Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a86l)

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=-19/4

http://functions.wolfram.com/07.23.03.a86l.01

Input Form

Hypergeometric2F1[-(21/4), -(19/4), 6, -z] == (1/(6845630929362225 Pi z^5)) (16384 (1 + z)^(1/4) (-2 (661504 + 16403232 z + 219914873 z^2 + 2335613000 z^3 + 28642302780 z^4 - 1350649369032 z^5 + 4355265320214 z^6 - 3995755814664 z^7 + 1173043434300 z^8 - 92999609880 z^9 + 891866745 z^10) EllipticE[1/2 - 1/(2 Sqrt[1 + z])] - Sqrt[1 + z] (-661504 - 15907104 z - 208062065 z^2 - 2181401495 z^3 - 27029949765 z^4 + 535026686157 z^5 - 1234220219763 z^6 + 798858720315 z^7 - 149879948055 z^8 + 5752130415 z^9) EllipticK[1/2 - 1/(2 Sqrt[1 + z])] + (661504 + 16403232 z + 219914873 z^2 + 2335613000 z^3 + 28642302780 z^4 - 1350649369032 z^5 + 4355265320214 z^6 - 3995755814664 z^7 + 1173043434300 z^8 - 92999609880 z^9 + 891866745 z^10) EllipticK[1/2 - 1/(2 Sqrt[1 + z])]))

Standard Form

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MathML Form

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16403232 </cn> <ci> z </ci> </apply> <cn type='integer'> 661504 </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]