Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a99l)

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=-5/4

http://functions.wolfram.com/07.23.03.a99l.01

Input Form

Hypergeometric2F1[-(19/4), -(5/4), 6, -z] == (1/(127442359719675 Pi z^5)) (16384 (1 + z)^(1/4) (2 (-428032 - 6440544 z - 48746115 z^2 - 265344310 z^3 - 1456379925 z^4 + 20723056536 z^5 - 13417398813 z^6 + 64940850 z^7 + 6016725 z^8 + 331500 z^9) EllipticE[1/2 - 1/(2 Sqrt[1 + z])] - (-428032 - 6440544 z - 48746115 z^2 - 265344310 z^3 - 1456379925 z^4 + 20723056536 z^5 - 13417398813 z^6 + 64940850 z^7 + 6016725 z^8 + 331500 z^9) EllipticK[1/2 - 1/(2 Sqrt[1 + z])] + Sqrt[1 + z] (428032 + 6119520 z + 44206635 z^2 + 232887655 z^3 + 1286635350 z^4 - 6105623706 z^5 + 1770889575 z^6 + 121279275 z^7 + 11536200 z^8 + 663000 z^9) EllipticK[1/2 - 1/(2 Sqrt[1 + z])]))

Standard Form

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MathML Form

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<times /> <cn type='integer'> 232887655 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 44206635 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6119520 </cn> <ci> z </ci> </apply> <cn type='integer'> 428032 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </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]