Gauss hypergeometric function 2F1: Identities (formula 07.23.17.0060)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Identities

Functional identities

Generic general cases

http://functions.wolfram.com/07.23.17.0060.01

Input Form

Hypergeometric2F1[a, b, c, z] == (((Gamma[c] Gamma[c - a - b])/(Gamma[c - a] Gamma[c - b])) Hypergeometric2F1[a, 1 + a - c, a + b - c + 1, 1 - 1/z])/z^a + ((Gamma[c] Gamma[a + b - c])/(Gamma[a] Gamma[b])) z^(a - c) (1 - z)^(c - a - b) Hypergeometric2F1[c - a, 1 - a, 1 + c - a - b, 1 - 1/z] /; !Element[c - a - b, Integers] && !IntervalMemberQ[Interval[{-Infinity, 0}], z]

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]