Gauss hypergeometric function 2F1: Series representations (formula 07.23.06.0011)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Series representations

Generalized power series

Expansions at z==1

For the function itself

Logarithmic cases

http://functions.wolfram.com/07.23.06.0011.01

Input Form

Hypergeometric2F1[a, b, a + b - n, z] \[Proportional] ((((n - 1)! Gamma[a + b - n])/(Gamma[a] Gamma[b])) (1 + O[z - 1]))/ (1 - z)^n + (-1)^(n - 1) (Gamma[a + b - n]/(n! Gamma[a - n] Gamma[b - n])) (Log[1 - z] + EulerGamma + PolyGamma[a] + PolyGamma[b] - PolyGamma[1 + n]) (1 + O[z - 1]) /; (z -> 1) && Element[n, Integers] && n > 0

Standard Form

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MathML Form

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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]