Generalized hypergeometric function 3F2: General characteristics (formula 07.27.04.0021)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃},{b₁,b₂},z]

General characteristics

Branch cuts

With respect to z

http://functions.wolfram.com/07.27.04.0021.01

Input Form

Limit[HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, x + I \[Epsilon]], \[Epsilon] -> Plus[0]] == ((Gamma[Subscript[a, 2] - Subscript[a, 1]] Gamma[Subscript[a, 3] - Subscript[a, 1]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[Subscript[b, 1] - Subscript[a, 1]] Gamma[Subscript[b, 2] - Subscript[a, 1]])) (-(1/x))^Subscript[a, 1] HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 1] - Subscript[b, 1] + 1, Subscript[a, 1] - Subscript[b, 2] + 1}, {Subscript[a, 1] - Subscript[a, 2] + 1, Subscript[a, 1] - Subscript[a, 3] + 1}, 1/x] + ((Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[Subscript[a, 3] - Subscript[a, 2]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 3]] Gamma[Subscript[b, 1] - Subscript[a, 2]] Gamma[Subscript[b, 2] - Subscript[a, 2]])) (-(1/x))^Subscript[a, 2] HypergeometricPFQ[{Subscript[a, 2], Subscript[a, 2] - Subscript[b, 1] + 1, Subscript[a, 2] - Subscript[b, 2] + 1}, {Subscript[a, 2] - Subscript[a, 1] + 1, Subscript[a, 2] - Subscript[a, 3] + 1}, 1/x] + ((Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[b, 1] - Subscript[a, 3]] Gamma[Subscript[b, 2] - Subscript[a, 3]])) (-(1/x))^Subscript[a, 3] HypergeometricPFQ[{Subscript[a, 3], Subscript[a, 3] - Subscript[b, 1] + 1, Subscript[a, 3] - Subscript[b, 2] + 1}, {Subscript[a, 3] - Subscript[a, 1] + 1, Subscript[a, 3] - Subscript[a, 2] + 1}, 1/x] /; ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= 3 && 1 <= k <= 3, !Element[Subscript[a, j] - Subscript[a, k], Integers]] && x > 1

Standard Form

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MathML Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29

HypergeometricPFQ[{},{},z]
HypergeometricPFQ[{},{b},z]
HypergeometricPFQ[{a},{},z]
HypergeometricPFQ[{a},{b},z]
HypergeometricPFQ[{a₁},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂},z]
HypergeometricPFQ[{a₁,a₂},{b₁,b₂,b₃},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]