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

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=-33/8, b>=a

For fixed z and a=-33/8, b=-5/8

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

Input Form

Hypergeometric2F1[-(33/8), -(5/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-30638080 + 390755200 z - 2432678325 z^2 + 10420420725 z^3 - 41856350250 z^4 - 658599935734 z^5 - 200355136345 z^6 + 19836663665 z^7 - 2347337720 z^8 + 156228800 z^9) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-30638080 + 402244480 z - 2576069925 z^2 + 11294276400 z^3 - 45534757125 z^4 - 76144855384 z^5 + 108079181249 z^6 + 5065718840 z^7 - 594157655 z^8 + 39057200 z^9) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-30638080 + 390755200 z - 2432678325 z^2 + 10420420725 z^3 - 41856350250 z^4 - 658599935734 z^5 - 200355136345 z^6 + 19836663665 z^7 - 2347337720 z^8 + 156228800 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-30638080 + 390755200 z - 2432678325 z^2 + 10420420725 z^3 - 41856350250 z^4 - 658599935734 z^5 - 200355136345 z^6 + 19836663665 z^7 - 2347337720 z^8 + 156228800 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (148821264070253775 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) 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]