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

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=1/4

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

Input Form

Hypergeometric2F1[-(15/4), 1/4, 5, -z] == (4096 Sqrt[2] (8 Sqrt[1 + z] (-16 - 146 z - 639 z^2 - 2069 z^3 + 2069 z^4 + 639 z^5 + 146 z^6 + 16 z^7) EllipticE[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] + 8 (-16 - 162 z - 785 z^2 - 2708 z^3 + 2708 z^5 + 785 z^6 + 162 z^7 + 16 z^8) EllipticE[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] - 8 Sqrt[1 + z] (-16 - 146 z - 639 z^2 - 2069 z^3 + 2069 z^4 + 639 z^5 + 146 z^6 + 16 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (-128 - 1264 z - 5979 z^2 - 20308 z^3 - 64298 z^4 + 1356 z^5 + 301 z^6 + 32 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (140821065 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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</apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 140821065 </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]