Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bbe1)

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=-37/8, b>=a

For fixed z and a=-37/8, b=-17/8

http://functions.wolfram.com/07.23.03.bbe1.01

Input Form

Hypergeometric2F1[-(37/8), -(17/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-7770374144 + 132612361856 z - 1162825060839 z^2 + 7552430420626 z^3 - 51635405559829 z^4 - 1868940680764080 z^5 - 2536482054716225 z^6 - 600639425961190 z^7 - 275367856995 z^8 + 9360946980 z^9) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-7770374144 + 132612361856 z - 1162825060839 z^2 + 7552430420626 z^3 - 51635405559829 z^4 - 1868940680764080 z^5 - 2536482054716225 z^6 - 600639425961190 z^7 - 275367856995 z^8 + 9360946980 z^9) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-7770374144 + 132612361856 z - 1162825060839 z^2 + 7552430420626 z^3 - 51635405559829 z^4 - 1868940680764080 z^5 - 2536482054716225 z^6 - 600639425961190 z^7 - 275367856995 z^8 + 9360946980 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-7770374144 + 135526252160 z - 1211757929655 z^2 + 7975315153585 z^3 - 54355278860155 z^4 - 382359663013251 z^5 + 92131882807195 z^6 + 303791076476195 z^7 + 34982638943175 z^8 - 1119413243025 z^9 + 37443787920 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/(384809918203371986325 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]