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 Zeta

 http://functions.wolfram.com/10.02.20.0033.01

 Input Form

 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

 Standard Form

 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"]]]]]]]]

 MathML Form

 ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( 1 - 2 n , p q ) ζ ( 1 - 2 n ) q - 2 n + B TagBox["B", BernoulliB] 2 n ( p q ) ( ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) - log ( 2 π q ) ) 2 n + ( - 1 ) n + 1 π ( 2 π q ) 2 n j = 1 q - 1 sin ( 2 π p j q ) ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( j q ) + ( - 1 ) n + 1 2 ( 2 n - 1 ) ! ( 2 π q ) 2 n j = 1 q - 1 cos ( 2 π p j q ) ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( 2 n , j q ) - ( ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) - log ( 2 π ) ) B TagBox["B", BernoulliB] 2 n q 2 n 2 n /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + p TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] 0 < p < q q TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] q > 1 Condition 1 0 Zeta 1 -1 2 n p q -1 D Zeta 1 -1 2 n 1 -1 2 n q -2 n BernoulliB 2 n p q -1 PolyGamma 2 n -1 2 q 2 n -1 -1 n 1 2 q 2 n -1 j 1 q -1 2 p j q -1 PolyGamma 2 n -1 j q -1 -1 n 1 2 2 n -1 2 q 2 n -1 j 1 q -1 2 p j q -1 1 0 Zeta 2 n j q -1 -1 PolyGamma 2 n -1 2 BernoulliB 2 n q 2 n 2 n -1 n SuperPlus p 0 p q q q 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", 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_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[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"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", "\[Pi]"]], ")"]], " ", 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"]]], "]"]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "q"]], ")"]], RowBox[List["2", " ", "n"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", "2", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], "!"]]]], ")"]], " ", 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"]]], "]"]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "q"]], ")"]], RowBox[List["2", " ", "n"]]]], "+", 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"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["p", "\[Element]", "Integers"]], "&&", RowBox[List["0", "<", "p", "<", "q"]], "&&", RowBox[List["q", "\[Element]", "Integers"]], "&&", RowBox[List["q", ">", "1"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02