Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.aac1)

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=-17/4, b>=a

For fixed z and a=-17/4, b=-9/4

http://functions.wolfram.com/07.23.03.aac1.01

Input Form

Hypergeometric2F1[-(17/4), -(9/4), 5, -z] == (4096 Sqrt[2] (-4 (96 + 1644 z + 15600 z^2 + 138075 z^3 - 5314775 z^4 + 8129234 z^5 - 2144294 z^6 + 18375 z^7 + 525 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - 4 Sqrt[1 + z] (96 + 1644 z + 15600 z^2 + 138075 z^3 - 5314775 z^4 + 8129234 z^5 - 2144294 z^6 + 18375 z^7 + 525 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + Sqrt[1 + z] (384 + 6480 z + 60825 z^2 + 537825 z^3 - 11898950 z^4 + 15625906 z^5 - 3451875 z^6 + 525 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + 4 (96 + 1644 z + 15600 z^2 + 138075 z^3 - 5314775 z^4 + 8129234 z^5 - 2144294 z^6 + 18375 z^7 + 525 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (19431286875 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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/> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1644 </cn> <ci> z </ci> </apply> <cn type='integer'> 96 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 19431286875 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </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 </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]