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

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=-13/8

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

Input Form

Hypergeometric2F1[-(31/8), -(13/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[1 - z] (-23363584 + 330649472 z - 2357374075 z^2 + 12184902625 z^3 - 65134546400 z^4 - 439711139630 z^5 - 206553569355 z^6 - 2983063395 z^7 + 110894550 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (-23363584 + 330649472 z - 2357374075 z^2 + 12184902625 z^3 - 65134546400 z^4 - 439711139630 z^5 - 206553569355 z^6 - 2983063395 z^7 + 110894550 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[1 - z] (-23363584 + 330649472 z - 2357374075 z^2 + 12184902625 z^3 - 65134546400 z^4 - 439711139630 z^5 - 206553569355 z^6 - 2983063395 z^7 + 110894550 z^8) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (-23363584 + 345251712 z - 2561634315 z^2 + 13625479910 z^3 - 72523597125 z^4 + 1064619632160 z^5 + 1543003172795 z^6 + 275927819310 z^7 - 6088110795 z^8 + 221789100 z^9) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (383979880204735815 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]