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variants of this functions
LegendreP






Mathematica Notation

Traditional Notation









Polynomials > LegendreP[n,mu,2,z] > Identities > Recurrence identities > Distant neighbors





http://functions.wolfram.com/05.07.17.0013.01









  


  










Input Form





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










Standard Form





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MathML Form







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</mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> &#956; </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> <mo> + </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> <mo> + </mo> <mi> &#956; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> &#119966; </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> n </mi> <mo> , </mo> <mi> &#956; 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</ci> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> n </ci> <ci> &#956; </ci> <ci> z </ci> </apply> <apply> <ci> LegendreP </ci> <ci> n </ci> <apply> <plus /> <ci> &#956; </ci> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </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> n </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> n </ci> <ci> &#956; </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> &#956; </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#119966; 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Rule Form





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










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





2007-05-02





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