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

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=-33/8, b>=a

For fixed z and a=-33/8, b=-3/8

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

Input Form

Hypergeometric2F1[-(33/8), -(3/8), 6, z] == (1/(26149610598336225 Pi z^5)) (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (-8355840 + 102065280 z - 602352925 z^2 + 2406077745 z^3 - 8743241610 z^4 - 128873203686 z^5 - 21122823393 z^6 + 3139473645 z^7 - 433317192 z^8 + 31706136 z^9) EllipticE[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 12 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (261120 - 2999820 z + 16662380 z^2 - 63286155 z^3 - 8084415450 z^4 - 1720606902 z^5 + 255913812 z^6 - 35669403 z^7 + 2642178 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-1044480 + 12586800 z - 73281305 z^2 + 289339065 z^3 - 20999358630 z^4 - 10501500222 z^5 + 519085035 z^6 - 71904987 z^7 + 5284356 z^8) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (-8355840 + 102065280 z - 602352925 z^2 + 2406077745 z^3 - 8743241610 z^4 - 128873203686 z^5 - 21122823393 z^6 + 3139473645 z^7 - 433317192 z^8 + 31706136 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]