Gauss hypergeometric function 2F1: Series representations (formula 07.23.06.0068)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Series representations

Residue representations

General case

http://functions.wolfram.com/07.23.06.0068.01

Input Form

Hypergeometric2F1[a, b, c, z] == ((Csc[(a - b) Pi] Gamma[c])/(Gamma[a] Gamma[b])) ((Sin[(a - c) Pi] Sum[Residue[((Gamma[a - s] Gamma[1 + a - c - s])/ ((-(1/z))^s Gamma[1 + a - b - s])) Gamma[s], {s, -j}], {j, 0, Infinity}])/(-z)^a + (Sin[(b - c) Pi] Sum[Residue[((Gamma[b - s] Gamma[1 + b - c - s])/ ((-(1/z))^s Gamma[1 + b - a - s])) Gamma[s], {s, -j}], {j, 0, Infinity}])/(-z)^b) /; Abs[z] > 1 && !Element[a - b, Integers]

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)

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]