Gauss hypergeometric function 2F1: Series representations (formula 07.23.06.0020)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[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/07.23.06.0020.02

Input Form

Hypergeometric2F1[a, b, c, z] \[Proportional] (((Gamma[b - a] Gamma[c])/(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[c])/(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> Hypergeometric2F1 </ci> <ci> a </ci> <ci> b </ci> <ci> c </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> b </ci> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </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> a </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <ci> a </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> 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Rule Form

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

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅},{b₁,b₂,b₃,b₄},z]
HypergeometricPFQ[{a₁,a₂,a₃,a₄,a₅,a₆},{b₁,b₂,b₃,b₄,b₅},z]
HypergeometricPFQ[{a₁,...,a_p},{b₁,...,b_q},z]