Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9gk)

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=37/8

http://functions.wolfram.com/07.23.03.b9gk.01

Input Form

Hypergeometric2F1[-(41/8), 37/8, 5, z] == (1/(20110981631115375 Pi z^4)) (32768 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-8 (176647680 + 344094960 z + 1127394015 z^2 + 6970338045 z^3 - 417986597015 z^4 + 1764906846779 z^5 - 3154999782432 z^6 + 2890154603952 z^7 - 1343470512384 z^8 + 252351348480 z^9) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (44161920 + 93269055 z + 299358015 z^2 - 120950183475 z^3 + 544844783005 z^4 - 1004730958744 z^5 + 938928739824 z^6 - 442816532928 z^7 + 84117116160 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-44161920 - 118110135 z - 370776120 z^2 - 206601307275 z^3 + 980131062410 z^4 - 1871025100016 z^5 + 1797189321312 z^6 - 867607969536 z^7 + 168234232320 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 (176647680 + 344094960 z + 1127394015 z^2 + 6970338045 z^3 - 417986597015 z^4 + 1764906846779 z^5 - 3154999782432 z^6 + 2890154603952 z^7 - 1343470512384 z^8 + 252351348480 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - 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)

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]