Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9gl)

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=37/8

http://functions.wolfram.com/07.23.03.b9gl.01

Input Form

Hypergeometric2F1[-(41/8), 37/8, 6, z] == (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (-1256161280 + 1133489280 z + 1166495715 z^2 + 2555986125 z^3 + 11802618135 z^4 - 553790431089 z^5 + 1951092382554 z^6 - 2994213729168 z^7 + 2403297721440 z^8 - 994184201472 z^9 + 168234232320 z^10) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-157020160 + 115925040 z + 156981825 z^2 + 346970085 z^3 - 81314170245 z^4 + 303680785527 z^5 - 479092479768 z^6 + 391420026000 z^7 - 164028376512 z^8 + 28039038720 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 6 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (78510080 - 13800600 z - 77398365 z^2 - 214254315 z^3 - 69943165875 z^4 + 275422560579 z^5 - 449003236320 z^6 + 376215740208 z^7 - 161024193792 z^8 + 28039038720 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (-1256161280 + 1133489280 z + 1166495715 z^2 + 2555986125 z^3 + 11802618135 z^4 - 553790431089 z^5 + 1951092382554 z^6 - 2994213729168 z^7 + 2403297721440 z^8 - 994184201472 z^9 + 168234232320 z^10) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (108599300808023025 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]