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http://functions.wolfram.com/10.02.20.0033.01
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Derivative[1, 0][Zeta][-2 n + 1, p/q] ==
((PolyGamma[2 n] - Log[2 Pi q]) BernoulliB[2 n, p/q])/(2 n) -
((PolyGamma[2 n] - Log[2 Pi]) BernoulliB[2 n])/(q^(2 n) 2 n) +
(((-1)^(n + 1) Pi)/(2 Pi q)^(2 n))
Sum[Sin[(2 Pi p j)/q] PolyGamma[2 n - 1, j/q], {j, 1, q - 1}] +
(((-1)^(n + 1) 2 (2 n - 1)!)/(2 Pi q)^(2 n))
Sum[Cos[(2 Pi p j)/q] Derivative[1, 0][Zeta][2 n, j/q], {j, 1, q - 1}] +
Derivative[1][Zeta][-2 n + 1]/q^(2 n) /; Element[n, Integers] && n > 0 &&
Element[p, Integers] && 0 < p < q && Element[q, Integers] && q > 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "+", "1"]], ",", FractionBox["p", "q"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["2", "n"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["2", "\[Pi]", " ", "q"]], "]"]]]], ")"]], RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["2", "n"]], ",", FractionBox["p", "q"]]], "]"]]]], RowBox[List["2", "n"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["2", "n"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["2", "\[Pi]"]], "]"]]]], ")"]], RowBox[List["BernoulliB", "[", RowBox[List["2", "n"]], "]"]]]], RowBox[List[SuperscriptBox["q", RowBox[List["2", "n"]]], "2", "n"]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], "\[Pi]"]], SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]", " ", "q"]], ")"]], RowBox[List["2", "n"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List[RowBox[List["Sin", "[", FractionBox[RowBox[List["2", "\[Pi]", " ", "p", " ", "j"]], "q"], " ", "]"]], RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["2", "n"]], "-", "1"]], ",", FractionBox["j", "q"]]], "]"]]]]]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], "2", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "n"]], "-", "1"]], ")"]], "!"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]", " ", "q"]], ")"]], RowBox[List["2", "n"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List["q", "-", "1"]]], RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["2", "\[Pi]", " ", "p", " ", "j"]], "q"], " ", "]"]], RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["2", "n"]], ",", FractionBox["j", "q"]]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["q", RowBox[List[RowBox[List["-", "2"]], "n"]]], RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "n"]], "+", "1"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["p", "\[Element]", "Integers"]], "\[And]", RowBox[List["0", "<", "p", "<", "q"]], "\[And]", RowBox[List["q", "\[Element]", "Integers"]], "\[And]", RowBox[List["q", ">", "1"]]]]]]]]
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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> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> , </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <msup> <mi> ζ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> q </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> <mo> ( </mo> <mfrac> <mi> p </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <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> ⁢ </mo> <mi> π </mi> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> p </mi> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> j </mi> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <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> ⁢ </mo> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> p </mi> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mi> q </mi> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> ζ </mi> <semantics> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> , </mo> <mfrac> <mi> j </mi> <mi> q </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msub> </mrow> <mrow> <msup> <mi> q </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] </annotation> </semantics> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> p </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mi> p </mi> <mo> < </mo> <mi> q </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> q </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> q </mi> <mo> > </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <ci> p </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> q </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <pi /> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <sin /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> p </ci> <ci> j </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> j </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> q </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> p </ci> <ci> j </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <apply> <partialdiff /> <list> <cn type='integer'> 1 </cn> <cn type='integer'> 0 </cn> </list> <ci> Zeta </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <times /> <ci> j </ci> <apply> <power /> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> </apply> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> q </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <integers /> </apply> </apply> <apply> <in /> <ci> p </ci> <integers /> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> p </ci> <ci> q </ci> </apply> <apply> <in /> <ci> q </ci> <integers /> </apply> <apply> <gt /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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