Integrate[LegendreP[n, m, 2, t], {t, -1, 1}] == (((-1)^m + (-1)^n) 2^(m - 2) m Gamma[n/2] Gamma[(1/2) (n + m + 1)])/ (((n - m)/2)! Gamma[(n + 3)/2]) /; Element[m, Integers]

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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> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 1 </mn> </msubsup> <mrow> <mrow> <msubsup> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mi> n </mi> <mi> m </mi> </msubsup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <mi> &#8484; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <plus /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> -1 </cn> </lowlimit> <uplimit> <cn type='integer'> 1 </cn> </uplimit> <apply> <ci> LegendreP </ci> <ci> n </ci> <ci> m </ci> <ci> t </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> <ci> m </ci> <apply> <ci> Gamma </ci> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <times /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> n </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> m </ci> <ci> &#8484; </ci> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18