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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Specific values
Specialized values
Cases with arbitrary {m,n,p,q}
Case {m,n,p,q}={1,p,p,q}
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http://functions.wolfram.com/07.34.03.0016.01
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MeijerG[{{0, (Subscript[a, 1] - m)/n, (Subscript[a, 1] - m + 1)/n,
\[Ellipsis], (Subscript[a, 1] - m + n - 1)/n, \[Ellipsis],
(Subscript[a, p] - m)/n, (Subscript[a, p] - m + 1)/n, \[Ellipsis],
(Subscript[a, p] - m + n - 1)/n}, {}},
{{0}, {-(m/n), (-m + 1)/n, \[Ellipsis], (-m + n - 1)/n,
(Subscript[b, 1] - m)/n, (Subscript[b, 1] - m + 1)/n, \[Ellipsis],
(Subscript[b, 1] - m + n - 1)/n, \[Ellipsis], (Subscript[b, q] - m)/n,
(Subscript[b, q] - m + 1)/n, \[Ellipsis], (Subscript[b, q] - m + n - 1)/
n}}, z] == ((2 Pi)^((q - p + 1) ((1 - n)/2))
n^((1/2) (-1 - p + q) + Sum[Subscript[a, j], {j, 1, p}] -
Sum[Subscript[b, j], {j, 1, q}]) (Product[Gamma[1 - Subscript[a, j]],
{j, 1, p}]/Product[Gamma[1 - Subscript[b, j]], {j, 1, q}])
Sum[Exp[-((2 Pi I k m)/n)] HypergeometricPFQ[{1 - Subscript[a, 1],
\[Ellipsis], 1 - Subscript[a, p]}, {1 - Subscript[b, 1], \[Ellipsis],
1 - Subscript[b, q]}, n^(q - p + 1) Exp[(2 Pi I k)/n] (-z)^(1/n)],
{k, 0, n - 1}])/(-z)^(m/n) /; Element[m, Integers] && m > 0 &&
Element[n, Integers] && n > 0
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> , </mo> <mrow> <mrow> <mi> q </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> n </mi> </mrow> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <mi> p </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> m </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> - </mo> <mi> m </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> m </mi> <mi> n </mi> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> m </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> m </mi> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mfrac> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List[RowBox[List["p", " ", "n"]], ",", RowBox[List[RowBox[List["q", " ", "n"]], "+", "n"]]]], RowBox[List["1", ",", RowBox[List["p", " ", "n"]]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "1"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["a", "p"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["m", "n"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "1"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m", "+", "1"]], "n"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List[SubscriptBox["b", "q"], "-", "m", "+", "n", "-", "1"]], "n"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> n </mi> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> p </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <msub> <mi> a </mi> <mi> j </mi> </msub> </mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> p </mi> </munderover> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - 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Date Added to functions.wolfram.com (modification date)
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| MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] | | |
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){kind=small}
