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http://functions.wolfram.com/07.12.17.0013.01
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LegendreQ[\[Nu], \[Mu], 3, z] ==
Subscript[\[ScriptCapitalC], n][\[Nu], \[Mu], z]
LegendreQ[\[Nu] + n, \[Mu], 3, z] +
((-n - 1 + \[Mu] - \[Nu])/(n + \[Mu] + \[Nu]))
Subscript[\[ScriptCapitalC], n - 1][\[Nu], \[Mu], z]
LegendreQ[\[Nu] + n + 1, \[Mu], 3, z] /;
Subscript[\[ScriptCapitalC], 0][\[Nu], \[Mu], z] == 1 &&
Subscript[\[ScriptCapitalC], 1][\[Nu], \[Mu], z] ==
((2 \[Nu] + 3) z)/(\[Nu] + \[Mu] + 1) &&
Subscript[\[ScriptCapitalC], n][\[Nu], \[Mu], z] ==
((z (1 + 2 n + 2 \[Nu]))/(n + \[Mu] + \[Nu]))
Subscript[\[ScriptCapitalC], n - 1][\[Nu], \[Mu], z] +
((-n + \[Mu] - \[Nu])/(n - 1 + \[Mu] + \[Nu]))
Subscript[\[ScriptCapitalC], n - 2][\[Nu], \[Mu], z] &&
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "-", "1", "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " "]], RowBox[List["n", "+", "\[Mu]", "+", "\[Nu]"]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n", "+", "1"]], ",", "\[Mu]", ",", "3", ",", "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["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]], " ", "z"]], RowBox[List["\[Nu]", "+", "\[Mu]", "+", "1"]]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["n", "+", "\[Mu]", "+", "\[Nu]"]]], RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], RowBox[List[" ", RowBox[List["(", RowBox[List["n", "-", "1", "+", "\[Mu]", "+", "\[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"]]]]]]]]
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<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["\[GothicCapitalQ]", 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["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> + </mo> <mi> n </mi> </mrow> <mi> μ </mi> </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> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </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["\[GothicCapitalQ]", LegendreQ] </annotation> </semantics> <mrow> <mi> ν </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> μ </mi> </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> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </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> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </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> <mrow> <mi> μ </mi> <mo> - </mo> <mi> ν </mi> <mo> - </mo> <mi> n </mi> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </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> <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> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <ci> ν </ci> </apply> <ci> μ </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreQ </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </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> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <apply> <plus /> <ci> ν </ci> <ci> n </ci> </apply> </apply> <ci> μ </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreQ </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <ci> μ </ci> <ci> ν </ci> </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> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔔 </ci> </apply> <apply> <plus /> <ci> ν </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> μ </ci> </apply> <apply> <apply> <ci> HoldComplete </ci> <ci> LegendreQ </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> </apply> </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 /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 3 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> μ </ci> <ci> ν </ci> <cn type='integer'> 1 </cn> </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 /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <ci> μ </ci> <ci> ν </ci> </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 /> <apply> <plus /> <ci> μ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <ci> μ </ci> <ci> ν </ci> <cn type='integer'> -1 </cn> </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>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LegendreQ", "[", RowBox[List["\[Nu]_", ",", "\[Mu]_", ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "-", "1", "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["LegendreQ", "[", RowBox[List[RowBox[List["\[Nu]", "+", "n", "+", "1"]], ",", "\[Mu]", ",", "3", ",", "z"]], "]"]]]], RowBox[List["n", "+", "\[Mu]", "+", "\[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[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "3"]], ")"]], " ", "z"]], RowBox[List["\[Nu]", "+", "\[Mu]", "+", "1"]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[ScriptCapitalC]", "n"], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "n"]], "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "1"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], RowBox[List["n", "+", "\[Mu]", "+", "\[Nu]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "n"]], "+", "\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List[SubscriptBox["\[ScriptCapitalC]", RowBox[List["n", "-", "2"]]], "[", RowBox[List["\[Nu]", ",", "\[Mu]", ",", "z"]], "]"]]]], RowBox[List["n", "-", "1", "+", "\[Mu]", "+", "\[Nu]"]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
