Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8gk)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-21/4, b>=a

For fixed z and a=-21/4, b=-1/4

http://functions.wolfram.com/07.23.03.a8gk.01

Input Form

Hypergeometric2F1[-(21/4), -(1/4), 11/2, z] == (1/(127536944535 Pi^(3/2) z^(9/2))) (8 (2 Sqrt[z] (-185640 + 2386800 z - 15241707 z^2 + 70835583 z^3 + 3559913994 z^4 + 517506990 z^5 - 136680775 z^6 + 32965779 z^7 - 5397216 z^8 + 428032 z^9) EllipticE[(1/2) (1 - Sqrt[z])] + 2 Sqrt[z] (-185640 + 2386800 z - 15241707 z^2 + 70835583 z^3 + 3559913994 z^4 + 517506990 z^5 - 136680775 z^6 + 32965779 z^7 - 5397216 z^8 + 428032 z^9) EllipticE[(1/2) (1 + Sqrt[z])] - (371280 - 185640 Sqrt[z] - 5052060 z + 2386800 z^(3/2) + 34021845 z^2 - 15241707 z^(5/2) - 164017581 z^3 + 70835583 z^(7/2) + 954310266 z^4 + 3559913994 z^(9/2) + 3232279050 z^5 + 517506990 z^(11/2) - 32047015 z^6 - 136680775 z^(13/2) + 7878255 z^7 + 32965779 z^(15/2) - 1319208 z^8 - 5397216 z^(17/2) + 107008 z^9 + 428032 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])] - (-371280 - 185640 Sqrt[z] + 5052060 z + 2386800 z^(3/2) - 34021845 z^2 - 15241707 z^(5/2) + 164017581 z^3 + 70835583 z^(7/2) - 954310266 z^4 + 3559913994 z^(9/2) - 3232279050 z^5 + 517506990 z^(11/2) + 32047015 z^6 - 136680775 z^(13/2) - 7878255 z^7 + 32965779 z^(15/2) + 1319208 z^8 - 5397216 z^(17/2) - 107008 z^9 + 428032 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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<plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 428032 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5397216 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 32965779 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 136680775 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 517506990 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3559913994 </cn> <apply> <power /> <ci> z </ci> 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type='rational'> 17 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1319208 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 32965779 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 15 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7878255 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 136680775 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 32047015 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 517506990 </cn> <apply> <power /> <ci> z </ci> <cn 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type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -371280 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </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]