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variants of this functions
LaguerreL






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LaguerreL[nu,z] > Identities > Recurrence identities > Distant neighbors





http://functions.wolfram.com/07.02.17.0008.01









  


  










Input Form





LaguerreL[\[Nu], z] == Subscript[\[ScriptCapitalC], n][\[Nu], z] LaguerreL[\[Nu] - n, z] - ((\[Nu] - n)/(1 - n + \[Nu])) Subscript[\[ScriptCapitalC], n - 1][\[Nu], z] LaguerreL[\[Nu] - n - 1, z] /; Subscript[\[ScriptCapitalC], 0][\[Nu], z] == 1 && Subscript[\[ScriptCapitalC], 1][\[Nu], z] == (2 \[Nu] - z - 1)/\[Nu] && Subscript[\[ScriptCapitalC], n][\[Nu], z] == ((1 - 2 n - z + 2 \[Nu])/(1 - n + \[Nu])) Subscript[\[ScriptCapitalC], n - 1][\[Nu], z] - ((1 - n + \[Nu])/(2 - n + \[Nu])) Subscript[\[ScriptCapitalC], n - 2][ \[Nu], z] && Element[n, Integers] && n > 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["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n"]], ",", "z"]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", "n"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["LaguerreL", "[", RowBox[List[RowBox[List["\[Nu]", "-", "n", "-", "1"]], ",", "z"]], "]"]]]], RowBox[List["1", "-", "n", "+", "\[Nu]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "-", "z", "-", "1"]], "\[Nu]"]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]], "-", "z", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List["1", "-", "n", "+", "\[Nu]"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "n", "+", "\[Nu]"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], RowBox[List["2", "-", "n", "+", "\[Nu]"]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02