Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bmf6)

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=-19/8, b>=a

For fixed z and a=-19/8, b=-1/8

http://functions.wolfram.com/07.23.03.bmf6.01

Input Form

Hypergeometric2F1[-(19/8), -(1/8), 6, z] == (1/(267173127992925 Pi z^5)) (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (-622592 + 5853824 z - 25519641 z^2 + 71073794 z^3 - 164795246 z^4 - 1221096264 z^5 - 7266225 z^6 + 522750 z^7) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 6 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (38912 - 337592 z + 1352439 z^2 - 3481940 z^3 - 162009694 z^4 - 3554700 z^5 + 261375 z^6) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (77824 - 718960 z + 3075891 z^2 - 8413504 z^3 + 223192354 z^4 + 54073260 z^5 - 3126045 z^6 + 209100 z^7) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (-622592 + 5853824 z - 25519641 z^2 + 71073794 z^3 - 164795246 z^4 - 1221096264 z^5 - 7266225 z^6 + 522750 z^7) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3075891 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 718960 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 77824 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 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</annotation-xml> </semantics> </math>

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]