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variants of this functions
GegenbauerC






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/05.09.06.0054.01









  


  










Input Form





GegenbauerC[n, \[Lambda], z] \[Proportional] ((2^(1 - \[Lambda]) n^(\[Lambda] - 1))/(Gamma[\[Lambda]] (1 - z^2)^(\[Lambda]/2))) (Cos[(Pi \[Lambda])/2 - (n + \[Lambda]) ArcCos[z]] + (((-1 + \[Lambda]) \[Lambda])/(2 n)) (Cos[(Pi \[Lambda])/2 - (n + \[Lambda]) ArcCos[z]] + Sin[(Pi \[Lambda])/2 - (-1 + n + \[Lambda]) ArcCos[z]]/Sqrt[1 - z^2]) + (((1 - \[Lambda]) (2 - \[Lambda]) \[Lambda])/(24 n^2)) (-((3 (1 + \[Lambda]) Cos[(1/2) Pi (2 + \[Lambda]) - (-2 + n + \[Lambda]) ArcCos[z]])/(-1 + z^2)) - (6 (-1 + \[Lambda]) Cos[(1/2) Pi (1 + \[Lambda]) - (-1 + n + \[Lambda]) ArcCos[z]])/Sqrt[1 - z^2] + (-1 + 3 \[Lambda]) Cos[(Pi \[Lambda])/2 - (n + \[Lambda]) ArcCos[z]]) + \[Ellipsis]) /; (n -> Infinity)










Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> n </ci> <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["2", RowBox[List["1", "-", "\[Lambda]"]]], " ", SuperscriptBox["n", RowBox[List["\[Lambda]", "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[Lambda]"]], "2"], "-", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "\[Lambda]"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Lambda]"]], ")"]], " ", "\[Lambda]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[Lambda]"]], "2"], "-", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "\[Lambda]"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]], "]"]], "+", FractionBox[RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[Lambda]"]], "2"], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n", "+", "\[Lambda]"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]], "]"]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], ")"]]]], RowBox[List["2", " ", "n"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[Lambda]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "-", "\[Lambda]"]], ")"]], " ", "\[Lambda]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["3", " ", RowBox[List["(", RowBox[List["1", "+", "\[Lambda]"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["2", "+", "\[Lambda]"]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "n", "+", "\[Lambda]"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]], "-", FractionBox[RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Lambda]"]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Lambda]"]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n", "+", "\[Lambda]"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["3", " ", "\[Lambda]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[Lambda]"]], "2"], "-", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "\[Lambda]"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["24", " ", SuperscriptBox["n", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List[RowBox[List["Gamma", "[", "\[Lambda]", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["\[Lambda]", "/", "2"]]]]]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.