Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a86s)

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=-21/4, b>=a

For fixed z and a=-21/4, b=-9/2

http://functions.wolfram.com/07.23.03.a86s.01

Input Form

Hypergeometric2F1[-(21/4), -(9/2), 4, z] == (Sqrt[2] (2 (121856 - 3754688 z + 88364640 z^2 + 7867691680 z^3 + 36887105600 z^4 + 44982139332 z^5 + 16253884394 z^6 + 1430303875 z^7 + 9461760 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] + 2 Sqrt[1 - z] (121856 - 3754688 z + 88364640 z^2 + 7867691680 z^3 + 36887105600 z^4 + 44982139332 z^5 + 16253884394 z^6 + 1430303875 z^7 + 9461760 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - (1 - z)^(3/4) (121856 - 3693760 z + 86536800 z^2 + 3839075200 z^3 + 14264749600 z^4 + 13739687172 z^5 + 3705382010 z^6 + 206580605 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - (121856 - 3754688 z + 88364640 z^2 + 7867691680 z^3 + 36887105600 z^4 + 44982139332 z^5 + 16253884394 z^6 + 1430303875 z^7 + 9461760 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - (1 - z)^(1/4) (121856 - 3754688 z + 88364640 z^2 + 7867691680 z^3 + 36887105600 z^4 + 44982139332 z^5 + 16253884394 z^6 + 1430303875 z^7 + 9461760 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])] - Sqrt[1 - z] (121856 - 3754688 z + 88364640 z^2 + 7867691680 z^3 + 36887105600 z^4 + 44982139332 z^5 + 16253884394 z^6 + 1430303875 z^7 + 9461760 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(1 + Sqrt[1 - z])]))/ (2035658625 Pi Sqrt[1 + Sqrt[1 - z]] z^3)

Standard Form

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MathML 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]