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

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=-7/4, b>=a

For fixed z and a=-7/4, b=21/4

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

Input Form

Hypergeometric2F1[-(7/4), 21/4, -(9/2), z] == (1/(318240 Pi^(3/2))) (((1/(-1 + z)^8) (2 Sqrt[z] (159120 - 888420 z + 1754961 z^2 - 1021020 z^3 - 714714 z^4 + 11747736 z^5 - 18378591 z^6 + 13298560 z^7 - 4843520 z^8 + 720896 z^9) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^8) (2 Sqrt[z] (159120 - 888420 z + 1754961 z^2 - 1021020 z^3 - 714714 z^4 + 11747736 z^5 - 18378591 z^6 + 13298560 z^7 - 4843520 z^8 + 720896 z^9) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^7)) ((318240 - 477360 Sqrt[z] - 1538160 z + 2426580 z^(3/2) + 2380170 z^2 - 4135131 z^(5/2) - 357357 z^3 + 1378377 z^(7/2) - 1582581 z^4 + 2297295 z^(9/2) - 4339335 z^5 - 7408401 z^(11/2) + 11234223 z^6 + 7144368 z^(13/2) - 10155904 z^7 - 3142656 z^(15/2) + 4302848 z^8 + 540672 z^(17/2) - 720896 z^9) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^8)) ((-318240 - 477360 Sqrt[z] + 1538160 z + 2426580 z^(3/2) - 2380170 z^2 - 4135131 z^(5/2) + 357357 z^3 + 1378377 z^(7/2) + 1582581 z^4 + 2297295 z^(9/2) + 4339335 z^5 - 7408401 z^(11/2) - 11234223 z^6 + 7144368 z^(13/2) + 10155904 z^7 - 3142656 z^(15/2) - 4302848 z^8 + 540672 z^(17/2) + 720896 z^9) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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</cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1021020 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1754961 </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'> 888420 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 159120 </cn> </apply> <apply> <ci> EllipticE </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> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> z 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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]