma function:

<span class="mw-page-title-main">Digamma function</span> Mathematical function

In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:

<span class="mw-page-title-main">Polylogarithm</span> Special mathematical function

In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams.

<span class="mw-page-title-main">Stable distribution</span> Distribution of variables which satisfies a stability property under linear combinations

In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.

In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887.

<span class="mw-page-title-main">Inverse-gamma distribution</span> Two-parameter family of continuous probability distributions

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.

<span class="mw-page-title-main">Trigamma function</span> Mathematical function

In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by

<span class="mw-page-title-main">Barnes G-function</span>

In mathematics, the Barnes G-functionG(z) 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. It can be written in terms of the double gamma function.

In complex analysis, functional analysis and operator theory, a Bergman space, named after Stefan Bergman, is a function space of holomorphic functions in a domain D of the complex plane that are sufficiently well-behaved at the boundary that they are absolutely integrable. Specifically, for 0 < p < ∞, the Bergman space Ap(D) is the space of all holomorphic functions in D for which the p-norm is finite:

<span class="mw-page-title-main">Reciprocal gamma function</span>

In mathematics, the reciprocal gamma function is the function

<span class="mw-page-title-main">Anatoly Karatsuba</span> Russian mathematician

Anatoly Alexeyevich Karatsuba was a Russian mathematician working in the field of analytic number theory, p-adic numbers and Dirichlet series.

In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll.

A product distribution is a probability distribution constructed as the distribution of the product of random variables having two other known distributions. Given two statistically independent random variables X and Y, the distribution of the random variable Z that is formed as the product is a product distribution.

In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the convergence/divergence of these expansions. Some functions can actually be expanded directly as infinite compositions. In addition, it is possible to use ICAF to evaluate solutions of fixed point equations involving infinite expansions. Complex dynamics offers another venue for iteration of systems of functions rather than a single function. For infinite compositions of a single function see Iterated function. For compositions of a finite number of functions, useful in fractal theory, see Iterated function system.

References

  1. Adamchik, Victor S. (1998), "PolyGamma Functions of Negative Order", Journal of Computational and Applied Mathematics, 100: 191–199, archived from the original on 2016-03-03
  2. Espinosa, Olivier; Moll, Victor Hugo, "A Generalized polygamma function" (PDF), Integral Transforms and Special Functions, 15 (2): 101–115, archived (PDF) from the original on 2023-05-14
  3. "A Generalization of Bohr-Mollerup's Theorem for Higher Order Convex Functions: a Tutorial" (PDF). Bitstream: 14. Archived (PDF) from the original on 2023-04-05.