  
| 
 | 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
Hypergeometric Functions
 
MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z]
 
Specific values
 
Specialized values
 
Cases with m==2
 
Case {m,n,p,q}={2,1,2,2}
 
 | 
 
 
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   http://functions.wolfram.com/07.34.03.0756.01
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
 | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
    MeijerG[{{a}, {c}}, {{b, (a + c - 1)/2}, {}}, z] == 
  ((2 E^(((Pi I)/2) (1 - a + 2 b - c)) Gamma[1 - a + b])/Gamma[-b + c]) 
   z^((1/4) (-1 + a + 2 b + c)) (1 + z)^((1/4) (-1 + a - 2 b + c)) 
   LegendreQ[(1/2) (-1 - a + c), (1/2) (-1 + a + c) - b, 3, 1 + 2 z] /; 
  !IntervalMemberQ[Interval[{-1, 0}], z] 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "a", "}"]], ",", RowBox[List["{", "c", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b", ",", FractionBox[RowBox[List["a", "+", "c", "-", "1"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List["(", RowBox[List["1", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]], ")"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]], " "]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], "]"]]], SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "-", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]]]]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "a", "+", "c"]], ")"]]]], ",", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "c"]], ")"]]]], "-", "b"]], ",", "3", ",", RowBox[List["1", "+", RowBox[List["2", " ", "z"]]]]]], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]], "]"]]]]]] 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
 
  | 
   
   <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> 2 </mn>  <mo> , </mo>  <mn> 1 </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>  <mfrac>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  </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["2", ",", "1"]]], "\[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[FractionBox[RowBox[List["a", "+", "c", "-", "1"]], "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mfrac>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </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>  </mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <msubsup>  <semantics>  <mi> 𝔔 </mi>  <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation>  </semantics>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mi> b </mi>  </mrow>  </msubsup>  <mo> ( </mo>  <semantics>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]], HoldComplete[LegendreQ, 3]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mn> 0 </mn>  </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> 2 </mn>  <mo> , </mo>  <mn> 1 </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>  <mfrac>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  </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["2", ",", "1"]]], "\[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[FractionBox[RowBox[List["a", "+", "c", "-", "1"]], "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msup>  <mi> ⅇ </mi>  <mrow>  <mfrac>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </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>  </mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <msubsup>  <semantics>  <mi> 𝔔 </mi>  <annotation encoding='Mathematica'> TagBox["\[GothicCapitalQ]", LegendreQ] </annotation>  </semantics>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mi> a </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <mrow>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mi> b </mi>  </mrow>  </msubsup>  <mo> ( </mo>  <semantics>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List["2", " ", "z"]], "+", "1"]], HoldComplete[LegendreQ, 3]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mi> z </mi>  <mo> ∉ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </annotation-xml>  </semantics>  </math> 
   
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
 
  |   
  |  
  |  
  |   
  |  |  
  |   
  |  
  |  
  |   
  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "a_", "}"]], ",", RowBox[List["{", "c_", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a_", "+", "c_", "-", "1"]], ")"]]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", "a", "+", RowBox[List["2", " ", "b"]], "-", "c"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a", "+", "b"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "-", RowBox[List["2", " ", "b"]], "+", "c"]], ")"]]]]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "a", "+", "c"]], ")"]]]], ",", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a", "+", "c"]], ")"]]]], "-", "b"]], ",", "3", ",", RowBox[List["1", "+", RowBox[List["2", " ", "z"]]]]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "b"]], "+", "c"]], "]"]]], "/;", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "1"]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]]]]]]]]  |  
  |  
  |   
  |  
  |  
  | 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Date Added to functions.wolfram.com (modification date)
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
 | 
  | 
  
| MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] |  |   |  
  |  
  
  
  
 |  
 
 |