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

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=-31/8, b>=a

For fixed z and a=-31/8, b=-21/8

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

Input Form

Hypergeometric2F1[-(31/8), -(21/8), 6, z] == (524288 2^(1/4) (2 Sqrt[1 - z] (23363584 - 382487424 z + 3230519579 z^2 - 20429111290 z^3 + 139665523725 z^4 + 1454346898420 z^5 + 1306392507525 z^6 + 201265286310 z^7 + 676456755 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (23363584 - 382487424 z + 3230519579 z^2 - 20429111290 z^3 + 139665523725 z^4 + 1454346898420 z^5 + 1306392507525 z^6 + 201265286310 z^7 + 676456755 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[1 - z] (23363584 - 382487424 z + 3230519579 z^2 - 20429111290 z^3 + 139665523725 z^4 + 1454346898420 z^5 + 1306392507525 z^6 + 201265286310 z^7 + 676456755 z^8) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - (23363584 - 397089664 z + 3467178539 z^2 - 22410089100 z^3 + 152120164945 z^4 - 2885793211970 z^5 - 6629771502355 z^6 - 2771721948600 z^7 - 186025607625 z^8 + 1352913510 z^9) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/(1115370128213756415 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]