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variants of this functions
LaguerreL






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/05.08.06.0032.01









  


  










Input Form





LaguerreL[n, \[Lambda], z] \[Proportional] ((Gamma[n + \[Lambda] + 1]/n!) E^(z/2) Sum[Subscript[A, k] (z/(2 (2 n + \[Lambda] + 1)))^(k/2) BesselJ[\[Lambda] + k, Sqrt[2 (2 n + \[Lambda] + 1) z]], {k, 0, Infinity}])/(((2 n + \[Lambda] + 1)/2)^(\[Lambda]/2) z^(\[Lambda]/2)) /; (n -> Infinity) && Subscript[A, 0] == 1 && Subscript[A, 1] == 0 && Subscript[A, 2] == (\[Lambda] + 1)/2 && Subscript[A, m] == ((m + \[Lambda] - 1)/m) Subscript[A, m - 2] - (2 n + \[Lambda] + 1) Subscript[A, m - 3] && Element[m, Integers] && m > 0










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LaguerreL", "[", RowBox[List["n_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["n", "+", "\[Lambda]", "+", "1"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "\[Lambda]", "+", "1"]], ")"]]]], ")"]], RowBox[List["-", FractionBox["\[Lambda]", "2"]]]], " ", SuperscriptBox["z", RowBox[List["-", FractionBox["\[Lambda]", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["z", "/", "2"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["A", "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "\[Lambda]", "+", "1"]], ")"]]]]], ")"]], RowBox[List["k", "/", "2"]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["\[Lambda]", "+", "k"]], ",", SqrtBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "\[Lambda]", "+", "1"]], ")"]], " ", "z"]]]]], "]"]]]]]]]], RowBox[List["n", "!"]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List[SubscriptBox["A", "0"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["A", "1"], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["A", "2"], "\[Equal]", FractionBox[RowBox[List["\[Lambda]", "+", "1"]], "2"]]], "&&", RowBox[List[SubscriptBox["A", "m"], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "\[Lambda]", "-", "1"]], ")"]], " ", SubscriptBox["A", RowBox[List["m", "-", "2"]]]]], "m"], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "+", "\[Lambda]", "+", "1"]], ")"]], " ", SubscriptBox["A", RowBox[List["m", "-", "3"]]]]]]]]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]










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





2007-05-02





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