Gauss hypergeometric function 2F1: Identities (formula 07.23.17.0080)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Identities

Functional identities

Relations including three Kummer's solutions

http://functions.wolfram.com/07.23.17.0080.01

Input Form

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