Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9sp)

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=-39/8, b>=a

For fixed z and a=-39/8, b=-21/8

http://functions.wolfram.com/07.23.03.b9sp.01

Input Form

Hypergeometric2F1[-(39/8), -(21/8), 5, z] == (65536 2^(1/4) (-4 Sqrt[1 - z] (14237184 - 271248016 z + 2943417295 z^2 - 31173037350 z^3 - 539885152595 z^4 - 769775038900 z^5 - 202417896135 z^6 - 1885207350 z^7 + 55447275 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - 2 Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (14237184 - 271248016 z + 2943417295 z^2 - 31173037350 z^3 - 539885152595 z^4 - 769775038900 z^5 - 202417896135 z^6 - 1885207350 z^7 + 55447275 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 2 Sqrt[1 - z] (14237184 - 271248016 z + 2943417295 z^2 - 31173037350 z^3 - 539885152595 z^4 - 769775038900 z^5 - 202417896135 z^6 - 1885207350 z^7 + 55447275 z^8) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (28474368 - 560292512 z + 6222974875 z^2 - 65971087910 z^3 + 1748649982685 z^4 + 6151387961720 z^5 + 4038797302085 z^6 + 468041537730 z^7 - 7662813405 z^8 + 221789100 z^9) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (91423781001127575 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] z^4)

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]