Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8e6)

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=-5/4

http://functions.wolfram.com/07.23.03.a8e6.01

Input Form

Hypergeometric2F1[-(21/4), -(5/4), 11/2, z] == (1/(586669944861 Pi^(3/2) z^(9/2))) (8 (-8 Sqrt[z] (46410 - 725985 z + 5911308 z^2 - 37754535 z^3 - 4226619306 z^4 - 4420162383 z^5 - 200548040 z^6 + 23924439 z^7 - 2595780 z^8 + 153824 z^9) EllipticE[(1/2) (1 - Sqrt[z])] - 8 Sqrt[z] (46410 - 725985 z + 5911308 z^2 - 37754535 z^3 - 4226619306 z^4 - 4420162383 z^5 - 200548040 z^6 + 23924439 z^7 - 2595780 z^8 + 153824 z^9) EllipticE[(1/2) (1 + Sqrt[z])] + (-371280 + 185640 Sqrt[z] + 6086340 z - 2903940 z^(3/2) - 51604605 z^2 + 23645232 z^(5/2) + 336871626 z^3 - 151018140 z^(7/2) - 3075437547 z^4 - 16906477224 z^(9/2) - 20030101728 z^5 - 17680649532 z^(11/2) - 12639742115 z^6 - 802192160 z^(13/2) + 23217810 z^7 + 95697756 z^(15/2) - 2552517 z^8 - 10383120 z^(17/2) + 153824 z^9 + 615296 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (371280 + 185640 Sqrt[z] - 6086340 z - 2903940 z^(3/2) + 51604605 z^2 + 23645232 z^(5/2) - 336871626 z^3 - 151018140 z^(7/2) + 3075437547 z^4 - 16906477224 z^(9/2) + 20030101728 z^5 - 17680649532 z^(11/2) + 12639742115 z^6 - 802192160 z^(13/2) - 23217810 z^7 + 95697756 z^(15/2) + 2552517 z^8 - 10383120 z^(17/2) - 153824 z^9 + 615296 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='integer'> 185640 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -371280 </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> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 615296 </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'> 153824 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10383120 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 17 <sep /> 2 </cn> </apply> 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type='integer'> 2 </cn> </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]