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Hypergeometric Functions
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
Representations through more general functions
Through Fox H
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http://functions.wolfram.com/07.34.26.0008.01
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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] ==
FoxH[{{{Subscript[a, 1], 1}, \[Ellipsis], {Subscript[a, n], 1}},
{{Subscript[a, n + 1], 1}, \[Ellipsis], {Subscript[a, p], 1}}},
{{{Subscript[b, 1], 1}, \[Ellipsis], {Subscript[b, m], 1}},
{{Subscript[b, m + 1], 1}, \[Ellipsis], {Subscript[b, q], 1}}}, z]
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Cell[BoxData[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["a", "p"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "m"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["b", "q"]]], "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List["FoxH", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "1"]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["a", "n"], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "1"]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["a", "p"], ",", "1"]], "}"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "1"]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "m"], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], ",", "1"]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["b", "q"], ",", "1"]], "}"]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]]
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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> <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> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", 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Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List["{", RowBox[List[SubscriptBox["a", "p"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]], 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> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["{", RowBox[List[SubscriptBox["b", "m"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["{", RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], ",", "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> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["{", RowBox[List[SubscriptBox["b", "q"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <semantics> <mrow> <msubsup> <mi> G </mi> <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> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> p </mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> q </mi> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["p", ",", "q"]], RowBox[List["m", ",", "n"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[SubscriptBox["a", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "n"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", RowBox[List["n", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["a", "p"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[SubscriptBox["b", "1"], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "m"], MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", RowBox[List["m", "+", "1"]]], MeijerG, Rule[Editable, True]], ",", TagBox["\[Ellipsis]", MeijerG, Rule[Editable, True]], ",", TagBox[SubscriptBox["b", "q"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <mstyle scriptlevel='0'> <msubsup> <mi> H </mi> <mrow> <mi> p </mi> <mo> , </mo> <mi> q </mi> </mrow> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msubsup> </mstyle> <mo> ( </mo> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", "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> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List["{", RowBox[List[SubscriptBox["a", "n"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List["{", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]], 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> <mrow> <mo> { </mo> <mrow> <msub> <mi> a </mi> <mi> p </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List["{", RowBox[List[SubscriptBox["a", "p"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List["{", RowBox[List[SubscriptBox["b", "1"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]], 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> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mi> m </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["{", RowBox[List[SubscriptBox["b", "m"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]] </annotation> </semantics> <mo> , </mo> <semantics> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["{", RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], ",", "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> <mrow> <mo> { </mo> <mrow> <msub> <mi> b </mi> <mi> q </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> } </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["{", RowBox[List[SubscriptBox["b", "q"], ",", "1"]], "}"]], MeijerG, Rule[Editable, True]] </annotation> </semantics> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> ) </mo> </mstyle> </mrow> </mrow> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "n_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["a_", RowBox[List["n_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["a_", "p_"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b_", "1"], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "m_"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["b_", RowBox[List["m_", "+", "1"]]], ",", "\[Ellipsis]_", ",", SubscriptBox["b_", "q_"]]], "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["FoxH", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["aa", "1"], ",", "1"]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["aa", "n"], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], ",", "1"]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["aa", "p"], ",", "1"]], "}"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["bb", "1"], ",", "1"]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "m"], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["b", RowBox[List["m", "+", "1"]]], ",", "1"]], "}"]], ",", "\[Ellipsis]", ",", RowBox[List["{", RowBox[List[SubscriptBox["bb", "q"], ",", "1"]], "}"]]]], "}"]]]], "}"]], ",", "z"]], "]"]]]]]] |
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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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