Meijer G-function: Series representations (formula 07.34.06.0049)

MeijerG

$Mathematica Notation$

Hypergeometric Functions

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z]

Series representations

Generalized power series

Expansions at generic point z==z₀

For the function itself

http://functions.wolfram.com/07.34.06.0049.01

Input Form

MeijerG[{{Subscript[a, 1], \[Ellipsis], Subscript[a, n]}, {Subscript[a, n + 1], \[Ellipsis], Subscript[a, p]}}, {{Subscript[b, 1], \[Ellipsis], Subscript[b, m]}, {Subscript[b, m + 1], \[Ellipsis], Subscript[b, q]}}, z] == Sum[(Product[If[j == k, 1, Gamma[Subscript[a, k] - Subscript[a, j]]] Product[Gamma[1 - Subscript[a, k] + Subscript[b, j]], {j, 1, m}], {j, 1, n}]/Product[Gamma[Subscript[a, k] - Subscript[b, j]] Product[Gamma[1 + Subscript[a, j] - Subscript[a, k]], {j, n + 1, p}], {j, m + 1, q}]) (1/Subscript[z, 0])^(Subscript[a, k] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Subscript[z, 0]^ (Subscript[a, k] (1 + Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) - 1) Sum[(Pochhammer[1 - Subscript[a, k], u]/((-Subscript[z, 0])^u u!)) HypergeometricPFQ[{1 - Subscript[a, k] + u, 1 - Subscript[a, k] + Subscript[b, 1], \[Ellipsis], 1 - Subscript[a, k] + Subscript[b, q]}, {1 - Subscript[a, k], 1 + Subscript[a, 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, k - 1] - Subscript[a, k], 1 + Subscript[a, k + 1] - Subscript[a, k], \[Ellipsis], 1 + Subscript[a, p] - Subscript[a, k]}, (-1)^(q - m - n)/Subscript[z, 0]] (z - Subscript[z, 0])^u, {u, 0, Infinity}], {k, 1, n}] /; (p > q || (p == q && m + n == p + 1 && !IntervalMemberQ[Interval[{-1, 0}], Subscript[z, 0]]) || (p == q && m + n > p + 1) || (p == q && m + n == p && Abs[Subscript[z, 0]] > 1)) && ForAll[{j, k}, Element[{j, k}, Integers] && j != k && 1 <= j <= n && 1 <= k <= n, !Element[Subscript[a, j] - Subscript[a, k], Integers]]

