PlanetMath: hypergeometric function

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hypergeometric function

(Definition)

Let be a triple of complex numbers with not belonging to the set of negative integers. For a complex number and a non negative integer , use Pochhammer symbol , to denote the expression :

$\displaystyle (w)_n=w(w+1)\dots(w+n-1).$

The Gauss hypergeometric function, $_{2}F_{1}$ , is then defined by the following power series expansion :

$\displaystyle _2F_1(a,b;\,c\,;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_{n}n!}z^n.$

"hypergeometric function" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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Also defines:

Gauss hypergeometric function

Keywords:

hypergeometric, Gauss

Attachments:: integral representation of the hypergeometric function (Theorem) by rspuzio
Barnes' integral representation of the hypergeometric function (Theorem) by rspuzio
differential-difference equations for hypergeometric function (Theorem) by rspuzio
global characterization of hypergeometric function (Definition) by rspuzio

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Cross-references: power series, expression, Pochhammer symbol, integers, negative, complex numbers
There are 11 references to this entry.

This is version 6 of hypergeometric function, born on 2004-07-05, modified 2006-12-12.
Object id is 5983, canonical name is HypergeometricFunction.
Accessed 6561 times total.

Classification:

33C05 (Special functions :: Hypergeometric functions :: Classical hypergeometric functions, $_2F_1$)

Pending Errata and Addenda

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