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Meijer G-function

In mathematics, the G-function was introduced by Cornelis Simon Meijer as a very general function intended to include most of the known special functions as particular cases. This was not the only att

Confluent hypergeometric function

In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three re

Appell series

In mathematics, Appell series are a set of four hypergeometric series F1, F2, F3, F4 of two variables that were introduced by Paul Appell and that generalize Gauss's hypergeometric series 2F1 of one v

Schwarz's list

In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by Hermann Schwarz when hypergeometric functions can be expressed algebraically. Mor

Dixon's identity

In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of thr

Lauricella hypergeometric series

In 1893 Giuseppe Lauricella defined and studied four hypergeometric series FA, FB, FC, FD of three variables. They are: for |x1| + |x2| + |x3| < 1 and for |x1| < 1, |x2| < 1, |x3| < 1 and for |x1|½ +

List of hypergeometric identities

Below is a list of hypergeometric identities.
* Hypergeometric function lists identities for the Gaussian hypergeometric function
* Generalized hypergeometric function lists identities for more gene

Bilateral hypergeometric series

In mathematics, a bilateral hypergeometric series is a series Σan summed over all integers n, and such that the ratio an/an+1 of two terms is a rational function of n. The definition of the generalize

Riemann's differential equation

In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere o

Barnes integral

In mathematics, a Barnes integral or Mellin–Barnes integral is a contour integral involving a product of gamma functions. They were introduced by Ernest William Barnes . They are closely related to ge

Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or

Picard–Fuchs equation

In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves.

MacRobert E function

In mathematics, the E-function was introduced by Thomas Murray MacRobert to extend the generalized hypergeometric series pFq(·) to the case p > q + 1. The underlying objective was to define a very gen

Egorychev method

The Egorychev method is a collection of techniques introduced by Georgy Egorychev for finding identities among sums of binomial coefficients, Stirling numbers, Bernoulli numbers, Harmonic numbers, Cat

Askey scheme

In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in ,

General hypergeometric function

In mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced by . The general hypergeometric funct

Generalized hypergeometric function

In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a gener

Gosper's algorithm

In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n) = S

Askey–Wilson polynomials

In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Askey and Wilson as q-analogs of the Wilson polynomials. They include many o

Hypergeometric function of a matrix argument

In mathematics, the hypergeometric function of a matrix argument is a generalization of the classical hypergeometric series. It is a function defined by an infinite summation which can be used to eval

Rogers–Ramanujan identities

In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Roger

Hypergeometric identity

In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These identities occur frequently in soluti

Legendre function

In physical science and mathematics, the Legendre functions Pλ, Qλ and associated Legendre functions Pμλ, Qμλ, and Legendre functions of the second kind, Qn, are all solutions of Legendre's differenti

Frobenius solution to the hypergeometric equation

In the following we solve the second-order differential equation called the hypergeometric differential equation using Frobenius method, named after Ferdinand Georg Frobenius. This is a method that us

Kampé de Fériet function

In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet. The Kampé de Fériet function is given by

Binomial transform

In combinatorics, the binomial transform is a sequence transformation (i.e., a transform of a sequence) that computes its forward differences. It is closely related to the Euler transform, which is th

Dougall's formula

No description available.

Elliptic hypergeometric series

In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratiocn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational

Fox H-function

In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox.It is defined by a Mellin–Barnes integral where L is a certai

Basic hypergeometric series

In mathematics, basic hypergeometric series, or q-hypergeometric series, are q-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric ser

Wilson polynomials

In mathematics, Wilson polynomials are a family of orthogonal polynomials introduced by James A. Wilsonthat generalize Jacobi polynomials, Hahn polynomials, and Charlier polynomials. They are defined

Fox–Wright function

In mathematics, the Fox–Wright function (also known as Fox–Wright Psi function, not to be confused with Wright Omega function) is a generalisation of the generalised hypergeometric function pFq(z) bas

Humbert series

In mathematics, Humbert series are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variables that generalize Kummer's confluent hypergeometric series 1F1 of one variable and the

Horn function

In the theory of special functions in mathematics, the Horn functions (named for Jakob Horn) are the 34 distinct convergent hypergeometric series of order two (i.e. having two independent variables),

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