Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a90b)

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=-19/4, b>=a

For fixed z and a=-19/4, b=-19/4

http://functions.wolfram.com/07.23.03.a90b.01

Input Form

Hypergeometric2F1[-(19/4), -(19/4), 5/2, z] == (1/(1923114375 Pi^(3/2) z^(3/2))) (4 (-4 (-917301 + 71549478 z + 1863946525 z^2 + 7050106420 z^3 + 6883554645 z^4 + 1767718598 z^5 + 80781731 z^6) EllipticE[(1/2) (1 - Sqrt[z])] + 4 (-917301 + 71549478 z + 1863946525 z^2 + 7050106420 z^3 + 6883554645 z^4 + 1767718598 z^5 + 80781731 z^6) EllipticE[(1/2) (1 + Sqrt[z])] + (-1834602 - 917301 Sqrt[z] + 143098956 z + 552251630 z^(3/2) + 3727893050 z^2 + 6148800245 z^(5/2) + 14100212840 z^3 + 15359019780 z^(7/2) + 13767109290 z^4 + 11115130885 z^(9/2) + 3535437196 z^5 + 2186447278 z^(11/2) + 161563462 z^6 + 72747675 z^(13/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (1834602 - 917301 Sqrt[z] - 143098956 z + 552251630 z^(3/2) - 3727893050 z^2 + 6148800245 z^(5/2) - 14100212840 z^3 + 15359019780 z^(7/2) - 13767109290 z^4 + 11115130885 z^(9/2) - 3535437196 z^5 + 2186447278 z^(11/2) - 161563462 z^6 + 72747675 z^(13/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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</apply> </apply> </apply> <apply> <times /> <cn type='integer'> 11115130885 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 13767109290 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 15359019780 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 14100212840 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 6148800245 </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'> 3727893050 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> 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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]