Sum[(1/Subscript[z, k]) (-1 + (1 + Subscript[z, k]^(n - 1)) Product[Subscript[z, j]/(Subscript[z, j] - Subscript[z, k]), {j, 1, k - 1}] Product[Subscript[z, j]/(Subscript[z, j] - Subscript[z, k]), {j, k + 1, n}]), {k, 1, n}] == 0 /; Element[n, Integers] && n > 0 && Subscript[z, j] != 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[FractionBox["1", SubscriptBox["z", "k"]], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SubsuperscriptBox["z", "k", RowBox[List["n", "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", "1"]], RowBox[List["k", "-", "1"]]], FractionBox[SubscriptBox["z", "j"], RowBox[List[SubscriptBox["z", "j"], "-", SubscriptBox["z", "k"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["k", "+", "1"]]]], "n"], FractionBox[SubscriptBox["z", "j"], RowBox[List[SubscriptBox["z", "j"], "-", SubscriptBox["z", "k"]]]]]], ")"]]]]]], ")"]]]]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[SubscriptBox["z", "j"], "\[NotEqual]", "0"]]]]]]]]

<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> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msub> <mi> z </mi> <mi> k </mi> </msub> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msubsup> <mi> z </mi> <mi> k </mi> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <msub> <mi> z </mi> <mi> j </mi> </msub> <mrow> <msub> <mi> z </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mi> n </mi> </munderover> <mfrac> <msub> <mi> z </mi> <mi> j </mi> </msub> <mrow> <msub> <mi> z </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mi> j </mi> </msub> <mo> &#8800; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <product /> <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> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <neq /> <apply> <ci> Subscript </ci> <ci> z </ci> <ci> j </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "n_"], FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SubsuperscriptBox["z_", "k_", RowBox[List["n_", "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j_", "=", "1"]], RowBox[List["k_", "-", "1"]]], FractionBox[SubscriptBox["z_", "j_"], RowBox[List[SubscriptBox["z_", "j_"], "-", SubscriptBox["z_", "k_"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j_", "=", RowBox[List["k_", "+", "1"]]]], "n_"], FractionBox[SubscriptBox["z_", "j_"], RowBox[List[SubscriptBox["z_", "j_"], "-", SubscriptBox["z_", "k_"]]]]]]]]]], SubscriptBox["z_", "k_"]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List[SubscriptBox["z", "j"], "\[NotEqual]", "0"]]]]]]]]]]

2002-12-18