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

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=-37/8, b>=a

For fixed z and a=-37/8, b=-17/8

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

Input Form

Hypergeometric2F1[-(37/8), -(17/8), 5, z] == (65536 2^(1/4) (-8 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (60706048 - 1064252904 z + 10362237834 z^2 - 94355457833 z^3 - 4340210998095 z^4 - 7060260249210 z^5 - 1948197004460 z^6 - 1081162005 z^7 + 41056785 z^8) EllipticE[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 4 Sqrt[1 - z] (60706048 - 1064252904 z + 10362237834 z^2 - 94355457833 z^3 - 4340210998095 z^4 - 7060260249210 z^5 - 1948197004460 z^6 - 1081162005 z^7 + 41056785 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (60706048 - 1064252904 z + 10362237834 z^2 - 94355457833 z^3 - 4340210998095 z^4 - 7060260249210 z^5 - 1948197004460 z^6 - 1081162005 z^7 + 41056785 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (242824192 - 4348070688 z + 43020431727 z^2 - 392541905678 z^3 - 3845043397083 z^4 + 212820501180 z^5 + 3516173681545 z^6 + 486632387010 z^7 - 17613360765 z^8 + 656908560 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (438379947827947125 Pi (1 + Sqrt[1 - z])^(1/4) (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]