2 (m - n) (n - 1) n D[w[n], {n, 3}] + (8 n + 8 m n - 13 n^2 - 3 m) D[w[n], {n, 2}] + 4 (1 + m - 4 n) D[w[n], n] - 2 w[n] == 0 /; w[n] == EllipticPi[n, m]

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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> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> n </mi> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mn> 3 </mn> </msup> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> n </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 13 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> m </mi> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mn> 2 </mn> </msup> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <mo> &#8706; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <ci> n </ci> <apply> <partialdiff /> <bvar> <ci> n </ci> <degree> <cn type='integer'> 3 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -13 </cn> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> m </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> n </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> n </ci> </bvar> <apply> <ci> w </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> w </ci> <ci> n </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> n </ci> </apply> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29