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http://functions.wolfram.com/05.09.06.0015.02
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GegenbauerC[n, \[Lambda], z] \[Proportional]
((2^n Pochhammer[\[Lambda], n])/n!) (z - 1)^n
(1 - n/(1 - z) - (n (1 - n) (3 - 2 \[Lambda] - 2 n))/
(4 (1 - \[Lambda] - n) (1 - z)^2) - \[Ellipsis]) /;
(Abs[z] -> Infinity) && !Element[2 \[Lambda] + 2 n, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", "n"]], "]"]], " "]], RowBox[List["n", "!"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[" ", "n"]], RowBox[List["1", "-", "z"]]], "-", FractionBox[RowBox[List["n", RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", "\[Lambda]"]], "-", RowBox[List["2", "n"]]]], ")"]]]], RowBox[List[" ", RowBox[List["4", RowBox[List["(", RowBox[List["1", "-", "\[Lambda]", "-", "n"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "2"]]]]]], "-", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[RowBox[List[RowBox[List["2", "\[Lambda]"]], "+", RowBox[List["2", "n"]]]], ",", "Integers"]], "]"]], "]"]]]]]]]]
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<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> <semantics> <msub> <mrow> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mi> n </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Lambda]", ")"]], "n"], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mtext> </mtext> <mi> n </mi> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> n </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mtext> </mtext> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> λ </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> </mfrac> <mo> - </mo> <mo> … </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <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> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </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> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> λ </ci> <ci> n </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <ci> n </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> n </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> … </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <notin /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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"], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", "n"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox["n", RowBox[List["1", "-", "z"]]], "-", FractionBox[RowBox[List["n", " ", RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Lambda]"]], "-", RowBox[List["2", " ", "n"]]]], ")"]]]], RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "-", "\[Lambda]", "-", "n"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "2"]]]], "-", "\[Ellipsis]"]], ")"]]]], RowBox[List["n", "!"]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["!", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[Lambda]"]], "+", RowBox[List["2", " ", "n"]]]], "\[Element]", "Integers"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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