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

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=-23/8, b>=a

For fixed z and a=-23/8, b=45/8

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

Input Form

Hypergeometric2F1[-(23/8), 45/8, 5, z] == (65536 2^(1/4) (4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-35328 - 143152 z - 557865 z^2 - 3187800 z^3 + 39437376 z^4 - 67212288 z^5 + 31629312 z^6) EllipticE[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-35328 - 143152 z - 557865 z^2 - 3187800 z^3 + 39437376 z^4 - 67212288 z^5 + 31629312 z^6) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 Sqrt[1 - z] (-35328 - 143152 z - 557865 z^2 - 3187800 z^3 + 39437376 z^4 - 67212288 z^5 + 31629312 z^6) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (-70656 - 259808 z - 1001121 z^2 - 5923995 z^3 - 132795168 z^4 + 546239232 z^5 - 658944000 z^6 + 253034496 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (7018685759085 Pi (1 + Sqrt[1 - z])^(1/4) 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]