Gauss hypergeometric function 2F1: Series representations (formula 07.23.06.0054)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Series representations

Generalized power series

Expansions at z==1

For the function itself

Generic formulas for main term

http://functions.wolfram.com/07.23.06.0054.01

Input Form

Hypergeometric2F1[a, b, c, z] \[Proportional] Piecewise[{{ComplexInfinity, Element[-c, Integers] && -c >= 0 && ((Element[-a, Integers] && -a >= 0 && c - a > 0) || (Element[-b, Integers] && -b >= 0 && c - b > 0))}, {(((a + b - c - 1)! Gamma[c])/(Gamma[a] Gamma[b])) (1 - z)^(c - a - b) + (((-1)^(a + b - c - 1) Gamma[c])/((a + b - c)! Gamma[c - a] Gamma[c - b])) (Log[1 - z] + EulerGamma + PolyGamma[a] + PolyGamma[b] - PolyGamma[1 + a + b - c]), Element[a + b - c, Integers] && a + b - c > 0}, {(-(Gamma[a + b]/(Gamma[a] Gamma[b]))) (Log[1 - z] + 2 EulerGamma + PolyGamma[a] + PolyGamma[b]), c == a + b}, {((c - a - b - 1)! Gamma[c])/(Gamma[c - a] Gamma[c - b]) - (Gamma[c]/((c - a - b)! Gamma[a] Gamma[b])) (z - 1)^(c - a - b) (Log[1 - z] + EulerGamma - PolyGamma[1 + c - a - b] + PolyGamma[c - a] + PolyGamma[c - b]), Element[c - a - b, Integers] && c - a - b > 0}}, (Gamma[c] Gamma[c - a - b])/ (Gamma[c - a] Gamma[c - b]) + ((Gamma[c] Gamma[a + b - c])/ (Gamma[a] Gamma[b])) (1 - z)^(c - a - b)] /; (z -> 1)

Standard Form

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MathML Form

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<apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> ℕ </ci> </apply> <apply> <gt /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </piece> <piece> <apply> <plus /> <apply> <times /> <apply> <factorial /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <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> <ci> b 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type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </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> <apply> <in /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </piece> <piece> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> a </ci> </apply> <apply> 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type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Gamma </ci> <ci> c </ci> </apply> <apply> <plus /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <eulergamma 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Rule Form

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

2007-05-02

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]