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

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=17/4

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

Input Form

Hypergeometric2F1[-(7/4), 17/4, -(11/2), z] == (1/(411840 Pi^(3/2))) (((1/(-1 + z)^8) (4 Sqrt[z] (102960 - 645840 z + 1596933 z^2 - 1817634 z^3 + 630630 z^4 + 360360 z^5 - 3253635 z^6 + 3268914 z^7 - 1385472 z^8 + 225280 z^9) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^8) (4 Sqrt[z] (102960 - 645840 z + 1596933 z^2 - 1817634 z^3 + 630630 z^4 + 360360 z^5 - 3253635 z^6 + 3268914 z^7 - 1385472 z^8 + 225280 z^9) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^7)) ((411840 - 617760 Sqrt[z] - 2274480 z + 3566160 z^(3/2) + 4712760 z^2 - 7906626 z^(5/2) - 3887247 z^3 + 7522515 z^(7/2) - 135135 z^4 - 1126125 z^(9/2) + 1216215 z^5 - 1936935 z^(11/2) + 3198195 z^6 + 3309075 z^(13/2) - 4765860 z^7 - 1771968 z^(15/2) + 2433024 z^8 + 337920 z^(17/2) - 450560 z^9) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^7 (1 + Sqrt[z])^8)) ((-411840 - 617760 Sqrt[z] + 2274480 z + 3566160 z^(3/2) - 4712760 z^2 - 7906626 z^(5/2) + 3887247 z^3 + 7522515 z^(7/2) + 135135 z^4 - 1126125 z^(9/2) - 1216215 z^5 - 1936935 z^(11/2) - 3198195 z^6 + 3309075 z^(13/2) + 4765860 z^7 - 1771968 z^(15/2) - 2433024 z^8 + 337920 z^(17/2) + 450560 z^9) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 7522515 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3887247 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7906626 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4712760 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3566160 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2274480 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 617760 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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]