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http://functions.wolfram.com/07.09.23.0001.01
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Sum[(-1)^m LegendreP[n, m, 3, z] LegendreP[n, -m, 3, z], {m, -n, n}] == 1 /;
Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", RowBox[List["-", "n"]]]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "m", ",", "3", ",", "z"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", RowBox[List["-", "m"]], ",", "3", ",", "z"]], "]"]]]]]], "\[Equal]", "1"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> <munderover> <mo> ∑ </mo> <mrow> <mi> m </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> 𝔓 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalP]", LegendreQ] </annotation> </semantics> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msubsup> <semantics> <mi> 𝔓 </mi> <annotation encoding='Mathematica'> TagBox["\[GothicCapitalP]", LegendreQ] </annotation> </semantics> <mi> n </mi> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mn> 1 </mn> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> m </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔓 </ci> </apply> <ci> n </ci> </apply> <ci> m </ci> </apply> <ci> z </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <ci> LegendreQ </ci> <ci> 𝔓 </ci> </apply> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m_", "=", RowBox[List["-", "n_"]]]], "n_"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m_"], " ", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", "m_", ",", "3", ",", "z_"]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n_", ",", RowBox[List["-", "m_"]], ",", "3", ",", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["1", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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LegendreP[n,z] | LegendreP[nu,z] | LegendreP[nu,mu,z] | LegendreP[n,mu,2,z] | LegendreP[nu,mu,2,z] | |
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