Assosiated Legendre function of the second kind of type 2: Identities (formula 07.11.17.0019)

LegendreQ

$Mathematica Notation$

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

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "2", ",", "z"]], "]"]], "\[Equal]", 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"]], "]"]]]], "+", RowBox[List[FractionBox["1", 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[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[SubscriptBox["\[ScriptCapitalC]", "0"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", "1"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "1"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", FractionBox[RowBox[List[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"]]]]]]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["n", "+", "\[Mu]"]], ")"]], " "]], 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]"]], ")"]]]]]], ")"]]]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", 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[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]]]]]], " ", "\[And]", RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mi> μ </mi> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <msub> <mi> 𝒞 </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mi> n </mi> </mrow> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> μ </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> 𝒞 </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> Q </mi> <annotation encoding='Mathematica'> TagBox["Q", LegendreQ] </annotation> </semantics> <mi> ν </mi> <mrow> <mi> μ </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <semantics> <mi> z </mi> <annotation encoding='Mathematica'> TagBox["z", HoldComplete[LegendreQ, 2]] </annotation> </semantics> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mi> 𝒞 </mi> <mn> 0 </mn> </msub> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msub> <mi> 𝒞 </mi> <mn> 1 </mn> </msub> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> μ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> μ </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ∧ </mo> <mrow> <mrow> <msub> <mi> 𝒞 </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> 𝒞 </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </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> n </mi> <mo> - </mo> <mn> 2 </mn> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> <mo> + </mo> <mi> μ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> 𝒞 </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> , </mo> <mi> μ </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LegendreQ </ci> <ci> ν </ci> <ci> μ </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <ci> n </ci> </apply> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <ci> LegendreQ </ci> <ci> ν </ci> <apply> <plus /> <ci> μ </ci> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <ci> μ </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <ci> μ </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <ci> LegendreQ </ci> <ci> ν </ci> <apply> <plus /> <ci> μ </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> 𝒞 </ci> <cn type='integer'> 0 </cn> </apply> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <cn type='integer'> 1 </cn> </apply> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> μ </ci> <apply> <plus /> <ci> μ </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> ν </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> 𝒞 </ci> <ci> n </ci> </apply> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <ci> n </ci> <ci> μ </ci> </apply> <apply> <power /> <apply> <times /> <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> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> <apply> <plus /> <ci> n </ci> <ci> μ </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> <ci> μ </ci> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> <ci> μ </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> 𝒞 </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <ci> ν </ci> <ci> μ </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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

LegendreQ[nu,z]
LegendreQ[nu,mu,z]
LegendreQ[nu,mu,3,z]