WeierstrassP[z, WeierstrassInvariants[{Subscript[\[Omega], 1]/n, Subscript[\[Omega], 3]}]] == WeierstrassP[z, {Subscript[g, 2], Subscript[g, 3]}] + Sum[WeierstrassP[z + (2 k Subscript[\[Omega], 1])/n, {Subscript[g, 2], Subscript[g, 3]}] - WeierstrassP[ (2 k Subscript[\[Omega], 1])/n, {Subscript[g, 2], Subscript[g, 3]}], {k, 1, n - 1}] /; Element[n, Integers] && 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> <mi> &#8472; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <mrow> <msub> <mi> g </mi> <mn> 2 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mi> n </mi> </mfrac> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <msub> <mi> g </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> <mi> n </mi> </mfrac> <mo> , </mo> <msub> <mi> &#969; </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8472; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mi> n </mi> </mfrac> </mrow> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8472; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <msub> <mi> &#969; </mi> <mn> 1 </mn> </msub> </mrow> <mi> n </mi> </mfrac> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#8472; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <msub> <mi> g </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <msub> <mi> g </mi> <mn> 3 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> WeierstrassP </ci> <ci> z </ci> <list> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 3 </cn> </apply> </apply> </list> </apply> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <plus /> <apply> <ci> WeierstrassP </ci> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> WeierstrassP </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> &#969; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> </apply> </apply> </apply> <apply> <ci> WeierstrassP </ci> <ci> z </ci> <list> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 2 </cn> </apply> <apply> <ci> Subscript </ci> <ci> g </ci> <cn type='integer'> 3 </cn> </apply> </list> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29