Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









Polynomials > GegenbauerC[n,lambda,z] > Series representations > Generalized power series > Expansions at lambda==infinity





http://functions.wolfram.com/05.09.06.0049.01









  


  










Input Form





GegenbauerC[n, \[Lambda], z] \[Proportional] (((2 z)^n \[Lambda]^n)/n!) (1 + ((n - 1) n (2 z^2 - 1))/(4 z^2 \[Lambda]) + ((n - 2) (n - 1) n (3 n (1 - 2 z^2)^2 - (3 - 2 z^2)^2))/ (96 z^4 \[Lambda]^2) + \[Ellipsis]) /; (Abs[\[Lambda]] -> Infinity)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "z"]], ")"]], "n"], SuperscriptBox["\[Lambda]", "n"]]], RowBox[List["n", "!"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", "1"]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", "\[Lambda]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], ")"]], "2"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], ")"]], "2"]]], ")"]]]], RowBox[List["96", " ", SuperscriptBox["z", "4"], " ", SuperscriptBox["\[Lambda]", "2"]]]], "+", "\[Ellipsis]"]], " ", ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Lambda]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]










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> <msubsup> <mi> C </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mi> n </mi> </msup> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mi> &#955; </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mrow> <mn> 96 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> &#955; </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> &#955; </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> n </ci> </apply> <ci> &#955; </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <ci> n </ci> </apply> <apply> <power /> <ci> &#955; </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#955; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> n </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 96 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> &#955; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> &#955; </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], "n"], " ", SuperscriptBox["\[Lambda]", "n"]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", "1"]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", "\[Lambda]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", "n", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "n", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], ")"]], "2"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], ")"]], "2"]]], ")"]]]], RowBox[List["96", " ", SuperscriptBox["z", "4"], " ", SuperscriptBox["\[Lambda]", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["n", "!"]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Lambda]", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.