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http://functions.wolfram.com/01.07.23.0009.01
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Sum[Cos[(2 Pi p k)/n + \[Alpha]]^(2 q), {k, 1, n}] ==
(n (2 q - 1)!!)/(2^q q!) /; Element[p, Integers] && p > 0 &&
Element[q, Integers] && q > 0 && 2 p q < n
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "p", " ", "k"]], "n"], "+", "\[Alpha]"]], "]"]], RowBox[List["2", " ", "q"]]]]], "\[Equal]", FractionBox[RowBox[List["n", " ", SuperscriptBox["2", RowBox[List["-", "q"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "q"]], "-", "1"]], ")"]], "!!"]]]], RowBox[List["q", "!"]]]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["p", ">", "0"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", ">", "0"]], "\[And]", RowBox[List[RowBox[List["2", "p", " ", "q"]], "<", "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> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mi> cos </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> q </mi> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> p </mi> </mrow> <mi> n </mi> </mfrac> <mo> + </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> n </mi> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> q </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mrow> <mrow> <mi> q </mi> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> p </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> q </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> p </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> < </mo> <mi> n </mi> </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> <power /> <apply> <cos /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <pi /> <ci> p </ci> <apply> <power /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> <ci> α </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> </apply> </apply> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <factorial /> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> p </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <in /> <ci> q </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <lt /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> p </ci> <ci> q </ci> </apply> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "n_"], SuperscriptBox[RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "p_", " ", "k_"]], "n_"], "+", "\[Alpha]_"]], "]"]], RowBox[List["2", " ", "q_"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["n", " ", SuperscriptBox["2", RowBox[List["-", "q"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "q"]], "-", "1"]], ")"]], "!!"]]]], RowBox[List["q", "!"]]], "/;", RowBox[List[RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["p", ">", "0"]], "&&", RowBox[List["q", "\[Element]", "Integers"]], "&&", RowBox[List["q", ">", "0"]], "&&", RowBox[List[RowBox[List["2", " ", "p", " ", "q"]], "<", "n"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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