Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a90y)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-19/4, b>=a

For fixed z and a=-19/4, b=-9/2

http://functions.wolfram.com/07.23.03.a90y.01

Input Form

Hypergeometric2F1[-(19/4), -(9/2), 5, z] == (8 Sqrt[2] (2 (1 - z)^(1/4) (-233472 + 5700608 z - 81996096 z^2 + 1199936640 z^3 + 41647040800 z^4 + 119146482432 z^5 + 89395836732 z^6 + 18485559216 z^7 + 728927865 z^8) EllipticE[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + 2 (1 - z)^(3/4) (-233472 + 5700608 z - 81996096 z^2 + 1199936640 z^3 + 41647040800 z^4 + 119146482432 z^5 + 89395836732 z^6 + 18485559216 z^7 + 728927865 z^8) EllipticE[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - (1 - z)^(1/4) (-233472 + 5700608 z - 81996096 z^2 + 1199936640 z^3 + 41647040800 z^4 + 119146482432 z^5 + 89395836732 z^6 + 18485559216 z^7 + 728927865 z^8) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - Sqrt[1 - z] (-233472 + 5700608 z - 81996096 z^2 + 1199936640 z^3 + 41647040800 z^4 + 119146482432 z^5 + 89395836732 z^6 + 18485559216 z^7 + 728927865 z^8) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - (1 - z)^(3/4) (-233472 + 5700608 z - 81996096 z^2 + 1199936640 z^3 + 41647040800 z^4 + 119146482432 z^5 + 89395836732 z^6 + 18485559216 z^7 + 728927865 z^8) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (233472 - 5817344 z + 84824512 z^2 - 1240411200 z^3 - 2152680160 z^4 + 63228403328 z^5 + 133572406764 z^6 + 68104381068 z^7 + 8772975405 z^8 + 162938880 z^9) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/ (2 z)]))/(155607276825 Pi Sqrt[1 + Sqrt[1 - z]] z^4)

Standard Form

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MathML 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]