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

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=-1/8

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

Input Form

Hypergeometric2F1[-(37/8), -(1/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-457080832 + 5601025664 z - 33000028087 z^2 + 130318741735 z^3 - 457884154910 z^4 - 5238621340370 z^5 - 36952371555 z^6 + 8430211515 z^7 - 1375459800 z^8 + 110404800 z^9) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-457080832 + 5601025664 z - 33000028087 z^2 + 130318741735 z^3 - 457884154910 z^4 - 5238621340370 z^5 - 36952371555 z^6 + 8430211515 z^7 - 1375459800 z^8 + 110404800 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-457080832 + 5601025664 z - 33000028087 z^2 + 130318741735 z^3 - 457884154910 z^4 - 5238621340370 z^5 - 36952371555 z^6 + 8430211515 z^7 - 1375459800 z^8 + 110404800 z^9) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-457080832 + 5772430976 z - 35053544071 z^2 + 142144327312 z^3 - 503658705095 z^4 - 348485119400 z^5 + 872104880955 z^6 - 163405544280 z^7 + 36310183635 z^8 - 5713448400 z^9 + 441619200 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (1239988995284764725 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]