# Category: Hypergeometric functions

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
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
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),