Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b907)

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=5/8

http://functions.wolfram.com/07.23.03.b907.01

Input Form

Hypergeometric2F1[-(41/8), 5/8, 6, z] == (1/(49258568801517075 Pi z^5)) (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (-114196480 + 1259283840 z - 6433933815 z^2 + 20593955805 z^3 - 50258021400 z^4 - 186076577862 z^5 + 101380936041 z^6 - 46980912951 z^7 + 15300396342 z^8 - 3055095120 z^9 + 280193760 z^10) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-14274560 + 155068560 z - 779514345 z^2 + 2453736225 z^3 - 43496166510 z^4 + 16490037186 z^5 - 7690956021 z^6 + 2520978957 z^7 - 506402820 z^8 + 46698960 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 6 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (7137280 - 73519560 z + 349207455 z^2 - 1038338325 z^3 - 28908806850 z^4 + 15845653782 z^5 - 7460298909 z^6 + 2470669047 z^7 - 501399360 z^8 + 46698960 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (-114196480 + 1259283840 z - 6433933815 z^2 + 20593955805 z^3 - 50258021400 z^4 - 186076577862 z^5 + 101380936041 z^6 - 46980912951 z^7 + 15300396342 z^8 - 3055095120 z^9 + 280193760 z^10) 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]