Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a90o)

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.a90o.01

Input Form

Hypergeometric2F1[-(19/4), -(19/4), 6, z] == (1/(3207049499298645 Pi z^5)) (16384 (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[(1/2) (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[(1/2) (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[(1/2) (1 - Sqrt[1 - z])]))

Standard Form

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MathML Form

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<power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 131685675 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 10252704 </cn> <ci> z </ci> </apply> <cn type='integer'> -428032 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </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]