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

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=-29/8, b>=a

For fixed z and a=-29/8, b=-9/8

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

Input Form

Hypergeometric2F1[-(29/8), -(9/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-12353536 + 162188416 z - 1050317099 z^2 + 4762447599 z^3 - 20925554650 z^4 - 397748043490 z^5 - 209869189775 z^6 - 198843645 z^7 + 9660420 z^8) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-12353536 + 162188416 z - 1050317099 z^2 + 4762447599 z^3 - 20925554650 z^4 - 397748043490 z^5 - 209869189775 z^6 - 198843645 z^7 + 9660420 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-12353536 + 162188416 z - 1050317099 z^2 + 4762447599 z^3 - 20925554650 z^4 - 397748043490 z^5 - 209869189775 z^6 - 198843645 z^7 + 9660420 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-12353536 + 166820992 z - 1109871035 z^2 + 5140358808 z^3 - 22612168579 z^4 - 55880187160 z^5 + 58788740815 z^6 + 16293908400 z^7 - 813890385 z^8 + 38641680 z^9) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (87675989565589425 Pi (1 + Sqrt[1 - z])^(1/4) (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]