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

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=-1/4, b>=a

For fixed z and a=-1/4, b=23/4

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

Input Form

Hypergeometric2F1[-(1/4), 23/4, -(9/2), z] == (1/(176964480 Pi^(3/2))) (((1/(-1 + z)^10) (4 (22120560 - 211374240 z + 894324585 z^2 - 2174151133 z^3 + 3204596164 z^4 - 1833241410 z^5 + 2071168905 z^6 - 1350275745 z^7 + 549890874 z^8 - 129672192 z^9 + 13578240 z^10) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^10) (4 (22120560 - 211374240 z + 894324585 z^2 - 2174151133 z^3 + 3204596164 z^4 - 1833241410 z^5 + 2071168905 z^6 - 1350275745 z^7 + 549890874 z^8 - 129672192 z^9 + 13578240 z^10) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^10 (1 + Sqrt[z])^9)) ((-44241120 + 22120560 Sqrt[z] + 400627920 z - 191097060 z^(3/2) - 1597552110 z^2 + 719920355 z^(5/2) + 3628381911 z^3 - 1520548568 z^(7/2) - 4888643760 z^4 + 1833241410 z^(9/2) + 1833241410 z^5 - 2714772840 z^(11/2) - 1427564970 z^6 + 2034564675 z^(13/2) + 665986815 z^7 - 923731380 z^(15/2) - 176050368 z^8 + 238977024 z^(17/2) + 20367360 z^9 - 27156480 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^9 (1 + Sqrt[z])^10)) ((44241120 + 22120560 Sqrt[z] - 400627920 z - 191097060 z^(3/2) + 1597552110 z^2 + 719920355 z^(5/2) - 3628381911 z^3 - 1520548568 z^(7/2) + 4888643760 z^4 + 1833241410 z^(9/2) - 1833241410 z^5 - 2714772840 z^(11/2) + 1427564970 z^6 + 2034564675 z^(13/2) - 665986815 z^7 - 923731380 z^(15/2) + 176050368 z^8 + 238977024 z^(17/2) - 20367360 z^9 - 27156480 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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type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 10 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -27156480 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 19 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20367360 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 238977024 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 17 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 176050368 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 923731380 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 15 <sep /> 2 </cn> </apply> 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</apply> </apply> <apply> <times /> <cn type='integer'> 4888643760 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1520548568 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3628381911 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 719920355 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1597552110 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 191097060 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> 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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]