Sum[Subscript[a, j] z^j, {j, 0, n}] == Product[z - Subscript[\[Alpha], k], {k, 1, n}] /; Subscript[a, 0] == (-1)^n Product[Subscript[\[Alpha], k], {k, 1, n}] && Subscript[a, 1] == (-1)^(n - 1) Sum[Product[If[j != k, Subscript[\[Alpha], j], 1], {j, 1, n}], {k, 1, n}] && \[Ellipsis] && Subscript[a, n - 2] == Sum[If[j != k, Subscript[\[Alpha], j] Subscript[\[Alpha], k], 0], {k, 1, n}, {j, 1, k - 1}] && Subscript[a, n - 1] == -Sum[Subscript[\[Alpha], k], {k, 1, n}]

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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> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> j </mi> </msup> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> If </mi> <mo> [ </mo> <mrow> <mrow> <mi> j </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> <mo> , </mo> <msub> <mi> &#945; </mi> <mi> j </mi> </msub> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ] </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mo> &#8230; </mo> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msub> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mi> If </mi> <mo> [ </mo> <mrow> <mrow> <mi> j </mi> <mo> &#8800; </mo> <mi> k </mi> </mrow> <mo> , </mo> <mrow> <msub> <mi> &#945; </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </mrow> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ] </mo> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> &#945; </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <power /> <ci> z </ci> <ci> j </ci> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> If </ci> <apply> <neq /> <ci> j </ci> <ci> k </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> j </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <ci> &#8230; </ci> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> If </ci> <apply> <neq /> <ci> j </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> &#945; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29