Hypergeometric functions

Hypergeometric functions are a class of special functions that generalize many series and functions in mathematics, primarily arising in the context of solving differential equations, combinatorics, and mathematical physics.

The Appell series is a type of mathematical series that generalizes the concept of power series and is related to certain types of functions known as Appell functions. The series is named after the French mathematician Paul Appell. A typical form of an Appell series can be represented as follows: \[ f(x) = \sum_{n=0}^{\infty} A_n x^n \] where \(A_n\) are the coefficients that depend on certain parameters.

The bilateral hypergeometric series is a generalization of the ordinary hypergeometric series, which allows for the summation of terms indexed by two parameters rather than one.

Dougall's formula is a result in the field of combinatorics and special functions, specifically related to partitions and q-series. It provides an expression for certain types of sums involving binomial coefficients and powers of variables, often used in the study of partitions and generating functions.

The elliptic hypergeometric series is a special class of hypergeometric series that incorporates elliptic functions and is closely related to the theory of elliptic integrals and modular forms. These series generalize the classical hypergeometric series by including parameters that arise from the elliptic functions, which are periodic functions that have two fundamental periods.

The Frobenius solution to the hypergeometric equation refers to the method of finding a series solution near a regular singular point of the hypergeometric differential equation.

The general hypergeometric function, often denoted as \(_pF_q\), is a special function defined by a series expansion that generalizes the concept of hypergeometric functions.

A Horn function is a special type of Boolean function that can be expressed in a specific standard form. More formally, a Boolean function is considered a Horn function if it can be represented as a disjunction (logical OR) of clauses, where each clause has at most one positive literal. In other words, a Horn clause is a disjunction of literals in which at most one literal is positive, while the others are negative.

The Humbert series is a type of mathematical series that arises in the context of certain types of convergent sequences. Specifically, it is often associated with the study of summability methods and can be used in various fields such as number theory and functional analysis. While there isn't a universally accepted definition that is widely recognized under the name "Humbert series," it may refer to specific series associated with Humbert transformations or may arise in particular mathematical contexts or problems.

The hypergeometric function is a special function that generalizes the concept of power series and appears in various areas of mathematics, physics, and statistics. In the context of matrix arguments, the hypergeometric function can be extended to accommodate matrices, leading to the concept of the matrix hypergeometric function.

The Kampé de Fériet function is a special function in the field of mathematical analysis, particularly in relation to hypergeometric functions. It is named after the mathematician Léon Kampé de Fériet. The function generalizes some properties of the hypergeometric functions and is often expressed in terms of series expansions or integrals.

The Lauricella hypergeometric series is a generalization of the classical hypergeometric series and is denoted as \( F_D \). It is a function of several variables and is defined for several complex variables. It generalizes the standard hypergeometric series, which is a function of one variable, to cases with multiple parameters and arguments.

The Legendre functions, often referred to in the context of Legendre polynomials and Legendre functions of the first and second kind, arise in the solution of a variety of problems in physics and engineering, particularly in the fields of potential theory and solving partial differential equations. 1. **Legendre Polynomials**: These are a sequence of orthogonal polynomials defined on the interval \([-1, 1]\) and are denoted as \(P_n(x)\).

The list of hypergeometric identities typically refers to a collection of mathematical equations involving hypergeometric functions, often expressed in terms of the generalized hypergeometric series.

The MacRobert E function, often denoted as \( E(x) \), is a special function in mathematics that is related to the theory of complex variables and is used primarily in the context of mathematical analysis and applied mathematics. It is particularly significant in the studies involving wave equations and stability analysis of certain differential equations. ### Definition The MacRobert E function can be defined in various contexts, including as part of integrals leading to special functions or as solutions to specific types of differential equations.

The Meijer G-function is a special function that generalizes many other special functions, including exponential functions, logarithmic functions, Bessel functions, and hypergeometric functions. It provides a powerful tool for solving a variety of problems in mathematical analysis, physics, engineering, and other fields.

Riemann's differential equation typically refers to a type of linear partial differential equation associated with Riemann surfaces and complex analysis. However, there isn't a single, universally recognized differential equation directly defined as "Riemann's differential equation." One prominent equation related to Riemann surfaces is the Riemann-Hilbert problem, which is a type of boundary value problem for holomorphic functions, involving a piecewise constant function defined on contours in the complex plane.

Schwarz's list is a classification of certain interesting or notable groups of mathematical objects, specifically in the context of algebraic topology and complex geometry. It is named after the mathematician Hermann Schwarz. In algebraic topology, Schwarz's list typically refers to specific examples or types of manifolds that exhibit particular properties or behaviors, often with an emphasis on those that are closely related to the study of Riemann surfaces, complex manifolds, or other geometric structures.