Regularized hypergeometric function 2F1: Series representations (formula 07.24.06.0021)

Hypergeometric2F1Regularized

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1Regularized[a,b,c,z]

Series representations

Generalized power series

Expansions at z==infinity

For the function itself

Case of simple poles

http://functions.wolfram.com/07.24.06.0021.02

Input Form

Hypergeometric2F1Regularized[a, b, c, z] \[Proportional] ((Gamma[b - a]/(Gamma[b] Gamma[c - a])) (1 + (a (1 + a - c))/((1 + a - b) z) + (a (1 + a) (1 + a - c) (2 + a - c))/(2 (1 + a - b) (2 + a - b) z^2) + \[Ellipsis]))/(-z)^a + ((Gamma[a - b]/(Gamma[a] Gamma[c - b])) (1 + (b (1 + b - c))/((1 - a + b) z) + (b (1 + b) (1 + b - c) (2 + b - c))/(2 (1 - a + b) (2 - a + b) z^2) + \[Ellipsis]))/(-z)^b /; (Abs[z] -> Infinity) && !Element[a - b, Integers]

Standard Form

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MathML Form

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/> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> b </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> b </ci> </apply> <ci> z </ci> </apply> <cn 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Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

HypergeometricPFQRegularized[{},{b},z]
HypergeometricPFQRegularized[{a₁},{b₁},z]
HypergeometricPFQRegularized[{a₁,...,a_p},{b₁,...,b_q},z]