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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/05.09.06.0055.01









  


  










Input Form





GegenbauerC[n, \[Lambda], z] \[Proportional] ((2^(1 - \[Lambda]) n^(\[Lambda] - 1))/(Gamma[\[Lambda]] (1 - z^2)^(\[Lambda]/2))) Sum[((((-1)^(j + k) Pochhammer[1 - \[Lambda], k] Pochhammer[\[Lambda], j])/ (2^j (1 - z^2)^(j/2) (j! (k - j)!))) Cos[(n - j + \[Lambda]) ArcCos[z] - (Pi (j + \[Lambda]))/2] NorlundB[k - j, \[Lambda] - j, \[Lambda] - j])/n^k, {k, 0, Infinity}, {j, 0, k}] /; (n -> Infinity)










Standard Form





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MathML Form







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</ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <ci> &#955; </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <ci> &#955; </ci> </apply> <apply> <arccos /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <plus /> <ci> j </ci> <ci> &#955; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> NorlundB </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <plus /> <ci> &#955; 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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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "k"]]], " ", SuperscriptBox["2", RowBox[List["-", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", FractionBox["j", "2"]]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "\[Lambda]"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j", "+", "\[Lambda]"]], ")"]], " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["j", "+", "\[Lambda]"]], ")"]]]]]], "]"]], " ", RowBox[List["NorlundB", "[", RowBox[List[RowBox[List["k", "-", "j"]], ",", RowBox[List["\[Lambda]", "-", "j"]], ",", RowBox[List["\[Lambda]", "-", "j"]]]], "]"]], " ", SuperscriptBox["n", RowBox[List["-", "k"]]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]]]], 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





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