Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8ya)

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=23/4

http://functions.wolfram.com/07.23.03.a8ya.01

Input Form

Hypergeometric2F1[-(21/4), 23/4, -(9/2), z] == (1/(8426880 Pi^(3/2))) (((1/(-1 + z)^5) (2 (-2106720 - 3862320 z - 8909670 z^2 - 26543209 z^3 - 117995548 z^4 - 1371124645 z^5 + 12194244640 z^6 - 30355650560 z^7 + 34515976192 z^8 - 18849202176 z^9 + 4026531840 z^10) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^5) (2 (-2106720 - 3862320 z - 8909670 z^2 - 26543209 z^3 - 117995548 z^4 - 1371124645 z^5 + 12194244640 z^6 - 30355650560 z^7 + 34515976192 z^8 - 18849202176 z^9 + 4026531840 z^10) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^4)) ((2106720 - 1053360 Sqrt[z] + 4915680 z - 2896740 z^(3/2) + 11806410 z^2 - 7146755 z^(5/2) + 33689964 z^3 - 19938182 z^(7/2) + 137933730 z^4 - 77571395 z^(9/2) + 1448696040 z^5 - 3275452960 z^(11/2) - 8918791680 z^6 + 14696806400 z^(13/2) + 15658844160 z^7 - 23115857920 z^(15/2) - 11400118272 z^8 + 15829303296 z^(17/2) + 3019898880 z^9 - 4026531840 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^4 (1 + Sqrt[z])^5)) ((2106720 + 1053360 Sqrt[z] + 4915680 z + 2896740 z^(3/2) + 11806410 z^2 + 7146755 z^(5/2) + 33689964 z^3 + 19938182 z^(7/2) + 137933730 z^4 + 77571395 z^(9/2) + 1448696040 z^5 + 3275452960 z^(11/2) - 8918791680 z^6 - 14696806400 z^(13/2) + 15658844160 z^7 + 23115857920 z^(15/2) - 11400118272 z^8 - 15829303296 z^(17/2) + 3019898880 z^9 + 4026531840 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

Standard Form

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MathML Form

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1448696040 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 77571395 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 137933730 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 19938182 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 33689964 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7146755 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 11806410 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2896740 </cn> <apply> <power /> <ci> z </ci> <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]