Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9v9)

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=-19/4, b>=a

For fixed z and a=-19/4, b=21/4

http://functions.wolfram.com/07.23.03.a9v9.01

Input Form

Hypergeometric2F1[-(19/4), 21/4, 6, -z] == (16384 Sqrt[2] (Sqrt[1 + z] (428032 - 501600 z + 806531 z^2 - 1916530 z^3 + 8791167 z^4 + 226254436 z^5 + 667217920 z^6 + 814276608 z^7 + 461373440 z^8 + 100663296 z^9) EllipticE[(-1 + Sqrt[1 + z])/ (1 + Sqrt[1 + z])] + (428032 - 73568 z + 304931 z^2 - 1109999 z^3 + 6874637 z^4 + 235045603 z^5 + 893472356 z^6 + 1481494528 z^7 + 1275650048 z^8 + 562036736 z^9 + 100663296 z^10) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (428032 - 180576 z + 400235 z^2 - 1262569 z^3 + 7272573 z^4 + 84862801 z^5 + 211147072 z^6 + 232765440 z^7 + 122421248 z^8 + 25165824 z^9) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (428032 - 501600 z + 806531 z^2 - 1916530 z^3 + 8791167 z^4 + 226254436 z^5 + 667217920 z^6 + 814276608 z^7 + 461373440 z^8 + 100663296 z^9) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (1213736759235 Pi z^5 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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</apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 100663296 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 461373440 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 814276608 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 667217920 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 226254436 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8791167 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> 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</apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1213736759235 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </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]