Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9l8)

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=-41/8, b>=a

For fixed z and a=-41/8, b=45/8

http://functions.wolfram.com/07.23.03.b9l8.01

Input Form

Hypergeometric2F1[-(41/8), 45/8, 6, z] == (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (1256161280 + 2046168960 z + 5027213565 z^2 + 17737106145 z^3 + 113663466675 z^4 - 6969968315181 z^5 + 29817230859468 z^6 - 53795782354608 z^7 + 49619485149696 z^8 - 23188285021440 z^9 + 4374090040320 z^10) EllipticE[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 10 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (78510080 + 140766120 z + 341219835 z^2 + 1173626025 z^3 - 503618740515 z^4 + 2299961130759 z^5 - 4281743074464 z^6 + 4029469316208 z^7 - 1910660209920 z^8 + 364507503360 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-157020160 - 369856080 z - 908194485 z^2 - 2908476450 z^3 - 1719525433725 z^4 + 8270334440532 z^5 - 15941219427504 z^6 + 15422272411392 z^7 - 7486423338240 z^8 + 1458030013440 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (1256161280 + 2046168960 z + 5027213565 z^2 + 17737106145 z^3 + 113663466675 z^4 - 6969968315181 z^5 + 29817230859468 z^6 - 53795782354608 z^7 + 49619485149696 z^8 - 23188285021440 z^9 + 4374090040320 z^10) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(1339391376632283975 Pi 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]