Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6yd)

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=-45/8, b>=a

For fixed z and a=-45/8, b=25/8

http://functions.wolfram.com/07.23.03.b6yd.01

Input Form

Hypergeometric2F1[-(45/8), 25/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[1 - z] (457080832 - 2026510720 z + 2140822775 z^2 + 1532995100 z^3 + 2221378250 z^4 - 34399202684 z^5 + 77621599335 z^6 - 85433762200 z^7 + 52405914000 z^8 - 17282126400 z^9 + 2403346176 z^10) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (457080832 - 2026510720 z + 2140822775 z^2 + 1532995100 z^3 + 2221378250 z^4 - 34399202684 z^5 + 77621599335 z^6 - 85433762200 z^7 + 52405914000 z^8 - 17282126400 z^9 + 2403346176 z^10) EllipticE[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (457080832 - 2312186240 z + 3360523335 z^2 + 380807700 z^3 + 1127776650 z^4 - 8262124444 z^5 + 16234643495 z^6 - 16492834480 z^7 + 9539185600 z^8 - 2999568000 z^9 + 400557696 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[1 - z] (457080832 - 2026510720 z + 2140822775 z^2 + 1532995100 z^3 + 2221378250 z^4 - 34399202684 z^5 + 77621599335 z^6 - 85433762200 z^7 + 52405914000 z^8 - 17282126400 z^9 + 2403346176 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (7270691817634245 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] 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]