| |  
 |  | 
 |  
 |  
 |  
 |  
 |  
 |  |  
 |  
 
 |  
 |  
 |  
 |  
 |  
 |  |  
 |  
 
 |  
 |  
 |  
 |  
 |  
 |  |  
 |  
 
 |  
 |  
 |  
 |  |   
 |  
 |  
 |  
 |  
 |  
 |  
 |  |  
 |  
 |  
 |  
 |  
 |  
 |  
 | | http://functions.wolfram.com/07.05.26.0011.01 | 
 |  
 |  
 |  
 |  
 |  
 |  |   
 |  
 |  |  |   
 |  
 |  
 |  
 |  
 |  
 |  
 |  |  
 |  
 |  
 |  
 |  
 |  
 |  
 | | (1 + z)^(-1 - \[Nu]/2) ChebyshevU[\[Nu], Sqrt[z/(1 + z)]] == 
  (2^\[Nu]/Gamma[1 + \[Nu]]) MeijerG[{{-(\[Nu]/2)}, {(1 - \[Nu])/2}}, 
    {{0, 1/2}, {}}, z] /;  !IntervalMemberQ[Interval[{-1, 0}], z] | 
 |  
 |  
 |  
 |  
 |  
 |  |   
 |  
 |  
 |  
 |  
 |  
 |  
 |  |  
 |  
 |  
 |  
 |  
 |  
 |  
 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]", ",", SqrtBox[FractionBox["z", RowBox[List["1", "+", "z"]]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["2", "\[Nu]"], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "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>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mi> ν </mi>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> U </mi>  <mi> ν </mi>  </msub>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <msqrt>  <mfrac>  <mi> z </mi>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mfrac>  </msqrt>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <msup>  <mn> 2 </mn>  <mi> ν </mi>  </msup>  <mrow>  <mi> Γ </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> ν </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <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>  <mrow>  <mo> - </mo>  <mfrac>  <mi> ν </mi>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mfrac>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> ν </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mfrac>  <mn> 1 </mn>  <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[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  </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>  <mrow>  <msup>  <mrow>  <mo> ( </mo>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mi> ν </mi>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <msub>  <mi> U </mi>  <mi> ν </mi>  </msub>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <msqrt>  <mfrac>  <mi> z </mi>  <mrow>  <mi> z </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  </mfrac>  </msqrt>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <msup>  <mn> 2 </mn>  <mi> ν </mi>  </msup>  <mrow>  <mi> Γ </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> ν </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <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>  <mrow>  <mo> - </mo>  <mfrac>  <mi> ν </mi>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mfrac>  <mrow>  <mn> 1 </mn>  <mo> - </mo>  <mi> ν </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mfrac>  <mn> 1 </mn>  <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[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  </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[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z_"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["\[Nu]_", "2"]]]], " ", RowBox[List["ChebyshevU", "[", RowBox[List["\[Nu]_", ",", SqrtBox[FractionBox["z_", RowBox[List["1", "+", "z_"]]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "z"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], "/;", 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) | 
 |  
 |  
 |  
 |  
 |  
 |  
 |  
 |  |  
 |   
 |  
 |  
 |  |  | 
 
 
 | 
 |