Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8xe)

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=-3/8

http://functions.wolfram.com/07.23.03.b8xe.01

Input Form

Hypergeometric2F1[-(41/8), -(3/8), 6, z] == (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (114196480 - 1548343680 z + 10266987735 z^2 - 46789156950 z^3 + 197833514265 z^4 + 3543719569272 z^5 + 730061730153 z^6 - 144620436366 z^7 + 29936820375 z^8 - 4380731124 z^9 + 317061360 z^10) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 20 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (3568640 - 47800260 z + 313179525 z^2 - 1413076890 z^3 + 140631281865 z^4 + 81146265426 z^5 - 5956172649 z^6 + 1236848130 z^7 - 181744101 z^8 + 13210890 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-14274560 + 183171600 z - 1151294145 z^2 + 5024578905 z^3 + 876621953250 z^4 + 236357199666 z^5 - 46704528045 z^6 + 9747938277 z^7 - 1442629188 z^8 + 105687120 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (114196480 - 1548343680 z + 10266987735 z^2 - 46789156950 z^3 + 197833514265 z^4 + 3543719569272 z^5 + 730061730153 z^6 - 144620436366 z^7 + 29936820375 z^8 - 4380731124 z^9 + 317061360 z^10) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (706039486155078075 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]