Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aa6v)

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=-9/2, b>=a

For fixed z and a=-9/2, b=23/4

http://functions.wolfram.com/07.23.03.aa6v.01

Input Form

Hypergeometric2F1[-(9/2), 23/4, 5, z] == (8 Sqrt[2] (2 (-36864 - 187392 z - 967488 z^2 - 7787520 z^3 + 210882400 z^4 - 772512640 z^5 + 1144775580 z^6 - 769849080 z^7 + 195634725 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + 2 Sqrt[1 - z] (-36864 - 187392 z - 967488 z^2 - 7787520 z^3 + 210882400 z^4 - 772512640 z^5 + 1144775580 z^6 - 769849080 z^7 + 195634725 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - (-36864 - 187392 z - 967488 z^2 - 7787520 z^3 + 210882400 z^4 - 772512640 z^5 + 1144775580 z^6 - 769849080 z^7 + 195634725 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - (1 - z)^(1/4) (-36864 - 187392 z - 967488 z^2 - 7787520 z^3 + 210882400 z^4 - 772512640 z^5 + 1144775580 z^6 - 769849080 z^7 + 195634725 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - Sqrt[1 - z] (-36864 - 187392 z - 967488 z^2 - 7787520 z^3 + 210882400 z^4 - 772512640 z^5 + 1144775580 z^6 - 769849080 z^7 + 195634725 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + (1 - z)^(3/4) (36864 + 205824 z + 1076160 z^2 + 8360640 z^3 - 6460000 z^4 - 239599360 z^5 + 654102540 z^6 - 613341300 z^7 + 195634725 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])]))/ (800224425 Pi Sqrt[1 + Sqrt[1 - z]] z^4)

Standard Form

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MathML Form

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type='integer'> 613341300 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 654102540 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 239599360 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6460000 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 8360640 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1076160 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 205824 </cn> <ci> z </ci> </apply> <cn type='integer'> 36864 </cn> </apply> <apply> 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type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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]