Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8qn)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-27/8

http://functions.wolfram.com/07.23.03.b8qn.01

Input Form

Hypergeometric2F1[-(41/8), -(27/8), 6, z] == (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (-73891840 + 1562985600 z - 17497435725 z^2 + 150878475825 z^3 - 1445357234475 z^4 - 93029563781931 z^5 - 232018004356425 z^6 - 143090126386605 z^7 - 21852672133545 z^8 - 309605699865 z^9 + 4476226986 z^10) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-9236480 + 193857840 z - 2155836375 z^2 + 18515568225 z^3 - 12630040353315 z^4 - 40744789879227 z^5 - 34948923244389 z^6 - 9375868764765 z^7 - 638864795745 z^8 + 746037831 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 6 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (4618240 - 94331160 z + 1025377815 z^2 - 8691394635 z^3 - 10292832149985 z^4 - 28741896419019 z^5 - 19196331729939 z^6 - 3161201302809 z^7 - 51476610339 z^8 + 746037831 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (-73891840 + 1562985600 z - 17497435725 z^2 + 150878475825 z^3 - 1445357234475 z^4 - 93029563781931 z^5 - 232018004356425 z^6 - 143090126386605 z^7 - 21852672133545 z^8 - 309605699865 z^9 + 4476226986 z^10) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (16321485899245752225 Pi 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]