Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6kp)

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=1/8

http://functions.wolfram.com/07.23.03.b6kp.01

Input Form

Hypergeometric2F1[-(45/8), 1/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[1 - z] (-105480192 + 1308201600 z - 7787517465 z^2 + 30978706395 z^3 - 109321102800 z^4 - 199462350346 z^5 + 68832403255 z^6 - 24908986545 z^7 + 6861605410 z^8 - 1207822000 z^9 + 100139424 z^10) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (-105480192 + 1308201600 z - 7787517465 z^2 + 30978706395 z^3 - 109321102800 z^4 - 199462350346 z^5 + 68832403255 z^6 - 24908986545 z^7 + 6861605410 z^8 - 1207822000 z^9 + 100139424 z^10) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (-105480192 + 1374126720 z - 8594327625 z^2 + 35716832970 z^3 - 127944063975 z^4 + 1029283996664 z^5 + 12572803145 z^6 - 4466823790 z^7 + 1201330135 z^8 - 206270900 z^9 + 16689904 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[1 - z] (-105480192 + 1308201600 z - 7787517465 z^2 + 30978706395 z^3 - 109321102800 z^4 - 199462350346 z^5 + 68832403255 z^6 - 24908986545 z^7 + 6861605410 z^8 - 1207822000 z^9 + 100139424 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/(304941368586659805 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] z^5)

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]