Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a90n)

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.a90n.01

Input Form

Hypergeometric2F1[-(19/4), -(19/4), 11/2, z] == (32 (-2 (46698960 - 1135785420 z + 15400677677 z^2 - 179649537067 z^3 + 3523581882513 z^4 + 35894689080305 z^5 + 67327574610215 z^6 + 37356911379999 z^7 + 5965393841291 z^8 + 180524342951 z^9) EllipticE[(1/2) (1 - Sqrt[z])] + 2 (46698960 - 1135785420 z + 15400677677 z^2 - 179649537067 z^3 + 3523581882513 z^4 + 35894689080305 z^5 + 67327574610215 z^6 + 37356911379999 z^7 + 5965393841291 z^8 + 180524342951 z^9) EllipticE[(1/2) (1 + Sqrt[z])] + (46698960 + 23349480 Sqrt[z] - 1135785420 z - 565946920 z^(3/2) + 15400677677 z^2 + 7653987341 z^(5/2) - 179649537067 z^3 - 89206104603 z^(7/2) + 3523581882513 z^4 + 9003275443745 z^(9/2) + 35894689080305 z^5 + 46920528498665 z^(11/2) + 67327574610215 z^6 + 63321207262215 z^(13/2) + 37356911379999 z^7 + 27382182692207 z^(15/2) + 5965393841291 z^8 + 3460179754019 z^(17/2) + 180524342951 z^9 + 78058255275 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (-46698960 + 23349480 Sqrt[z] + 1135785420 z - 565946920 z^(3/2) - 15400677677 z^2 + 7653987341 z^(5/2) + 179649537067 z^3 - 89206104603 z^(7/2) - 3523581882513 z^4 + 9003275443745 z^(9/2) - 35894689080305 z^5 + 46920528498665 z^(11/2) - 67327574610215 z^6 + 63321207262215 z^(13/2) - 37356911379999 z^7 + 27382182692207 z^(15/2) - 5965393841291 z^8 + 3460179754019 z^(17/2) - 180524342951 z^9 + 78058255275 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)/ (231957246217125 Pi^(3/2) z^(9/2))

Standard Form

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MathML Form

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type='integer'> 9003275443745 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3523581882513 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 89206104603 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 7 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 179649537067 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7653987341 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 15400677677 </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'> 565946920 </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'> 1135785420 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 23349480 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> 46698960 </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> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 78058255275 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 19 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <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]