The balanced polygamma function is a generalization of the classical polygamma function, which is itself the derivative of the logarithm of the gamma function.
The Barnes G-function is a special function in mathematical analysis and number theory, which generalizes the gamma function and is related to various areas such as complex analysis, combinatorics, and the theory of special functions. It was introduced by the mathematician W. R. Barnes in the early 20th century. The Barnes G-function, denoted as \( G(a; b) \), is defined for complex numbers and can be constructed from the Gamma function.
The Beta function is a special function in mathematics that is closely related to the gamma function and is defined for positive real numbers. It is often denoted as \( B(x, y) \) and defined as follows: \[ B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} \, dt \] for \( x > 0 \) and \( y > 0 \).
The Bohr–Mollerup theorem is a result in mathematical analysis that characterizes the gamma function among other functions. Specifically, it provides a characterization of the gamma function using properties of a specific class of functions. The theorem states that if a function \( f : (0, \infty) \to \mathbb{R} \) satisfies the following conditions: 1. \( f(x) \) is continuous on \( (0, \infty) \).
The term "Chebyshev integral" can refer to various concepts associated with the work of the Russian mathematician Pafnuty Chebyshev, particularly in the context of approximations, polynomials, and inequalities. One common interpretation relates to the Chebyshev polynomials and their application in numerical integration and approximation theory.
The Chowla–Selberg formula is a significant result in analytic number theory concerning the distribution of prime numbers. Named after the mathematicians Sang-chul Chowla and Atle Selberg, the formula provides an elegant expression for certain types of sums involving prime numbers and is often related to the theory of modular forms and Dirichlet series. In its more specific aspects, the Chowla–Selberg formula can be expressed in the context of the distribution of primes.
The digamma function, denoted as \( \psi(x) \), is the logarithmic derivative of the gamma function \( \Gamma(x) \). Mathematically, it is defined as: \[ \psi(x) = \frac{d}{dx} \ln(\Gamma(x)) = \frac{\Gamma'(x)}{\Gamma(x)} \] where \( \Gamma'(x) \) is the derivative of the gamma function.
The elliptic gamma function is a special function that generalizes the classical gamma function through the use of elliptic functions. It is a part of the theory of elliptic hypergeometric functions and has connections to various areas in mathematics and mathematical physics, including representation theory, combinatorics, and algebraic geometry.
The term "Euler integral" typically refers to a specific type of integral that is associated with the work of the mathematician Leonhard Euler. While there are several concepts related to integrals that are named after Euler, one of the most prominent is the Euler integral of the first kind, which relates to the gamma function.
The Fransén–Robinson constant, denoted by \( F \), is a mathematical constant that arises in the study of continued fractions and nested radicals. It is defined specifically in the context of the formula for the square root of a certain expression involving the golden ratio.
The Gamma function, denoted as \( \Gamma(n) \), is a mathematical function that generalizes the factorial function to complex and real number arguments. For any positive integer \( n \), the Gamma function satisfies the relation: \[ \Gamma(n) = (n-1)! \] The Gamma function is defined for all complex numbers except for the non-positive integers.
Gautschi's inequality is a result in the context of approximation theory and special functions, particularly dealing with the behavior of certain orthogonal polynomials such as the Hermite and Laguerre polynomials. It provides bounds on the values of these polynomials or their derivatives. The inequality is typically stated for polynomials that arise in certain contexts, such as exponential integrals and related functions.
The Generalized Gamma Distribution (GGD) is a flexible probability distribution that extends the gamma distribution by including additional shape parameters, thus allowing it to model a wider range of data behaviors.
Hadamard's gamma function is a special function related to the classical gamma function, denoted as \( \Gamma(z) \). It is defined for complex numbers and can be expressed in terms of an infinite product involving prime numbers. Hadamard's gamma function is particularly useful in number theory and complex analysis.
Hölder's theorem, often referred to in the context of measure theory and functional analysis, is related to the concept of measure and integration. It primarily states conditions under which the integral of the product of two functions can be bounded by the product of their respective norms. The specific version often cited is the Hölder inequality, which can be a key part of Hölder's theorem.
