Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8yv)

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=23/4

http://functions.wolfram.com/07.23.03.a8yv.01

Input Form

Hypergeometric2F1[-(21/4), 23/4, 3, -z] == (64 Sqrt[2] ((-204204 + 5564559 z + 594932735 z^2 + 4864493600 z^3 + 14984243200 z^4 + 21756903424 z^5 + 15074328576 z^6 + 4026531840 z^7) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + Sqrt[1 + z] (-204204 + 5564559 z + 594932735 z^2 + 4864493600 z^3 + 14984243200 z^4 + 21756903424 z^5 + 15074328576 z^6 + 4026531840 z^7) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - 2 Sqrt[1 + z] (-102102 + 2807805 z + 131933500 z^2 + 772012800 z^3 + 1662156800 z^4 + 1522532352 z^5 + 503316480 z^6) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (-204204 + 5564559 z + 594932735 z^2 + 4864493600 z^3 + 14984243200 z^4 + 21756903424 z^5 + 15074328576 z^6 + 4026531840 z^7) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (10548412875 Pi z^2 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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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> <apply> <power /> <apply> <times /> <cn type='integer'> 10548412875 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </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]