Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6tl)

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=-45/8, b>=a

For fixed z and a=-45/8, b=17/8

http://functions.wolfram.com/07.23.03.b6tl.01

Input Form

Hypergeometric2F1[-(45/8), 17/8, 5, z] == (65536 2^(1/4) (-16 Sqrt[1 - z] (1785472 - 9764300 z + 15134665 z^2 + 13670020 z^3 - 112065660 z^4 + 208586126 z^5 - 204960245 z^6 + 116473610 z^7 - 36340920 z^8 + 4845456 z^9) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - 8 Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (1785472 - 9764300 z + 15134665 z^2 + 13670020 z^3 - 112065660 z^4 + 208586126 z^5 - 204960245 z^6 + 116473610 z^7 - 36340920 z^8 + 4845456 z^9) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 8 Sqrt[1 - z] (1785472 - 9764300 z + 15134665 z^2 + 13670020 z^3 - 112065660 z^4 + 208586126 z^5 - 204960245 z^6 + 116473610 z^7 - 36340920 z^8 + 4845456 z^9) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (14283776 - 87041760 z + 168434175 z^2 + 41010060 z^3 - 206189270 z^4 + 342554548 z^5 - 313664225 z^6 + 168874280 z^7 - 50377360 z^8 + 6460608 z^9) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/(25158103175205 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] z^4)

Standard Form

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MathML Form

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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]