The incomplete gamma function is a mathematical function that generalizes the gamma function, which itself is a fundamental function in mathematics, particularly in the fields of statistics and probability theory. The incomplete gamma function is useful in various applications, including statistical distributions and hypothesis testing. The incomplete gamma function is defined in two forms: the lower incomplete gamma function and the upper incomplete gamma function.
The Inverse-Gamma distribution is a continuous probability distribution that is often used in Bayesian statistics, particularly in the context of prior distributions for variances. It is a two-parameter distribution that is defined over positive real numbers.
The inverse gamma function refers to the function that is defined as the inverse of the gamma function. The gamma function, denoted as \(\Gamma(z)\), is a generalization of the factorial function to complex numbers, except for the non-positive integers. It is defined for \(z > 0\) as: \[ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt.
The K-function, or K statistic, is a tool used in spatial statistics to analyze the distribution of points in a given space. It is particularly useful in evaluating whether the spatial pattern of points in a dataset is clustered, random, or dispersed. The K-function is defined for a specific radius \( r \) and is calculated as follows: 1. For each point in the dataset, determine how many other points lie within a distance \( r \).
The multiple gamma function, often denoted as \( \Gamma_p(z) \), generalizes the classical gamma function to multiple variables. It is closely associated with multivariable calculus and has applications in various fields such as statistics, number theory, and mathematical physics.
The Multiplication Theorem is a concept from probability theory that deals with the probabilities of events occurring in sequence or conjunction.
The multivariate gamma function is a generalization of the gamma function to multiple dimensions. It is used in various fields such as multivariate statistics, probability theory, and in the theory of random matrices. The multivariate gamma function can be used to describe distributions of multivariate random variables and often appears in the context of the Wishart distribution and other multivariate statistical models.
The Nu function is not a standard mathematical or scientific function widely recognized in literature or academia. However, if you are referring to a function or concept that is known by a specific name or acronym, please provide more context.
The gamma function, denoted as \(\Gamma(z)\), is a generalization of the factorial function that extends its definition to all complex numbers except the non-positive integers. It is defined for positive real numbers \(z\) by the following integral: \[ \Gamma(z) = \int_0^\infty t^{z-1} e^{-t} \, dt \] The gamma function has several important values, particularly at positive integers and half-integers.
The polygamma function is a special function that is defined as the \( n \)th derivative of the logarithm of the gamma function, denoted as \( \psi^{(n)}(x) \). Specifically, it is defined as: \[ \psi^{(n)}(x) = \frac{d^n}{dx^n} \ln(\Gamma(x)) \] where \( \Gamma(x) \) is the gamma function.
The Q-gamma function is a generalization of the gamma function that is typically encountered in the context of probability theory and special functions. To be more precise, the Q-gamma function can sometimes refer to a function that relates to quantile functions in statistics or may involve modifications of the standard gamma function to include additional parameters, often for applications in statistical distributions or advanced analytical methods.
The reciprocal gamma function is simply the reciprocal of the gamma function, which is a fundamental function in mathematics, particularly in statistics and probability theory. The gamma function, denoted as \(\Gamma(z)\), is defined for complex numbers \(z\) with a positive real part and is an extension of the factorial function, satisfying the relation \(\Gamma(n) = (n-1)!\) for any positive integer \(n\).
Stirling's approximation is a formula used to approximate the factorial of a large integer \( n \). It is particularly useful in combinatorics, statistical mechanics, and various areas of mathematics and physics where factorials of large numbers arise. The approximation is given by the formula: \[ n!
The trigamma function, denoted as \(\psi' (x)\) or sometimes as \(\mathrm{Trigamma}(x)\), is the derivative of the digamma function \(\psi(x)\), which is itself the logarithmic derivative of the gamma function \(\Gamma(x)\).
Wielandt's theorem is a result in the field of linear algebra, particularly concerning the properties of eigenvalues and eigenvectors of matrices. Specifically, it provides conditions under which the eigenvalues of a matrix can be related in a specific way to the eigenvalues of its perturbations. The theorem is often stated in the context of normal operators on a Hilbert space, but it can also be applied to matrices.
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