Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ab18)

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=-15/4, b>=a

For fixed z and a=-15/4, b=-15/4

http://functions.wolfram.com/07.23.03.ab18.01

Input Form

Hypergeometric2F1[-(15/4), -(15/4), 5, -z] == (4096 Sqrt[2] (-4 Sqrt[1 + z] (36960 + 719180 z + 8028405 z^2 + 88455675 z^3 - 1613858198 z^4 + 2568166362 z^5 - 878398527 z^6 + 50446943 z^7) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - 4 (36960 + 756140 z + 8747585 z^2 + 96484080 z^3 - 1525402523 z^4 + 954308164 z^5 + 1689767835 z^6 - 827951584 z^7 + 50446943 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + 4 Sqrt[1 + z] (36960 + 719180 z + 8028405 z^2 + 88455675 z^3 - 1613858198 z^4 + 2568166362 z^5 - 878398527 z^6 + 50446943 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (147840 + 2987600 z + 34260765 z^2 + 377710410 z^3 + 2191226383 z^4 - 17647322196 z^5 + 17869391979 z^6 - 4033363942 z^7 + 140821065 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (17169326707995 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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</apply> </apply> <apply> <times /> <cn type='integer'> 2987600 </cn> <ci> z </ci> </apply> <cn type='integer'> 147840 </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'> 17169326707995 </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]