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

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=-39/8, b>=a

For fixed z and a=-39/8, b=27/8

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

Input Form

Hypergeometric2F1[-(39/8), 27/8, 6, z] == (524288 2^(1/4) (2 Sqrt[1 - z] (-70090752 + 242853504 z - 135373745 z^2 - 177455005 z^3 - 403896675 z^4 + 3148480481 z^5 - 5489158832 z^6 + 4516037760 z^7 - 1854278400 z^8 + 307632000 z^9) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (-70090752 + 242853504 z - 135373745 z^2 - 177455005 z^3 - 403896675 z^4 + 3148480481 z^5 - 5489158832 z^6 + 4516037760 z^7 - 1854278400 z^8 + 307632000 z^9) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[1 - z] (-70090752 + 242853504 z - 135373745 z^2 - 177455005 z^3 - 403896675 z^4 + 3148480481 z^5 - 5489158832 z^6 + 4516037760 z^7 - 1854278400 z^8 + 307632000 z^9) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - (-70090752 + 286660224 z - 279970145 z^2 - 114379460 z^3 - 288729350 z^4 - 3330411484 z^5 + 11420222543 z^6 - 15527026840 z^7 + 10996411920 z^8 - 4046952000 z^9 + 615264000 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (1769561680481595 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]