Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6be)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-45/8, b>=a

For fixed z and a=-45/8, b=-25/8

http://functions.wolfram.com/07.23.03.b6be.01

Input Form

Hypergeometric2F1[-(45/8), -(25/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-175800320 + 3780393600 z - 43075558175 z^2 + 378565049525 z^3 - 3701853621375 z^4 - 217662022526263 z^5 - 542450453406785 z^6 - 330000877643745 z^7 - 46326245622965 z^8 - 12109068125 z^9 + 305913300 z^10) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-175800320 + 3780393600 z - 43075558175 z^2 + 378565049525 z^3 - 3701853621375 z^4 - 217662022526263 z^5 - 542450453406785 z^6 - 330000877643745 z^7 - 46326245622965 z^8 - 12109068125 z^9 + 305913300 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-175800320 + 3780393600 z - 43075558175 z^2 + 378565049525 z^3 - 3701853621375 z^4 - 217662022526263 z^5 - 542450453406785 z^6 - 330000877643745 z^7 - 46326245622965 z^8 - 12109068125 z^9 + 305913300 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-175800320 + 3846318720 z - 44475179375 z^2 + 394340322350 z^3 - 3839598252750 z^4 - 53206724102938 z^5 - 36142309070252 z^6 + 57650320762650 z^7 + 33051667053790 z^8 + 2180906901250 z^9 - 49022606325 z^10 + 1223653200 z^11) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (42756657578152442925 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^5)

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]