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

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.accv.01

Input Form

Hypergeometric2F1[-(13/4), -(1/4), -(11/2), z] == (1/(59136 Pi^(3/2))) ((-((1/(-1 + z)^2) (2 (-14784 + 29904 z - 15736 z^2 + 525 z^3 + 111 z^4 + 32 z^5) EllipticE[(1/2) (1 - Sqrt[z])])) - (2 (-14784 + 29904 z - 15736 z^2 + 525 z^3 + 111 z^4 + 32 z^5) EllipticE[(1/2) (1 + Sqrt[z])])/(-1 + z)^2 + (1/((-1 + Sqrt[z])^2 (1 + Sqrt[z]))) ((-14784 + 7392 Sqrt[z] + 22512 z - 8176 z^(3/2) - 7560 z^2 + 630 z^(5/2) - 105 z^3 + 135 z^(7/2) - 24 z^4 + 32 z^(9/2)) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/(-1 - Sqrt[z] + z + z^(3/2))) ((14784 + 7392 Sqrt[z] - 22512 z - 8176 z^(3/2) + 7560 z^2 + 630 z^(5/2) + 105 z^3 + 135 z^(7/2) + 24 z^4 + 32 z^(9/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

Standard Form

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MathML Form

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<power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 630 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7560 </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'> 8176 </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'> 22512 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 7392 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 14784 </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> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn 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]