  
| 
 | 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
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}={2,2,2,2}
 
 | 
 
 
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   http://functions.wolfram.com/07.34.03.0881.01
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
 | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
    MeijerG[{{a, 1/4 + a}, {}}, {{b, 1/4 + b}, {}}, z] == 
 ((2^(1/2 + a - b) Pi^2 Csc[(a - b) Pi] Gamma[3/2 - 2 a + 2 b])/
   (Gamma[a - b] Sqrt[1 + Sqrt[z]])) z^b ((-1 + z) Sign[-1 + Abs[z]])^
   (-(1/2) + a - b) LegendreP[-1 + a - b, -(1/2) + a - b, 2, 
   (2 z^(1/4))/(1 + Sqrt[z])] 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Cell[BoxData[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", ",", RowBox[List[FractionBox["1", "4"], "+", "a"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b", ",", RowBox[List[FractionBox["1", "4"], "+", "b"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "a", "-", "b"]]], " ", SuperscriptBox["\[Pi]", "2"], RowBox[List["Csc", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", "b"]]]], "]"]]]], RowBox[List[" ", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", "b"]], "]"]], SqrtBox[RowBox[List["1", "+", SqrtBox["z"]]]]]]]]], SuperscriptBox["z", "b"], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "z", "]"]]]], "]"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "a", "-", "b"]]], RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "a", "-", "b"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "a", "-", "b"]], ",", "2", ",", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], RowBox[List["1", "+", SqrtBox["z"]]]]]], "]"]], " "]]]]]] 
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
    
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
 
  | 
   
   <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> 2 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 2 </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>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  </mrow>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mi> b </mi>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 4 </mn>  </mfrac>  </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["2", ",", "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[RowBox[List["a", "+", FractionBox["1", "4"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["b", "+", FractionBox["1", "4"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <msup>  <mn> 2 </mn>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> + </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </msup>  <mo> ⁢ </mo>  <msup>  <mi> π </mi>  <mn> 2 </mn>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> csc </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  <mo> - </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> + </mo>  <mfrac>  <mn> 3 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <mtext>   </mtext>  <mrow>  <mrow>  <mi> Γ </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msqrt>  </mrow>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mi> b </mi>  </msup>  <mo> ⁢ </mo>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sgn </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation>  </semantics>  <mi> z </mi>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation>  </semantics>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </msup>  <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> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> - </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  </msubsup>  <mo> ( </mo>  <semantics>  <mfrac>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mroot>  <mi> z </mi>  <mn> 4 </mn>  </mroot>  </mrow>  <mrow>  <msqrt>  <mi> z </mi>  </msqrt>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mfrac>  <annotation encoding='Mathematica'> TagBox[FractionBox[RowBox[List["2", " ", RadicalBox["z", "4"]]], RowBox[List[SqrtBox["z"], "+", "1"]]], HoldComplete[LegendreP, 2]] </annotation>  </semantics>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <ci> MeijerG </ci>  <list>  <list>  <ci> a </ci>  <apply>  <plus />  <ci> a </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  </list>  <list />  </list>  <list>  <list>  <ci> b </ci>  <apply>  <plus />  <ci> b </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  </list>  <list />  </list>  <ci> z </ci>  </apply>  <apply>  <times />  <apply>  <times />  <apply>  <power />  <cn type='integer'> 2 </cn>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <power />  <pi />  <cn type='integer'> 2 </cn>  </apply>  <apply>  <csc />  <apply>  <times />  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  <pi />  </apply>  </apply>  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> b </ci>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  </apply>  </apply>  <cn type='rational'> 3 <sep /> 2 </cn>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <ci> Gamma </ci>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <power />  <ci> z </ci>  <ci> b </ci>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <plus />  <ci> z </ci>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <ci> Sign </ci>  <apply>  <plus />  <apply>  <abs />  <ci> z </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  <apply>  <ci> LegendreP </ci>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <cn type='integer'> 2 </cn>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 4 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
   
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
 
  |   
  |  
  |  
  |   
  |  |  
  |   
  |  
  |  
  |   
  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a_", ",", RowBox[List[FractionBox["1", "4"], "+", "a_"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", RowBox[List[FractionBox["1", "4"], "+", "b_"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "+", "a", "-", "b"]]], " ", SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["Csc", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "b"]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "-", RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", "b"]]]], "]"]]]], ")"]], " ", SuperscriptBox["z", "b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], " ", RowBox[List["Sign", "[", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Abs", "[", "z", "]"]]]], "]"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "a", "-", "b"]]], " ", RowBox[List["LegendreP", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "a", "-", "b"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "a", "-", "b"]], ",", "2", ",", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], RowBox[List["1", "+", SqrtBox["z"]]]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", "-", "b"]], "]"]], " ", SqrtBox[RowBox[List["1", "+", SqrtBox["z"]]]]]]]]]]]  |  
  |  
  |   
  |  
  |  
  | 
  
  
 |  
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  | 
   Date Added to functions.wolfram.com (modification date)
   |   
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
  
  | 
 
  
  | 
  
  | 
 
  
  | 
  
  | 
  
  | 
 
 | 
  | 
  
| MeijerG[{{a1,...,an},{an+1,...,ap}},{{b1,...,bm},{bm+1,...,bq}},z,r] |  |   |  
  |  
  
  
  
 |  
 
 |