Multivariate gamma function
In mathematics, the multivariate gamma function Γp is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and
Q-gamma function
In q-analog theory, the -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by
Elliptic gamma function
In mathematics, the elliptic gamma function is a generalization of the q-gamma function, which is itself the q-analog of the ordinary gamma function. It is closely related to a function studied by , a
Bohr–Mollerup theorem
In mathematical analysis, the Bohr–Mollerup theorem is a theorem proved by the Danish mathematicians Harald Bohr and Johannes Mollerup. The theorem characterizes the gamma function, defined for x > 0
Particular values of the gamma function
The gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known fo
Trigamma function
In mathematics, the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by . It follows from this definition that where ψ(z) is the digamma function.
Spouge's approximation
In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper. The formula is a m
Stirling's approximation
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of . It is named afte
Digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: It is the first of the polygamma functions. The digamma function is often denoted as or Ϝ (the uppe
Factorial
In mathematics, the factorial of a non-negative integer , denoted by , is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller
Chowla–Selberg formula
In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadra
Generalized Pochhammer symbol
In mathematics, the generalized Pochhammer symbol of parameter and partition generalizes the classical Pochhammer symbol, named after Leo August Pochhammer, and is defined as It is used in multivariat
Wielandt theorem
In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers for which by as the only function defined on the half-plane such that:
* is holomorphic on ;
*
Nu function
In mathematics, the nu function is a generalization of the reciprocal gamma function of the Laplace transform. Formally, it can be defined as where is the Gamma function.
Gautschi's inequality
In real analysis, a branch of mathematics, Gautschi's inequality is an inequality for ratios of gamma functions. It is named after Walter Gautschi.
Polygamma 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: Thus holds where ψ(z) is
Chebyshev integral
In mathematics, the Chebyshev integral, named after Pafnuty Chebyshev, is where is an incomplete beta function.
Falling and rising factorials
In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial The rising factorial (sometimes called th
Lanczos approximation
In mathematics, the Lanczos approximation is a method for computing the gamma function numerically, published by Cornelius Lanczos in 1964. It is a practical alternative to the more popular Stirling's
Gamma function
In mathematics, the gamma function (represented by Γ, the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function
Reciprocal gamma function
In mathematics, the reciprocal gamma function is the function where Γ(z) denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal
K-function
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.
Fransén–Robinson constant
The Fransén–Robinson constant, sometimes denoted F, is the mathematical constant that represents the area between the graph of the reciprocal Gamma function, 1/Γ(x), and the positive x axis. That is,
Incomplete gamma function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective name
Euler integral
In mathematics, there are two types of Euler integral: 1.
* The Euler integral of the first kind is the beta function 2.
* The Euler integral of the second kind is the gamma function For positive in
Hadamard's gamma function
In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function. This function, with its argument shifte
Beta function
In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by
Multiple gamma function
In mathematics, the multiple gamma function is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by . At the end of this paper he mentioned
Hölder's theorem
In mathematics, Hölder's theorem states that the gamma function does not satisfy any algebraic differential equation whose coefficients are rational functions. This result was first proved by Otto Höl
Balanced polygamma function
In mathematics, the generalized polygamma function or balanced negapolygamma function is a function introduced by Olivier Espinosa Aldunate and Victor Hugo Moll. It generalizes the polygamma function
Multiplication theorem
In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a
Pochhammer k-symbol
In the mathematical theory of special functions, the Pochhammer k-symbol and the k-gamma function, introduced by Rafael Díaz and Eddy Pariguan are generalizations of the Pochhammer symbol and gamma fu