Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9ty)

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=-19/8

http://functions.wolfram.com/07.23.03.b9ty.01

Input Form

Hypergeometric2F1[-(39/8), -(19/8), 5, z] == (65536 2^(1/4) (-8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (45084416 - 835470584 z + 8728061160 z^2 - 87429777617 z^3 - 1914384430945 z^4 - 2848841706186 z^5 - 755272421106 z^6 - 2220642765 z^7 + 78375627 z^8) EllipticE[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (45084416 - 835470584 z + 8728061160 z^2 - 87429777617 z^3 - 1914384430945 z^4 - 2848841706186 z^5 - 755272421106 z^6 - 2220642765 z^7 + 78375627 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[1 - z] (45084416 - 835470584 z + 8728061160 z^2 - 87429777617 z^3 - 1914384430945 z^4 - 2848841706186 z^5 - 755272421106 z^6 - 2220642765 z^7 + 78375627 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (180337664 - 3409508960 z + 36146958861 z^2 - 362478352418 z^3 + 6779685079223 z^4 + 23527749204036 z^5 + 13755426288555 z^6 + 1102640571054 z^7 - 36131164047 z^8 + 1254010032 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (468893118936817575 Pi (1 + Sqrt[1 - z])^(1/4) 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]