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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}={1,0,0,q}
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http://functions.wolfram.com/07.34.03.0004.01
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MeijerG[{{}, {}}, {{b}, {b + 1/q, b + 2/q, \[Ellipsis], b + (q - 1)/q}},
z] == (z^b/((2 Pi)^((q - 1)/2) Sqrt[q]))
Sum[Exp[q Exp[(2 Pi I k)/q] (-z)^(1/q)], {k, 0, q - 1}]
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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> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mi> b </mi> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> q </mi> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mfrac> <mn> 2 </mn> <mi> q </mi> </mfrac> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mfrac> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["0", ",", "q"]], RowBox[List["1", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox["1", "q"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox["2", "q"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox[RowBox[List["q", "-", "1"]], "q"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mi> b </mi> </msup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> q </mi> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mi> q </mi> </mfrac> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mfrac> <mn> 1 </mn> <mi> q </mi> </mfrac> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 0 </mn> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mi> b </mi> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> q </mi> </mfrac> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mfrac> <mn> 2 </mn> <mi> q </mi> </mfrac> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <mi> b </mi> <mo> + </mo> <mfrac> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mi> q </mi> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["0", ",", "q"]], RowBox[List["1", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox["1", "q"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox["2", "q"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox[RowBox[List["q", "-", "1"]], "q"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mi> b </mi> </msup> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <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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