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 LaguerreL

 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["n", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["n", "+", "\[Lambda]", "+", "1"]], "]"]], RowBox[List["n", "!"]]], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List["2", "n"]], "+", "\[Lambda]", "+", "1"]], "2"], ")"]], RowBox[List["-", FractionBox["\[Lambda]", "2"]]]], SuperscriptBox["z", RowBox[List["-", FractionBox["\[Lambda]", "2"]]]], SuperscriptBox["\[ExponentialE]", FractionBox["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"]], ")"]]]]], ")"]], FractionBox["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[RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List[SubscriptBox["A", "0"], "\[Equal]", "1"]], "\[And]", "\[IndentingNewLine]", RowBox[List[SubscriptBox["A", "1"], "\[Equal]", "0"]], "\[And]", "\[IndentingNewLine]", RowBox[List[SubscriptBox["A", "2"], "\[Equal]", FractionBox[RowBox[List["\[Lambda]", "+", "1"]], "2"]]], "\[And]", "\[IndentingNewLine]", RowBox[List[SubscriptBox["A", "m"], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["m", "+", "\[Lambda]", "-", "1"]], "m"], SubscriptBox["A", RowBox[List["m", "-", "2"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "+", "\[Lambda]", "+", "1"]], ")"]], SubscriptBox["A", RowBox[List["m", "-", "3"]]]]]]]]], "\[And]", RowBox[List["Element", "[", RowBox[List["m", ",", "Integers"]], "]"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 L n λ ( z ) Γ ( n + λ + 1 ) n ! ( 2 n + λ + 1 2 ) - λ 2 z - λ 2 z / 2 k = 0 A k ( z 2 ( 2 n + λ + 1 ) ) k / 2 J k + λ ( 2 ( 2 n + λ + 1 ) z ) /; ( n "\[Rule]" ) A 0 1 A 1 0 A 2 λ + 1 2 A m m + λ - 1 m A m - 2 - ( 2 n + λ + 1 ) A m - 3 m + Condition Proportional LaguerreL n λ z Gamma n λ 1 n -1 2 n λ 1 2 -1 -1 λ 2 -1 z -1 λ 2 -1 z 2 -1 k 0 Subscript A k z 2 2 n λ 1 -1 k 2 -1 BesselJ k λ 2 2 n λ 1 z 1 2 Rule n Subscript A 0 1 Subscript A 1 0 Subscript A 2 λ 1 2 -1 Subscript A m m λ -1 m -1 Subscript A m -2 -1 2 n λ 1 Subscript A m -3 m SuperPlus [/itex]

 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