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variants of this functions
LegendreQ






Mathematica Notation

Traditional Notation









Hypergeometric Functions > LegendreQ[nu,mu,2,z] > Identities > Recurrence identities > Distant neighbors





http://functions.wolfram.com/07.11.17.0019.01









  


  










Input Form





LegendreQ[\[Nu], \[Mu], 2, z] == Subscript[\[ScriptCapitalC], n][\[Nu], \[Mu], z] LegendreQ[\[Nu], \[Mu] + n, 2, z] + (1/((n - 1 + \[Mu]) (n + \[Mu]) - \[Nu] (1 + \[Nu]))) Subscript[\[ScriptCapitalC], n - 1][\[Nu], \[Mu], z] LegendreQ[\[Nu], \[Mu] + n + 1, 2, z] /; Subscript[\[ScriptCapitalC], 0][\[Nu], \[Mu], z] == 1 && Subscript[\[ScriptCapitalC], 1][\[Nu], \[Mu], z] == (2 (\[Mu] + 1) z)/((\[Mu] (\[Mu] + 1) - \[Nu] (1 + \[Nu])) Sqrt[1 - z^2]) && Subscript[\[ScriptCapitalC], n][\[Nu], \[Mu], z] == ((2 z (n + \[Mu]))/(Sqrt[1 - z^2] ((n - 1 + \[Mu]) (n + \[Mu]) - \[Nu] (1 + \[Nu])))) Subscript[\[ScriptCapitalC], n - 1][\[Nu], \[Mu], z] + (1/((n - 2 + \[Mu]) (n - 1 + \[Mu]) - \[Nu] (1 + \[Nu]))) Subscript[\[ScriptCapitalC], n - 2][\[Nu], \[Mu], z] && Element[n, Integers] && n > 0










Standard Form





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MathML Form







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</ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> <apply> <ci> LegendreQ </ci> <ci> &#957; </ci> <apply> <plus /> <ci> &#956; </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <cn type='integer'> 0 </cn> </apply> <ci> &#957; </ci> <ci> &#956; </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#957; </ci> <ci> &#956; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[Mu]", "+", "n"]], ",", "2", ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[Mu]", "+", "n", "+", "1"]], ",", "2", ",", "z"]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "\[Mu]"]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", "1"]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]], " ", "z"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Mu]", " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "1"]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]], ")"]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["n", "+", "\[Mu]"]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "+", "\[Mu]"]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]], ")"]]]]], "+", FractionBox[RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "2", "+", "\[Mu]"]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1", "+", "\[Mu]"]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02