Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a90p)

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=-19/4

http://functions.wolfram.com/07.23.03.a90p.01

Input Form

Hypergeometric2F1[-(19/4), -(19/4), 6, -z] == (16384 Sqrt[2] (Sqrt[1 + z] (-428032 - 9931680 z - 124267011 z^2 - 1230609765 z^3 - 14122410687 z^4 + 321766436103 z^5 - 742109370345 z^6 + 467568632529 z^7 - 82933653525 z^8 + 2818837613 z^9) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (-428032 - 10359712 z - 134198691 z^2 - 1354876776 z^3 - 15353020452 z^4 + 307644025416 z^5 - 420342934242 z^6 - 274540737816 z^7 + 384634979004 z^8 - 80114815912 z^9 + 2818837613 z^10) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (-428032 - 9931680 z - 124267011 z^2 - 1230609765 z^3 - 14122410687 z^4 + 321766436103 z^5 - 742109370345 z^6 + 467568632529 z^7 - 82933653525 z^8 + 2818837613 z^9) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - (-428032 - 10252704 z - 131685675 z^2 - 1323125496 z^3 - 15036942228 z^4 - 80228196552 z^5 + 947851951326 z^6 - 1490753365320 z^7 + 669614523564 z^8 - 83963955928 z^9 + 1830673845 z^10) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (3207049499298645 Pi z^5 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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<apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 19 <sep /> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 19 <sep /> 4 </cn> </apply> </list> <list> <cn type='integer'> 6 </cn> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 16384 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <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'> 2818837613 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 82933653525 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 467568632529 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 742109370345 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 321766436103 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 14122410687 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1230609765 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 124267011 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 9931680 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -428032 </cn> </apply> <apply> <ci> EllipticE </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> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2818837613 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 10 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn 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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]