Generalized hypergeometric function 4F3: Representations through more general functions (formula 07.28.26.0005)

HypergeometricPFQ

$Mathematica Notation$

Hypergeometric Functions

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

Representations through more general functions

Through Meijer G

Classical cases for the direct function itself

http://functions.wolfram.com/07.28.26.0005.01

Input Form

HypergeometricPFQ[{Subscript[a, 1], Subscript[a, 2], Subscript[a, 3], Subscript[a, 4]}, {Subscript[b, 1], Subscript[b, 2], Subscript[b, 3]}, z] == (Product[Gamma[Subscript[b, k]], {k, 1, 3}]/ (Pi Sin[Subscript[\[Psi], 3] Pi] Product[Gamma[Subscript[a, k]], {k, 1, 4}])) Sum[(Product[Sin[Pi (Subscript[b, j] - Subscript[a, k])], {k, 1, 4}]/Product[If[k != j, Sin[Pi (Subscript[b, j] - Subscript[b, k])], 1], {k, 1, 3}]) MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{0, 1 - Subscript[b, j]}, {1 - Subscript[b, 1], \[Ellipsis], 1 - Subscript[b, j - 1], 1 - Subscript[b, j + 1], \[Ellipsis], 1 - Subscript[b, 3]}}, z], {j, 1, 3}] - ((Pi Product[Gamma[Subscript[b, k]], {k, 1, 3}])/ (Sin[Subscript[\[Psi], 3] Pi] Product[Gamma[Subscript[a, k]], {k, 1, 4}])) (MeijerG[{{}, {1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}}, {{0, 1 - Subscript[b, 1], 1 - Subscript[b, 2], 1 - Subscript[b, 3]}, {}}, z] + ((1 - z)^Subscript[\[Psi], 3] MeijerG[{{1 - Subscript[a, 1], 1 - Subscript[a, 2], 1 - Subscript[a, 3], 1 - Subscript[a, 4]}, {}}, {{}, {0, 1 - Subscript[b, 1], 1 - Subscript[b, 2], 1 - Subscript[b, 3]}}, z])/(z - 1)^Subscript[\[Psi], 3]) /; Subscript[\[Psi], 3] == Sum[Subscript[b, j], {j, 1, 3}] - Sum[Subscript[a, j], {j, 1, 4}] && !IntervalMemberQ[Interval[{-1, 0}], z] && !Element[Subscript[\[Psi], 3], Integers]

Standard Form

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MathML Form

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<apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 2 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 3 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> + </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <apply> <ci> TagBox </ci> <ms> G </ms> <ci> MeijerG </ci> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> , </ms> <ms> 4 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <ms> , </ms> <ms> 0 </ms> </list> </apply> </apply> <ms> ⁡ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> z </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ❘ </ms> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 1 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 2 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 3 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 4 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> 0 </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox 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<ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> 3 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> 4 </ms> </apply> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> ∉ </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <ms> 0 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> ψ </ms> <ms> 3 </ms> </apply> <ms> ∉ </ms> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

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},{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]