Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9r6)

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=-27/8

http://functions.wolfram.com/07.23.03.b9r6.01

Input Form

Hypergeometric2F1[-(39/8), -(27/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (51937247232 - 1056801253248 z + 11373027712917 z^2 - 94391551641933 z^3 + 875654892290997 z^4 + 20364618396126683 z^5 + 37792811134113063 z^6 + 16970534628723873 z^7 + 1645049354871823 z^8 + 929456560593 z^9) EllipticE[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (51937247232 - 1056801253248 z + 11373027712917 z^2 - 94391551641933 z^3 + 875654892290997 z^4 + 20364618396126683 z^5 + 37792811134113063 z^6 + 16970534628723873 z^7 + 1645049354871823 z^8 + 929456560593 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (51937247232 - 1056801253248 z + 11373027712917 z^2 - 94391551641933 z^3 + 875654892290997 z^4 + 20364618396126683 z^5 + 37792811134113063 z^6 + 16970534628723873 z^7 + 1645049354871823 z^8 + 929456560593 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (51937247232 - 1076277720960 z + 11764002586245 z^2 - 98550902847735 z^3 + 909941260979205 z^4 - 17193534293592979 z^5 - 69135644541727365 z^6 - 57381973221372645 z^7 - 11888003963469065 z^8 - 357840775828305 z^9 + 3717826242372 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (9761948362228130002935 Pi (1 + Sqrt[1 - z])^(1/4) 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]