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

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=1/4, b>=a

For fixed z and a=1/4, b=17/4

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

Input Form

Hypergeometric2F1[1/4, 17/4, -(11/2), z] == (1/(411840 Pi^(3/2))) (((1/(-1 + z)^10) (4 Sqrt[z] (102960 - 1010880 z + 4476693 z^2 - 11821251 z^3 + 20823426 z^4 - 26475150 z^5 - 48959295 z^6 + 15416049 z^7 - 3191848 z^8 + 307648 z^9) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^10) (4 Sqrt[z] (102960 - 1010880 z + 4476693 z^2 - 11821251 z^3 + 20823426 z^4 - 26475150 z^5 - 48959295 z^6 + 15416049 z^7 - 3191848 z^8 + 307648 z^9) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^10 (1 + Sqrt[z])^9)) ((411840 - 617760 Sqrt[z] - 3734640 z + 5756400 z^(3/2) + 15136680 z^2 - 24090066 z^(5/2) - 36193287 z^3 + 59835789 z^(7/2) + 57119868 z^4 - 98766720 z^(9/2) - 65157690 z^5 + 118107990 z^(11/2) + 78738660 z^6 + 19179930 z^(13/2) - 26462535 z^7 - 4369563 z^(15/2) + 5922224 z^8 + 461472 z^(17/2) - 615296 z^9) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^9 (1 + Sqrt[z])^10)) ((-411840 - 617760 Sqrt[z] + 3734640 z + 5756400 z^(3/2) - 15136680 z^2 - 24090066 z^(5/2) + 36193287 z^3 + 59835789 z^(7/2) - 57119868 z^4 - 98766720 z^(9/2) + 65157690 z^5 + 118107990 z^(11/2) - 78738660 z^6 + 19179930 z^(13/2) + 26462535 z^7 - 4369563 z^(15/2) - 5922224 z^8 + 461472 z^(17/2) + 615296 z^9) 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]