Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bc2l)

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=31/8

http://functions.wolfram.com/07.23.03.bc2l.01

Input Form

Hypergeometric2F1[-(37/8), 31/8, 5, z] == (65536 2^(1/4) (4 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (7141888 - 2008656 z - 4226547 z^2 - 17994210 z^3 + 215601725 z^4 - 506881512 z^5 + 530380224 z^6 - 266652672 z^7 + 52715520 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 2 Sqrt[1 - z] (7141888 - 2008656 z - 4226547 z^2 - 17994210 z^3 + 215601725 z^4 - 506881512 z^5 + 530380224 z^6 - 266652672 z^7 + 52715520 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (7141888 - 2008656 z - 4226547 z^2 - 17994210 z^3 + 215601725 z^4 - 506881512 z^5 + 530380224 z^6 - 266652672 z^7 + 52715520 z^8) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (14283776 - 9373728 z - 8411247 z^2 - 33184671 z^3 - 701385605 z^4 + 3314771091 z^5 - 5883491808 z^6 + 5220365568 z^7 - 2335297536 z^8 + 421724160 z^9) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (37573790456475 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]