Generalized hypergeometric function 3F2: Representations through more general functions (formula 07.27.26.0005)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Representations through more general functions

Through Meijer G

Classical cases for the direct function itself

http://functions.wolfram.com/07.27.26.0005.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3]}, {Subscript[b, 1], Subscript[b, 2]}, z] == ((Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (Pi Sin[Subscript[\[Psi], 2] Pi] Sin[Pi (Subscript[b, 1] - Subscript[b, 2])] Product[Gamma[Subscript[a, k]], {k, 1, 3}])) (Product[Sin[Pi (Subscript[b, 1] - Subscript[a, k])] MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}, {}}, {{0, 1 - Subscript[b, 1]}, {1 - Subscript[b, 2]}}, z], {k, 1, 3}] - Product[Sin[Pi (Subscript[b, 2] - Subscript[a, k])] MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}, {}}, {{0, 1 - Subscript[b, 2]}, {1 - Subscript[b, 1]}}, z], {k, 1, 3}]) - ((Pi Gamma[Subscript[b, 1]] Gamma[Subscript[b, 2]])/ (Sin[Subscript[\[Psi], 2] Pi] Product[Gamma[Subscript[a, k]], {k, 1, 3}])) (MeijerG[{{}, {1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2]}, {}}, z] + ((1 - z)^Subscript[\[Psi], 2] MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3]}, {}}, {{}, {0, 1 - Subscript[b, 1], 1 - Subscript[b, 2]}}, z])/(z - 1)^Subscript[\[Psi], 2]) /; Subscript[\[Psi], 2] == Subscript[b, 1] + Subscript[b, 2] - Subscript[a, 1] - Subscript[a, 2] - Subscript[a, 3] && !IntervalMemberQ[Interval[{-1, 0}], z] && !Element[Subscript[\[Psi], 2], Integers]

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)

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]