Generalized hypergeometric function 4F3: Integral transforms (formula 07.28.22.0001)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

HypergeometricPFQ[{a₁,a₂,a₃,a₄},{b₁,b₂,b₃},z]

Integral transforms

Laplace transforms

http://functions.wolfram.com/07.28.22.0001.01

Input Form

LaplaceTransform[HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, -t], t, z] == (Pi z^(-1 + Subscript[a, 1]) Csc[Pi Subscript[a, 1]] Gamma[-Subscript[a, 1] + Subscript[a, 2]] Gamma[-Subscript[a, 1] + Subscript[a, 3]] Gamma[-Subscript[a, 1] + Subscript[a, 4]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]] HypergeometricPFQ[{1 + Subscript[a, 1] - Subscript[b, 1], 1 + Subscript[a, 1] - Subscript[b, 2], 1 + Subscript[a, 1] - Subscript[b, 3]}, {1 + Subscript[a, 1] - Subscript[a, 2], 1 + Subscript[a, 1] - Subscript[a, 3], 1 + Subscript[a, 1] - Subscript[a, 4]}, z])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[Subscript[a, 4]] Gamma[-Subscript[a, 1] + Subscript[b, 1]] Gamma[-Subscript[a, 1] + Subscript[b, 2]] Gamma[-Subscript[a, 1] + Subscript[b, 3]]) + (Pi z^(-1 + Subscript[a, 2]) Csc[Pi Subscript[a, 2]] Gamma[Subscript[a, 1] - Subscript[a, 2]] Gamma[-Subscript[a, 2] + Subscript[a, 3]] Gamma[-Subscript[a, 2] + Subscript[a, 4]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]] HypergeometricPFQ[{1 + Subscript[a, 2] - Subscript[b, 1], 1 + Subscript[a, 2] - Subscript[b, 2], 1 + Subscript[a, 2] - Subscript[b, 3]}, {1 - Subscript[a, 1] + Subscript[a, 2], 1 + Subscript[a, 2] - Subscript[a, 3], 1 + Subscript[a, 2] - Subscript[a, 4]}, z])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[Subscript[a, 4]] Gamma[-Subscript[a, 2] + Subscript[b, 1]] Gamma[-Subscript[a, 2] + Subscript[b, 2]] Gamma[-Subscript[a, 2] + Subscript[b, 3]]) + (Pi z^(-1 + Subscript[a, 3]) Csc[Pi Subscript[a, 3]] Gamma[Subscript[a, 1] - Subscript[a, 3]] Gamma[Subscript[a, 2] - Subscript[a, 3]] Gamma[-Subscript[a, 3] + Subscript[a, 4]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]] HypergeometricPFQ[{1 + Subscript[a, 3] - Subscript[b, 1], 1 + Subscript[a, 3] - Subscript[b, 2], 1 + Subscript[a, 3] - Subscript[b, 3]}, {1 - Subscript[a, 1] + Subscript[a, 3], 1 - Subscript[a, 2] + Subscript[a, 3], 1 + Subscript[a, 3] - Subscript[a, 4]}, z])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[Subscript[a, 4]] Gamma[-Subscript[a, 3] + Subscript[b, 1]] Gamma[-Subscript[a, 3] + Subscript[b, 2]] Gamma[-Subscript[a, 3] + Subscript[b, 3]]) + (Pi z^(-1 + Subscript[a, 4]) Csc[Pi Subscript[a, 4]] Gamma[Subscript[a, 1] - Subscript[a, 4]] Gamma[Subscript[a, 2] - Subscript[a, 4]] Gamma[Subscript[a, 3] - Subscript[a, 4]] Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]] Gamma[Subscript[b, 3]] HypergeometricPFQ[{1 + Subscript[a, 4] - Subscript[b, 1], 1 + Subscript[a, 4] - Subscript[b, 2], 1 + Subscript[a, 4] - Subscript[b, 3]}, {1 - Subscript[a, 1] + Subscript[a, 4], 1 - Subscript[a, 2] + Subscript[a, 4], 1 - Subscript[a, 3] + Subscript[a, 4]}, z])/(Gamma[Subscript[a, 1]] Gamma[Subscript[a, 2]] Gamma[Subscript[a, 3]] Gamma[Subscript[a, 4]] Gamma[-Subscript[a, 4] + Subscript[b, 1]] Gamma[-Subscript[a, 4] + Subscript[b, 2]] Gamma[-Subscript[a, 4] + Subscript[b, 3]]) + (HypergeometricPFQ[{1, 2 - Subscript[b, 1], 2 - Subscript[b, 2], 2 - Subscript[b, 3]}, {2 - Subscript[a, 1], 2 - Subscript[a, 2], 2 - Subscript[a, 3], 2 - Subscript[a, 4]}, z] (-1 + Subscript[b, 1]) (-1 + Subscript[b, 2]) (-1 + Subscript[b, 3]))/ ((-1 + Subscript[a, 1]) (-1 + Subscript[a, 2]) (-1 + Subscript[a, 3]) (-1 + Subscript[a, 4])) /; Re[z] > 0

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},{b₁,b₂},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]