Gauss hypergeometric function 2F1: Series representations (formula 07.23.06.0057)

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 double poles

http://functions.wolfram.com/07.23.06.0057.01

Input Form

Hypergeometric2F1[a, a + n, c, z] == ((Gamma[c] (-z)^(-a - n))/(Gamma[a + n] Gamma[c - a])) Sum[(((Pochhammer[a, k + n] Pochhammer[1 + a - c, k + n])/(k! (k + n)!)) (PolyGamma[k + n + 1] + PolyGamma[k + 1] - PolyGamma[a + n + k] - PolyGamma[c - a - n - k]))/z^k, {k, 0, Infinity}] + (Sin[(c - a) Pi]/(Pi n! Gamma[a])) Gamma[c] Gamma[1 + a - c + n] Log[-z] (-z)^(-a - n) Sum[(Pochhammer[a + n, k] Pochhammer[1 + a - c + n, k])/ (k! Pochhammer[1 + n, k])/z^k, {k, 0, Infinity}] + ((Gamma[c]/Gamma[a + n]) Sum[(Pochhammer[a, k] Gamma[n - k])/ (k! Gamma[c - a - k])/z^k, {k, 0, n - 1}])/(-z)^a /; Abs[z] > 1 && Element[n, Integers] && n >= 0 && !Element[c - a, Integers]

Standard Form

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MathML Form

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</apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <notin /> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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]