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <semantics> <mi> G </mi> <annotation encoding='Mathematica'> TagBox["G", MeijerG] </annotation> </semantics> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <semantics> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> a </mi> <mi> p </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <semantics> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mi> m </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mo> … </mo> <annotation encoding='Mathematica'> TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <msub> <mi> b </mi> <mi> q </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mrow> <mtext> </mtext> <mo> ∏ </mo> </mrow> <munder> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> </munder> <mi> n </mi> </munderover> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> q </mi> </munderover> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> p </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mrow> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mtext> </mtext> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> u </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> u </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mi> u </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["a", "k"]]], ")"]], "u"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> u </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mrow> <msub> <mo>   </mo> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> F </mi> <mi> p </mi> </msub> </mrow> <mo> ( </mo> <mrow> <semantics> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <mi> u </mi> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", "u"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["b", "q"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQRegularized, Rule[Editable, True]] </annotation> </semantics> <mo> ; </mo> <semantics> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> + </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", "1"]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", RowBox[List["k", "-", "1"]]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", RowBox[List["k", "+", "1"]]]]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", SubscriptBox["a", "k"], "+", SubscriptBox["a", "p"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]] </annotation> </semantics> <mo> ; </mo> <semantics> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["q", "-", "m", "-", "n"]]], SubscriptBox["z", "0"]], HypergeometricPFQRegularized, Rule[Editable, True]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> u </mi> </msup> </mrow> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> p </mi> <mo> > </mo> <mi> q </mi> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> p </mi> <mo> ⩵ </mo> <mi> q </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ⩵ </mo> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ∉ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> p </mi> <mo> ⩵ </mo> <mi> q </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> > </mo> <mrow> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ∨ </mo> <mrow> <mrow> <mi> p </mi> <mo> ⩵ </mo> <mi> q </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ⩵ </mo> <mi> p </mi> </mrow> <mo> ∧ </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <msub> <mo> ∀ </mo> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> j </mi> <mo> , </mo> <mi> k </mi> </mrow> <mo> } </mo> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> j </mi> <mo> ≠ </mo> <mi> k </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> j </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> </mrow> </mrow> </msub> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <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> p </ms> <ms> , </ms> <ms> q </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> , </ms> <ms> n </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> SubscriptBox </ci> <ms> a </ms> <ms> 1 </ms> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> … </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> n </ms> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> … </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> p </ms> </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> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> … </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> m </ms> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> … </ms> <ci> MeijerG </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </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> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∏ </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> ≠ </ms> <ms> k </ms> </list> </apply> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <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> m </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∏ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> q </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> j </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∏ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> p </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> Γ </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 0 </ms> </apply> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ⌊ </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> arg </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 0 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> π </ms> </list> </apply> </apply> <ms> ⌋ </ms> </list> </apply> </list> </apply> </apply> <apply> <ci> SubsuperscriptBox </ci> <ms> z </ms> <ms> 0 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <ms> ⌊ </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> arg </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 0 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> π </ms> </list> </apply> </apply> <ms> ⌋ </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> u </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> ∞ </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 0 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> u </ms> </list> </apply> </apply> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> u </ms> </apply> <ci> Pochhammer </ci> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> u </ms> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms>  </ms> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> TraditionalForm </ci> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> F </ms> <apply> <ci> FormBox </ci> <ms> p </ms> <ci> TraditionalForm </ci> </apply> </apply> </list> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <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> k </ms> </apply> <ms> + </ms> <ms> u </ms> </list> </apply> <ci> HypergeometricPFQ </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> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> 1 </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> … </ms> <ci> HypergeometricPFQ </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> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> b </ms> <ms> q </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <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> k </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </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> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 1 </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> … </ms> <ci> HypergeometricPFQ </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> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> - </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ci> HypergeometricPFQ </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> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> … </ms> <ci> HypergeometricPFQ </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> k </ms> </apply> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> p </ms> </apply> </list> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQ </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> q </ms> <ms> - </ms> <ms> m </ms> <ms> - </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 0 </ms> </apply> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 0 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> u </ms> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> > </ms> <ms> q </ms> </list> </apply> <ms> ∨ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> ⩵ </ms> <ms> q </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> n </ms> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 0 </ms> </apply> <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> </list> </apply> <ms> ∨ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> ⩵ </ms> <ms> q </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> n </ms> </list> </apply> <ms> > </ms> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ∨ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> p </ms> <ms> ⩵ </ms> <ms> q </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> n </ms> </list> </apply> <ms> ⩵ </ms> <ms> p </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms>  </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> 0 </ms> </apply> <ms>  </ms> </list> </apply> <ms> > </ms> <ms> 1 </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> ∀ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> , </ms> <ms> k </ms> </list> </apply> <ms> } </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> { </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> , </ms> <ms> k </ms> </list> </apply> <ms> } </ms> </list> </apply> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> ≠ </ms> <ms> k </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> ≤ </ms> <ms> j </ms> <ms> ≤ </ms> <ms> n </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> ≤ </ms> <ms> k </ms> <ms> ≤ </ms> <ms> n </ms> </list> </apply> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> j </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> k </ms> </apply> </list> </apply> <ms> ∉ </ms> <apply> <ci> TagBox </ci> <ms> ℤ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

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

2007-05-02

MeijerG[{{a₁,...,a_n},{a_n+1,...,a_p}},{{b₁,...,b_m},{b_m+1,...,b_q}},z,r]