Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9qy)

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=-39/8, b>=a

For fixed z and a=-39/8, b=-27/8

http://functions.wolfram.com/07.23.03.b9qy.01

Input Form

Hypergeometric2F1[-(39/8), -(27/8), 4, z] == (2048 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (67626624 - 1675343943 z + 30821890554 z^2 + 1290451036735 z^3 + 3598165009740 z^4 + 2257554422151 z^5 + 291224135258 z^6 + 235126881 z^7) EllipticE[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (67626624 - 1675343943 z + 30821890554 z^2 + 1290451036735 z^3 + 3598165009740 z^4 + 2257554422151 z^5 + 291224135258 z^6 + 235126881 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (67626624 - 1675343943 z + 30821890554 z^2 + 1290451036735 z^3 + 3598165009740 z^4 + 2257554422151 z^5 + 291224135258 z^6 + 235126881 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (67626624 - 1700703927 z + 31443210162 z^2 - 870720459749 z^3 - 5563529544720 z^4 - 6609236090289 z^5 - 1848768401158 z^6 - 72183952467 z^7 + 940507524 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (2201376145243275 Pi (1 + Sqrt[1 - z])^(1/4) z^3)

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]