Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9ub)

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=21/4

http://functions.wolfram.com/07.23.03.a9ub.01

Input Form

Hypergeometric2F1[-(19/4), 21/4, -(11/2), z] == (1/(7001280 Pi^(3/2))) (((1/(-1 + z)^6) (4 Sqrt[z] (1750320 - 1909440 z - 1255059 z^2 - 1284231 z^3 - 1891539 z^4 - 4592601 z^5 + 417343830 z^6 - 1206488064 z^7 + 1389809664 z^8 - 738197504 z^9 + 150994944 z^10) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^6) (4 Sqrt[z] (1750320 - 1909440 z - 1255059 z^2 - 1284231 z^3 - 1891539 z^4 - 4592601 z^5 + 417343830 z^6 - 1206488064 z^7 + 1389809664 z^8 - 738197504 z^9 + 150994944 z^10) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^6 (1 + Sqrt[z])^5)) ((7001280 - 10501920 Sqrt[z] - 2386800 z + 6205680 z^(3/2) - 6285240 z^2 + 8795358 z^(5/2) - 9701679 z^3 + 12270141 z^(7/2) - 15239718 z^4 + 19022796 z^(9/2) - 30763863 z^5 + 39949065 z^(11/2) - 153086700 z^6 - 681600960 z^(13/2) + 1137991680 z^7 + 1274984448 z^(15/2) - 1877581824 z^8 - 902037504 z^(17/2) + 1249902592 z^9 + 226492416 z^(19/2) - 301989888 z^10) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^5 (1 + Sqrt[z])^6)) ((-7001280 - 10501920 Sqrt[z] + 2386800 z + 6205680 z^(3/2) + 6285240 z^2 + 8795358 z^(5/2) + 9701679 z^3 + 12270141 z^(7/2) + 15239718 z^4 + 19022796 z^(9/2) + 30763863 z^5 + 39949065 z^(11/2) + 153086700 z^6 - 681600960 z^(13/2) - 1137991680 z^7 + 1274984448 z^(15/2) + 1877581824 z^8 - 902037504 z^(17/2) - 1249902592 z^9 + 226492416 z^(19/2) + 301989888 z^10) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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type='integer'> 10501920 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 7001280 </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 /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 6 </cn> </apply> </apply> <cn type='integer'> -1 </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]