Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ah0v)

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=-3/4, b>=a

For fixed z and a=-3/4, b=21/4

http://functions.wolfram.com/07.23.03.ah0v.01

Input Form

Hypergeometric2F1[-(3/4), 21/4, -(11/2), z] == (1/(7001280 Pi^(3/2))) (((1/(-1 + z)^10) (4 Sqrt[z] (1750320 - 15593760 z + 60324381 z^2 - 129973857 z^3 + 161151432 z^4 - 87297210 z^5 + 360183285 z^6 - 313544517 z^7 + 148023414 z^8 - 38308864 z^9 + 4280320 z^10) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^10) (4 Sqrt[z] (1750320 - 15593760 z + 60324381 z^2 - 129973857 z^3 + 161151432 z^4 - 87297210 z^5 + 360183285 z^6 - 313544517 z^7 + 148023414 z^8 - 38308864 z^9 + 4280320 z^10) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^10 (1 + Sqrt[z])^9)) ((7001280 - 10501920 Sqrt[z] - 57124080 z + 88311600 z^(3/2) + 198979560 z^2 - 319628322 z^(5/2) - 374616879 z^3 + 634564593 z^(7/2) + 376074816 z^4 - 698377680 z^(9/2) - 87297210 z^5 + 261891630 z^(11/2) - 407386980 z^6 - 312979590 z^(13/2) + 451637505 z^7 + 175451529 z^(15/2) - 244402092 z^8 - 51644736 z^(17/2) + 70197248 z^9 + 6420480 z^(19/2) - 8560640 z^10) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^9 (1 + Sqrt[z])^10)) ((-7001280 - 10501920 Sqrt[z] + 57124080 z + 88311600 z^(3/2) - 198979560 z^2 - 319628322 z^(5/2) + 374616879 z^3 + 634564593 z^(7/2) - 376074816 z^4 - 698377680 z^(9/2) + 87297210 z^5 + 261891630 z^(11/2) + 407386980 z^6 - 312979590 z^(13/2) - 451637505 z^7 + 175451529 z^(15/2) + 244402092 z^8 - 51644736 z^(17/2) - 70197248 z^9 + 6420480 z^(19/2) + 8560640 z^10) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

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]