Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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









Hypergeometric Functions > GegenbauerC[nu,lambda,z] > Series representations > Generalized power series > Expansions at z==infinity > For the function itself > Expansions in 1/z





http://functions.wolfram.com/07.14.06.0073.01









  


  










Input Form





GegenbauerC[n, \[Lambda], z] == ((2^n z^n Pochhammer[\[Lambda], n])/n!) Sum[(Pochhammer[-(n/2), k] Pochhammer[(1 - n)/2, k])/ (Pochhammer[1 - \[Lambda] - n, k] k!)/z^(2 k), {k, 0, Floor[n/2]}] /; !(Element[\[Lambda] + n, Integers] && \[Lambda] + n > 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", SuperscriptBox["z", "n"], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", "n"]], "]"]]]], RowBox[List["n", "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["n", "/", "2"]], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Lambda]", "-", "n"]], ",", "k"]], "]"]], RowBox[List["k", "!"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]]]]]]]]]], " ", "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["Element", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "n"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["\[Lambda]", "+", "n"]], ">", "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> <msubsup> <mi> C </mi> <mi> n </mi> <mi> &#955; </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> &#955; </mi> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;\[Lambda]&quot;, &quot;)&quot;]], &quot;n&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </munderover> <mrow> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, FractionBox[&quot;n&quot;, &quot;2&quot;]]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;n&quot;]], &quot;2&quot;], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mi> &#955; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;n&quot;, &quot;-&quot;, &quot;\[Lambda]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> &#955; </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> &#8713; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <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 /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> &#955; </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#955; </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <plus /> <ci> &#955; </ci> <ci> n </ci> </apply> <integers /> </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["2", "n"], " ", SuperscriptBox["z", "n"], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", "n"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["n", "2"]]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Lambda]", "-", "n"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]]]]]]]]], RowBox[List["n", "!"]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["\[Lambda]", "+", "n"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["\[Lambda]", "+", "n"]], ">", "0"]]]], ")"]]]]]]]]]]










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





2007-05-02