Gauss hypergeometric function 2F1: Identities (formula 07.23.17.0108)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Identities

Functional identities

Quadratic transformations with fixed a,b,z

http://functions.wolfram.com/07.23.17.0108.01

Input Form

Hypergeometric2F1[a, b, 1/2, z] == (((Gamma[a + 1/2] Gamma[1 - b])/(2 Sqrt[Pi] Gamma[a - b + 1])) (Hypergeometric2F1[2 a, 1 - 2 b, a - b + 1, (Sqrt[1 - z] + Sqrt[-z])/ (2 Sqrt[1 - z])] + Hypergeometric2F1[2 a, 1 - 2 b, a - b + 1, (Sqrt[1 - z] - Sqrt[-z])/(2 Sqrt[1 - z])]))/(1 - z)^a /; !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]