Gegenbauer polynomials: Series representations (formula 05.09.06.0001)

GegenbauerC

$Mathematica Notation$

Polynomials

GegenbauerC[n,lambda,z]

Series representations

Generalized power series

Expansions at z==0

For the function itself

http://functions.wolfram.com/05.09.06.0001.02

Input Form

GegenbauerC[n, \[Lambda], z] \[Proportional] (2^n Sqrt[Pi] Gamma[n/2 + \[Lambda]])/(Gamma[(1 - n)/2] n! Gamma[\[Lambda]]) + (2^n Sqrt[Pi] Gamma[(1 + n)/2 + \[Lambda]] z)/ (Gamma[1 - n/2] (n - 1)! Gamma[\[Lambda]]) - ((2 Cos[(n Pi)/2] Gamma[1 + n/2 + \[Lambda]])/(Gamma[\[Lambda]] Gamma[n/2])) z^2 - (2^n Sqrt[Pi] Gamma[(3 + n)/2 + \[Lambda]] z^3)/ (3 Gamma[1 - n/2] (n - 2)! Gamma[\[Lambda]]) + \[Ellipsis] /; (z -> 0)

Standard Form

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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> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> λ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mi> λ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mo> … </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </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> λ </ci> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> λ </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </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> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> Gamma </ci> <ci> λ </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> λ </ci> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <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> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <ci> λ </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <ci> n </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> λ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> λ </ci> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> λ </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <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> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <ci> λ </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> … </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </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[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["n", "2"], "+", "\[Lambda]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "n"]], "2"], "]"]], " ", RowBox[List["n", "!"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "n"]], "2"], "+", "\[Lambda]"]], "]"]], " ", "z"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["n", "2"]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["n", " ", "\[Pi]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["n", "2"], "+", "\[Lambda]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", RowBox[List["Gamma", "[", FractionBox["n", "2"], "]"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["3", "+", "n"]], "2"], "+", "\[Lambda]"]], "]"]], " ", SuperscriptBox["z", "3"]]], RowBox[List["3", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", FractionBox["n", "2"]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]]

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

2001-10-29

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