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| Hypergeometric Functions  MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]  Specific values  Specialized values  Cases with m==1  Case {m,n,p,q}={1,2,2,2}   |  |  
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 | | http://functions.wolfram.com/07.34.03.0410.01 | 
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 | | MeijerG[{{a, c}, {}}, {{b}, {b + c - a}}, z] == 
  z^(b + (1/2) (-a + c)) (1 + z)^(-1 + a - b) Gamma[1 - a + b] 
   Gamma[1 + b - c] LegendreP[-1 + a - b, -a + c, 2, (1 - z)/(1 + z)] /; 
  !IntervalMemberQ[Interval[{-Infinity, -1}], z] | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "c"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "b", "}"]], ",", RowBox[List["{", RowBox[List["b", "+", "c", "-", "a"]], "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["z", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ")"]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "a", "-", "b"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "c"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "a", "-", "b"]], ",", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ",", "2", ",", FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "+", "z"]]]]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "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>  <mrow>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 1 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> c </mi>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> a </mi>  <mo> + </mo>  <mi> c </mi>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["1", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox["c", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "a", "+", "c"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <msup>  <mi> z </mi>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mfrac>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msubsup>  <semantics>  <mi> P </mi>  <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation>  </semantics>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  </msubsup>  <mo> ( </mo>  <semantics>  <mfrac>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> z </mi>  </mrow>  <mrow>  <mn> 1 </mn>  <mo> + </mo>  <mi> z </mi>  </mrow>  </mfrac>  <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "+", "z"]]], HoldComplete[LegendreP, 2]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> ∞ </mi>  </mrow>  <mo> , </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <mrow>  <mrow>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 1 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> c </mi>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> a </mi>  <mo> + </mo>  <mi> c </mi>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "2"]], RowBox[List["1", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox["a", MeijerG, Rule[Editable, True]], ",", TagBox["c", MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "-", "a", "+", "c"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <msup>  <mi> z </mi>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mfrac>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msubsup>  <semantics>  <mi> P </mi>  <annotation encoding='Mathematica'> TagBox["P", LegendreP] </annotation>  </semantics>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  </mrow>  </msubsup>  <mo> ( </mo>  <semantics>  <mfrac>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> z </mi>  </mrow>  <mrow>  <mn> 1 </mn>  <mo> + </mo>  <mi> z </mi>  </mrow>  </mfrac>  <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "+", "z"]]], HoldComplete[LegendreP, 2]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mi> ∞ </mi>  </mrow>  <mo> , </mo>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </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["a_", ",", "c_"]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "b_", "}"]], ",", RowBox[List["{", RowBox[List["b_", "+", "c_", "-", "a_"]], "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ")"]]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "a", "-", "b"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "b", "-", "c"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "a", "-", "b"]], ",", RowBox[List[RowBox[List["-", "a"]], "+", "c"]], ",", "2", ",", FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "+", "z"]]]]], "]"]]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "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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