Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a87i)

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=-17/4

http://functions.wolfram.com/07.23.03.a87i.01

Input Form

Hypergeometric2F1[-(21/4), -(17/4), 11/2, z] == (8 (2 Sqrt[z] (-450840 + 10820160 z - 145023957 z^2 + 1679164851 z^3 + 622766766615 z^4 + 2362753469055 z^5 + 2329637577505 z^6 + 683219875353 z^7 + 47277756021 z^8 + 113998005 z^9) EllipticE[(1/2) (1 - Sqrt[z])] + 2 Sqrt[z] (-450840 + 10820160 z - 145023957 z^2 + 1679164851 z^3 + 622766766615 z^4 + 2362753469055 z^5 + 2329637577505 z^6 + 683219875353 z^7 + 47277756021 z^8 + 113998005 z^9) EllipticE[(1/2) (1 + Sqrt[z])] - (901680 - 450840 Sqrt[z] - 22316580 z + 10820160 z^(3/2) + 306176715 z^2 - 145023957 z^(5/2) - 3573481821 z^3 + 1679164851 z^(7/2) + 68499558855 z^4 + 622766766615 z^(9/2) + 1037878596615 z^5 + 2362753469055 z^(11/2) + 2566330267585 z^6 + 2329637577505 z^(13/2) + 1913075238345 z^7 + 683219875353 z^(15/2) + 441449420349 z^8 + 47277756021 z^(17/2) + 23369591025 z^9 + 113998005 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])] - (-901680 - 450840 Sqrt[z] + 22316580 z + 10820160 z^(3/2) - 306176715 z^2 - 145023957 z^(5/2) + 3573481821 z^3 + 1679164851 z^(7/2) - 68499558855 z^4 + 622766766615 z^(9/2) - 1037878596615 z^5 + 2362753469055 z^(11/2) - 2566330267585 z^6 + 2329637577505 z^(13/2) - 1913075238345 z^7 + 683219875353 z^(15/2) - 441449420349 z^8 + 47277756021 z^(17/2) - 23369591025 z^9 + 113998005 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)/ (20984732643105 Pi^(3/2) z^(9/2))

Standard Form

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MathML Form

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<ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 113998005 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 47277756021 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 683219875353 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2329637577505 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2362753469055 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> 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<apply> <times /> <cn type='integer'> 441449420349 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 683219875353 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 15 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1913075238345 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2329637577505 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2566330267585 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2362753469055 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1037878596615 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 622766766615 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 68499558855 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1679164851 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3573481821 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 145023957 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 306176715 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 10820160 </cn> <apply> <power /> <ci> z </ci> 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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]