Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bs7o)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-9/8, b>=a

For fixed z and a=-9/8, b=-3/8

http://functions.wolfram.com/07.23.03.bs7o.01

Input Form

Hypergeometric2F1[-(9/8), -(3/8), 6, z] == (1/(15952843246725 Pi z^5)) (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (98304 - 804480 z + 2987401 z^2 - 6887508 z^3 + 12735558 z^4 + 71041740 z^5 + 2830905 z^6) EllipticE[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 80 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (768 - 6159 z + 22367 z^2 - 50427 z^3 + 852885 z^4 + 205590 z^5) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-12288 + 91632 z - 307715 z^2 + 643401 z^3 + 19141815 z^4 + 943635 z^5) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (98304 - 804480 z + 2987401 z^2 - 6887508 z^3 + 12735558 z^4 + 71041740 z^5 + 2830905 z^6) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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</cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </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]