Gegenbauer function: Representations through more general functions (formula 07.14.26.0040)

GegenbauerC

$Mathematica Notation$

Hypergeometric Functions

GegenbauerC[nu,lambda,z]

Representations through more general functions

Through Meijer G

Generalized cases involving unit step theta

http://functions.wolfram.com/07.14.26.0040.01

Input Form

UnitStep[Abs[z] - 1] (-1 + z^2)^(-1 + 2 \[Lambda]) GegenbauerC[\[Nu], \[Lambda], (1 + z^2)/(2 z)] == (Gamma[2 \[Lambda] + \[Nu]]/Gamma[1 + \[Nu]]) MeijerG[{{\[Lambda] - \[Nu]/2, 2 \[Lambda] + \[Nu]/2}, {}}, {{}, {\[Lambda] + \[Nu]/2, -(\[Nu]/2)}}, z, 1/2] /; Re[z] > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "-", "1"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "\[Lambda]"]]]]], RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", FractionBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]], RowBox[List["2", " ", "z"]]]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <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> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <mi> C </mi> <mi> ν </mi> <mi> λ </mi> </msubsup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> Γ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <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> 0 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mi> λ </mi> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mi> λ </mi> <mo> + </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </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["0", ",", "2"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox["z", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List["\[Lambda]", "-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[RowBox[List["\[Lambda]", "+", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> UnitStep </ci> <apply> <plus /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> ν </ci> </apply> <ci> λ </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> Γ </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <times /> <ci> Γ </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <plus /> <ci> λ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> <list /> </list> <list> <list /> <list> <apply> <plus /> <ci> λ </ci> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> </list> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <times /> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List[RowBox[List["Abs", "[", "z_", "]"]], "-", "1"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z_", "2"]]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "\[Lambda]_"]]]]], " ", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", FractionBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]], RowBox[List["2", " ", "z_"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", "\[Nu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["\[Lambda]", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]

Date Added to functions.wolfram.com (modification date)

2001-10-29

GegenbauerC[nu,z]
GegenbauerC[n,lambda,z]