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

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=-13/4, b>=a

For fixed z and a=-13/4, b=-1/4

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

Input Form

Hypergeometric2F1[-(13/4), -(1/4), 9/2, z] == (1/(29981259 Pi^(3/2) z^(7/2))) (4 (2 Sqrt[z] (780 - 7605 z + 38844 z^2 + 1657074 z^3 + 170016 z^4 - 26565 z^5 + 2464 z^6) EllipticE[(1/2) (1 - Sqrt[z])] + 2 Sqrt[z] (780 - 7605 z + 38844 z^2 + 1657074 z^3 + 170016 z^4 - 26565 z^5 + 2464 z^6) EllipticE[(1/2) (1 + Sqrt[z])] - (-1560 + 780 Sqrt[z] + 16380 z - 7605 z^(3/2) - 88920 z^2 + 38844 z^(5/2) + 490152 z^3 + 1657074 z^(7/2) + 1424808 z^4 + 170016 z^(9/2) - 6468 z^5 - 26565 z^(11/2) + 616 z^6 + 2464 z^(13/2)) EllipticK[(1/2) (1 - Sqrt[z])] - (1560 + 780 Sqrt[z] - 16380 z - 7605 z^(3/2) + 88920 z^2 + 38844 z^(5/2) - 490152 z^3 + 1657074 z^(7/2) - 1424808 z^4 + 170016 z^(9/2) + 6468 z^5 - 26565 z^(11/2) - 616 z^6 + 2464 z^(13/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='integer'> 7605 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 16380 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 780 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1560 </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 /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2464 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 616 </cn> 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<cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 38844 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 88920 </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'> 7605 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 16380 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 780 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 1560 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </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]