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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==0  Case {m,n,p,q}={0,4,4,2}   |  |  
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 | | http://functions.wolfram.com/07.34.03.0225.01 | 
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 | | MeijerG[{{a, c, (a + c - 1)/2, (a + c)/2}, {}}, {{}, {b, a - b + c - 1}}, 
  z] == (2^(1 + a - c) Sqrt[Pi] z^(-1 + c) HypergeometricU[a - b, 1 + a - c, 
    2/Sqrt[z]] HypergeometricU[1 + b - c, 1 + a - c, 2/Sqrt[z]])/E^(2/Sqrt[z]) | 
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 | | Cell[BoxData[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "c", ",", FractionBox[RowBox[List["a", "+", "c", "-", "1"]], "2"], ",", FractionBox[RowBox[List["a", "+", "c"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["b", ",", RowBox[List["a", "-", "b", "+", "c", "-", "1"]]]], "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "a", "-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["2", SqrtBox["z"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "c"]]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["a", "-", "b"]], ",", RowBox[List["1", "+", "a", "-", "c"]], ",", FractionBox["2", SqrtBox["z"]]]], "]"]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["1", "+", "b", "-", "c"]], ",", RowBox[List["1", "+", "a", "-", "c"]], ",", FractionBox["2", SqrtBox["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>  <mn> 4 </mn>  <mo> , </mo>  <mn> 2 </mn>  </mrow>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mn> 4 </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>  <mo> , </mo>  <mfrac>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> , </mo>  <mfrac>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> c </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  <mo> + </mo>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["4", ",", "2"]], RowBox[List["0", ",", "4"]]], "\[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]], ",", TagBox[FractionBox[RowBox[List["a", "+", "c", "-", "1"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "c"]], "2"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["a", "-", "b", "+", "c", "-", "1"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <msup>  <mn> 2 </mn>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <msqrt>  <mi> π </mi>  </msqrt>  <mo> ⁢ </mo>  <msup>  <mi> z </mi>  <mrow>  <mi> c </mi>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <semantics>  <mrow>  <mrow>  <mi> exp </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mo> - </mo>  <mfrac>  <mn> 2 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mi> U </mi>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List["exp", "(", RowBox[List["-", FractionBox["2", SqrtBox["z"]]]], ")"]], "U"]], HypergeometricU] </annotation>  </semantics>  <mo> ( </mo>  <mrow>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mfrac>  <mn> 2 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <semantics>  <mi> U </mi>  <annotation encoding='Mathematica'> TagBox["U", HypergeometricU] </annotation>  </semantics>  <mo> ( </mo>  <mrow>  <mrow>  <mi> b </mi>  <mo> - </mo>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> c </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> , </mo>  <mfrac>  <mn> 2 </mn>  <msqrt>  <mi> z </mi>  </msqrt>  </mfrac>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <ci> MeijerG </ci>  <list>  <list>  <ci> a </ci>  <ci> c </ci>  <apply>  <times />  <apply>  <plus />  <ci> a </ci>  <ci> c </ci>  <cn type='integer'> -1 </cn>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <apply>  <plus />  <ci> a </ci>  <ci> c </ci>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </list>  <list />  </list>  <list>  <list />  <list>  <ci> b </ci>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  <ci> c </ci>  <cn type='integer'> -1 </cn>  </apply>  </list>  </list>  <ci> z </ci>  </apply>  <apply>  <times />  <apply>  <power />  <cn type='integer'> 2 </cn>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> c </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  </apply>  <apply>  <power />  <pi />  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <ci> z </ci>  <apply>  <plus />  <ci> c </ci>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <apply>  <ci> HypergeometricU </ci>  <apply>  <times />  <apply>  <exp />  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  <ci> U </ci>  </apply>  </apply>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> c </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  <apply>  <ci> HypergeometricU </ci>  <apply>  <plus />  <ci> b </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> c </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> c </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </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[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a_", "+", "c_", "-", "1"]], ")"]]]], ",", FractionBox[RowBox[List["a_", "+", "c_"]], "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["b_", ",", RowBox[List["a_", "-", "b_", "+", "c_", "-", "1"]]]], "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "a", "-", "c"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["2", SqrtBox["z"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "c"]]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["a", "-", "b"]], ",", RowBox[List["1", "+", "a", "-", "c"]], ",", FractionBox["2", SqrtBox["z"]]]], "]"]], " ", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["1", "+", "b", "-", "c"]], ",", RowBox[List["1", "+", "a", "-", "c"]], ",", FractionBox["2", SqrtBox["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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