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

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=-35/8, b>=a

For fixed z and a=-35/8, b=-25/8

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

Input Form

Hypergeometric2F1[-(35/8), -(25/8), 6, z] == (1/(31940285517114975 Pi z^5)) (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (294912 - 5602176 z + 55594971 z^2 - 417977217 z^3 + 3415848975 z^4 + 178896936531 z^5 + 350015190337 z^6 + 156001866573 z^7 + 14327351829 z^8 + 4757025 z^9) EllipticE[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-36864 + 673488 z - 6462639 z^2 + 47597850 z^3 + 40221341775 z^4 + 88304227068 z^5 + 42744199519 z^6 + 4257268218 z^7 + 4757025 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 10 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-18432 + 347112 z - 3418659 z^2 + 25579575 z^3 - 12393708075 z^4 - 32314559241 z^5 - 21333510769 z^6 - 4054107267 z^7 - 156981825 z^8 + 951405 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (294912 - 5602176 z + 55594971 z^2 - 417977217 z^3 + 3415848975 z^4 + 178896936531 z^5 + 350015190337 z^6 + 156001866573 z^7 + 14327351829 z^8 + 4757025 z^9) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

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]