Sum[Cot[(Pi*k)/n]^(2*m), {k, 1, n - 1}] == n*(-1)^m - n*(-1)^m*2^(2*m)* Sum[… Sum[KroneckerDelta[Sum[Subscript[l, r], {r, 0, 2*m}] - m]* n^(2*Subscript[l, 0] - 1)*Product[BernoulliB[2*Subscript[l, r]]/ (2*Subscript[l, r])!, {r, 0, 2*m}], {Subscript[l, 2*m], 0, m}], …, {Subscript[l, 0], 0, m}] /; n ∈ Integers && n > 0 && m ∈ Integers && m > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["n", "-", "1"]]], SuperscriptBox[RowBox[List["Cot", "[", FractionBox[RowBox[List["\[Pi]", " ", "k"]], "n"], "]"]], RowBox[List["2", "m"]]]]], "\[Equal]", RowBox[List[RowBox[List["n", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"]]], "-", RowBox[List["n", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], SuperscriptBox["2", RowBox[List["2", "m"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["l", "0"], "=", "0"]], "m"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["l", RowBox[List["2", "m"]]], "=", "0"]], "m"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], RowBox[List["2", "m"]]], SubscriptBox["l", "r"]]], "-", "m"]], "]"]], SuperscriptBox["n", RowBox[List[RowBox[List["2", SubscriptBox["l", "0"]]], "-", "1"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "0"]], RowBox[List["2", "m"]]], FractionBox[RowBox[List["BernoulliB", "[", RowBox[List["2", SubscriptBox["l", "r"]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["2", SubscriptBox["l", "r"]]], ")"]], "!"]]]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", " ", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["Element", "[", RowBox[List["m", ",", " ", "Integers"]], "]"]], "\[And]", RowBox[List["m", ">", "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> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mi> cot </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mi> n </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> - </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> l </mi> <mn> 0 </mn> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> l </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <msub> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </munderover> <msub> <mi> l </mi> <mi> r </mi> </msub> </mrow> <mo> - </mo> <mi> m </mi> </mrow> </msub> <mo> &#8290; </mo> <msup> <mi> n </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> l </mi> <mn> 0 </mn> </msub> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> r </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> </munderover> <mfrac> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox[&quot;B&quot;, BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> l </mi> <mi> r </mi> </msub> </mrow> </msub> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> l </mi> <mi> r </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> <annotation encoding='Mathematica'> TagBox[SuperscriptBox[&quot;\[DoubleStruckCapitalN]&quot;, &quot;+&quot;], Function[Integers]] </annotation> </semantics> </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> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <power /> <apply> <cot /> <apply> <times /> <pi /> <ci> k </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> l </ci> <cn type='integer'> 0 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> l </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <times /> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </uplimit> <apply> <ci> Subscript </ci> <ci> l </ci> <ci> r </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> </apply> <apply> <power /> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> l </ci> <cn type='integer'> 0 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <product /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </uplimit> <apply> <times /> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> l </ci> <ci> r </ci> </apply> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> l </ci> <ci> r </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], RowBox[List["n_", "-", "1"]]], SuperscriptBox[RowBox[List["Cot", "[", FractionBox[RowBox[List["\[Pi]", " ", "k_"]], "n_"], "]"]], RowBox[List["2", " ", "m_"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["n", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"]]], "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["2", RowBox[List["2", " ", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["l", "0"], "=", "0"]], "m"], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["l", RowBox[List["2", " ", "m"]]], "=", "0"]], "m"], RowBox[List[RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "0"]], RowBox[List["2", " ", "m"]]], SubscriptBox["l", "r"]]], "-", "m"]], "]"]], " ", SuperscriptBox["n", RowBox[List[RowBox[List["2", " ", SubscriptBox["l", "0"]]], "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["r", "=", "0"]], RowBox[List["2", " ", "m"]]], FractionBox[RowBox[List["BernoulliB", "[", RowBox[List["2", " ", SubscriptBox["l", "r"]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", SubscriptBox["l", "r"]]], ")"]], "!"]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

2001-10-29