About: Generalized hypergeometric function

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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 generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials.

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rdfs:label	Verallgemeinerte hypergeometrische Funktion (de) Generalized hypergeometric function (en) 一般化された超幾何関数 (ja) Generaliserad hypergeometrisk funktion (sv) 广义超几何函数 (zh)
rdfs:comment	Die verallgemeinerte hypergeometrische Funktion ist in der Mathematik eine Funktion, die die Gaußsche hypergeometrische Funktion und letztlich die geometrische Reihe verallgemeinert. Sie wird zur Klasse der speziellen Funktionen gezählt. Die verallgemeinerte hypergeometrische Funktion enthält viele wichtige Funktionen als Spezialfälle, allen voran die Exponentialfunktion und die trigonometrischen Funktionen. In der Tat gibt es eine große Zahl von Funktionen, die sich als eine hypergeometrische Funktion schreiben lassen. (de) 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 generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the classical orthogonal polynomials. (en) 数学において、一般化された超幾何関数（いっぱんかされたちょうきかかんすう、英: generalized hypergeometric function）は、一般にの形式で表される級数である。ただし、はポッホハマー記号である。古典的にはガウスの超幾何関数を単に超幾何級数という。なお、厳密にいうと、右辺の級数が超幾何級数であり、左辺の記号は級数の和によって定義される超幾何関数を表すものである。 (ja) Inom matematiken är en generaliserad hypergeometrisk serie en potensserie där kvoten av två konsekutiva koefficienter är en rationell funktion. Om serien konvergerar definierar den en generaliserad hypergeometrisk funktion. Många speciella funktioner kan skrivas som specialfall av generaliserade hypergeometriska funktionen. (sv) 广义超几何函数（generalized hypergeometric function），有时也称超几何函数，是一个用幂级数定义的函数，其中幂级数的系数由若干个升阶乘的积和商给出。下文中用“超几何函数”一词代指广义超几何函数，而用“高斯超几何函数”一词代指 p=2、 q=1 时的广义超几何函数。 (zh)
differentFrom	General hypergeometric function
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dcterms:subject	Hypergeometric functions Factorial and binomial topics Mathematical series Ordinary differential equations
Wikipage page ID	361184 (xsd:integer)
Wikipage revision ID	1117726772 (xsd:integer)
Link from a Wikipage to another Wikipage	Carl Friedrich Gauss Power series Meromorphic function Bessel function Appell series Hypergeometric functions Hypergeometric function Elementary functions Lie group Pochhammer symbol Hypergeometric differential equation Factorial and binomial topics Complex numbers Confluent hypergeometric function Continued fraction Analytic continuation Analytic function MathWorld Mathematics SO(3) Elliptic function Elliptic hypergeometric series Gauss's continued fraction General hypergeometric function Q-analog Clebsch–Gordan coefficients Generating function Geometric series Arrangement of hyperplanes Legendre polynomials Louis de Branges de Bourcia Symmetric space Zonal spherical function Lauricella hypergeometric series Eduard Heine Ernst Kummer Exponential function Basic hypergeometric series Fox–Wright function Joseph Kampé de Fériet Kampé de Fériet function Meijer G-function Rational function Mathematical series Hermite polynomials Asymptotic expansion Israel Gelfand Jacobi polynomials Hyperplane Paul Emile Appell Ordinary differential equations Chebyshev polynomial Laguerre polynomials Bilateral hypergeometric series Coefficient Egorychev method Dilogarithm Polynomial Special functions Spherical harmonics Humbert series Incomplete gamma function Orthogonal polynomials Radius of convergence Ratio test Exponential integral Mott polynomials Hahn polynomial

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