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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==2  Case {m,n,p,q}={2,0,2,2}   |  |  
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 | | http://functions.wolfram.com/07.34.03.0669.01 | 
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 | | MeijerG[{{}, {a, 1/2 + a}}, {{b, 2 a - b}, {}}, z] == 
  ((z^a UnitStep[1 - Abs[z]])/(Sqrt[Pi] Sqrt[1 - z])) 
   ChebyshevT[(a - b)/2, (8 - 8 z + z^2)/z^2] /; Re[z] > 0 | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["a", ",", RowBox[List[FractionBox["1", "2"], "+", "a"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b", ",", RowBox[List[RowBox[List["2", " ", "a"]], "-", "b"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "a"], RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z", "]"]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], SqrtBox[RowBox[List["1", "-", "z"]]]]]], RowBox[List["ChebyshevT", "[", RowBox[List[FractionBox[RowBox[List["a", "-", "b"]], "2"], ",", FractionBox[RowBox[List["8", "-", RowBox[List["8", " ", "z"]], "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]] | 
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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> 2 </mn>  <mo> , </mo>  <mn> 0 </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> 2 </mn>  </mfrac>  </mrow>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mi> b </mi>  <mo> , </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mo> - </mo>  <mi> b </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["2", ",", "0"]]], "\[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", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["b", MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "a"]], "-", "b"]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mrow>  <msup>  <mi> z </mi>  <mi> a </mi>  </msup>  <mo> ⁢ </mo>  <mrow>  <semantics>  <mi> θ </mi>  <annotation-xml encoding='MathML-Content'>  <ci> UnitStep </ci>  </annotation-xml>  </semantics>  <mo> ( </mo>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mrow>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation>  </semantics>  <mi> z </mi>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation>  </semantics>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mrow>  <msqrt>  <mi> π </mi>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> z </mi>  </mrow>  </msqrt>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> T </mi>  <mfrac>  <mrow>  <mi> a </mi>  <mo> - </mo>  <mi> b </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </msub>  <mo> ( </mo>  <mfrac>  <mrow>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  <mo> - </mo>  <mrow>  <mn> 8 </mn>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> + </mo>  <mn> 8 </mn>  </mrow>  <msup>  <mi> z </mi>  <mn> 2 </mn>  </msup>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <mi> Re </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> > </mo>  <mn> 0 </mn>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <eq />  <apply>  <ci> MeijerG </ci>  <list>  <list />  <list>  <ci> a </ci>  <apply>  <plus />  <ci> a </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </list>  </list>  <list>  <list>  <ci> b </ci>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> a </ci>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  </list>  <list />  </list>  <ci> z </ci>  </apply>  <apply>  <times />  <apply>  <times />  <apply>  <power />  <ci> z </ci>  <ci> a </ci>  </apply>  <apply>  <ci> UnitStep </ci>  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <abs />  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  <apply>  <power />  <apply>  <times />  <apply>  <power />  <pi />  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <cn type='integer'> 1 </cn>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> z </ci>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <ci> ChebyshevT </ci>  <apply>  <times />  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> -1 </cn>  <ci> b </ci>  </apply>  </apply>  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <times />  <apply>  <plus />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 8 </cn>  <ci> z </ci>  </apply>  </apply>  <cn type='integer'> 8 </cn>  </apply>  <apply>  <power />  <apply>  <power />  <ci> z </ci>  <cn type='integer'> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <gt />  <apply>  <real />  <ci> z </ci>  </apply>  <cn type='integer'> 0 </cn>  </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["{", RowBox[List["a_", ",", RowBox[List[FractionBox["1", "2"], "+", "a_"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["b_", ",", RowBox[List[RowBox[List["2", " ", "a_"]], "-", "b_"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "a"], " ", RowBox[List["UnitStep", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "z", "]"]]]], "]"]]]], ")"]], " ", RowBox[List["ChebyshevT", "[", RowBox[List[FractionBox[RowBox[List["a", "-", "b"]], "2"], ",", FractionBox[RowBox[List["8", "-", RowBox[List["8", " ", "z"]], "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]] | 
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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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