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

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=5/4, b>=a

For fixed z and a=5/4, b=17/4

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

Input Form

Hypergeometric2F1[5/4, 17/4, -(9/2), z] == (1/(56160 Pi^(3/2))) (((1/(-1 + z)^10) (2 Sqrt[z] (28080 - 303420 z + 1539759 z^2 - 5020587 z^3 + 12926550 z^4 + 276148830 z^5 + 18675195 z^6 - 2158343 z^7 + 153824 z^8) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^10) (2 Sqrt[z] (28080 - 303420 z + 1539759 z^2 - 5020587 z^3 + 12926550 z^4 + 276148830 z^5 + 18675195 z^6 - 2158343 z^7 + 153824 z^8) EllipticE[(1/2) (1 + Sqrt[z])]) - (1/((-1 + Sqrt[z])^10 (1 + Sqrt[z])^9)) ((-56160 + 84240 Sqrt[z] + 564720 z - 868140 z^(3/2) - 2660190 z^2 + 4199949 z^(5/2) + 8085753 z^3 - 13106340 z^(7/2) - 19963320 z^4 + 32889870 z^(9/2) + 74254830 z^5 + 201894000 z^(11/2) + 17160990 z^6 + 1514205 z^(13/2) - 2042975 z^7 - 115368 z^(15/2) + 153824 z^8) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^9 (1 + Sqrt[z])^10)) ((-56160 - 84240 Sqrt[z] + 564720 z + 868140 z^(3/2) - 2660190 z^2 - 4199949 z^(5/2) + 8085753 z^3 + 13106340 z^(7/2) - 19963320 z^4 - 32889870 z^(9/2) + 74254830 z^5 - 201894000 z^(11/2) + 17160990 z^6 - 1514205 z^(13/2) - 2042975 z^7 + 115368 z^(15/2) + 153824 z^8) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4199949 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2660190 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 868140 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 564720 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 84240 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -56160 </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> 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type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2042975 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1514205 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 17160990 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 201894000 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 74254830 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 32889870 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19963320 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 13106340 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8085753 </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'> 4199949 </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'> 2660190 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 868140 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </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]