Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8vs)

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=-21/4, b>=a

For fixed z and a=-21/4, b=19/4

http://functions.wolfram.com/07.23.03.a8vs.01

Input Form

Hypergeometric2F1[-(21/4), 19/4, 6, z] == (1/(1852856480475 Pi z^5)) (16384 (-2 (-1357824 + 784992 z + 1026987 z^2 + 2485587 z^3 + 12238317 z^4 - 351427183 z^5 + 1194104932 z^6 - 1809051648 z^7 + 1443684352 z^8 - 595591168 z^9 + 100663296 z^10) EllipticE[(1/2) (1 - Sqrt[1 - z])] - Sqrt[1 - z] (1357824 - 445536 z - 979251 z^2 - 2683161 z^3 - 12986181 z^4 + 121645969 z^5 - 267790656 z^6 + 268744704 z^7 - 130809856 z^8 + 25165824 z^9) EllipticK[(1/2) (1 - Sqrt[1 - z])] + (-1357824 + 784992 z + 1026987 z^2 + 2485587 z^3 + 12238317 z^4 - 351427183 z^5 + 1194104932 z^6 - 1809051648 z^7 + 1443684352 z^8 - 595591168 z^9 + 100663296 z^10) EllipticK[(1/2) (1 - Sqrt[1 - z])]))

Standard Form

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MathML Form

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12238317 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2485587 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1026987 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 784992 </cn> <ci> z </ci> </apply> <cn type='integer'> -1357824 </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 /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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]