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 PolyGamma

 http://functions.wolfram.com/06.15.03.0052.01

 Input Form

 PolyGamma[-2 n, p/q] == (1/(-1 + 2 n)!) (((-(p/q)^(-1 + 2 n)) (EulerGamma + (EulerGamma p)/(2 n q) - Pi I - Log[p/q] + PolyGamma[2 n] + Sum[(Binomial[-1 + 2 n, k] (Sum[(p/q)^j Binomial[k, j] PolyGamma[1 - j + k] (2 Zeta[j - k, 1 - p/q] - Zeta[j - k, 1 - p/q]), {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]))/(-(p/q))^k, {k, 0, -1 + 2 n}]) - (BernoulliB[2 n] (-Log[2 Pi] + PolyGamma[2 n]))/ (q^(2 n) (2 n)) + (BernoulliB[2 n, (-p + q)/q] (-Log[2 Pi q] + PolyGamma[2 n]))/(2 n) + ((-1)^(1 + n) Pi^(1 - 2 n) Sum[PolyGamma[-1 + 2 n, j/q] Sin[(2 j Pi (-p + q))/q], {j, 1, -1 + q}])/(2^(2 n) q^(2 n)) + ((-1)^(1 + n) 2^(1 - 2 n) (-1 + 2 n)! Sum[Cos[(2 j Pi (-p + q))/q] Derivative[1, 0][Zeta][2 n, j/q], {j, 1, -1 + q}])/ (Pi^(2 n) q^(2 n)) + Derivative[1][Zeta][1 - 2 n]/q^(2 n)) - (I (-1 + 2 n) Pi (p/q)^(2 n))/(2 n) - (p/q)^(-1 + 2 n) (-2 EulerGamma + Log[p/(Pi q)] + Pi I + Log[Sin[(p Pi)/q]] - 2 PolyGamma[2 n] + Sum[((-1)^j Binomial[-1 + 2 n, j])/(-1 - j + 2 n), {j, 0, -2 + 2 n}] + Sum[(-1)^j Binomial[-1 + 2 n, j] Sum[(Binomial[-1 - j + 2 n, k] k! PolyLog[1 + k, E^((2 I p Pi)/q)])/ (-((2 Pi I p)/q))^k, {k, 0, -1 - j + 2 n}], {j, 0, -2 + 2 n}]) - (2 I Pi)^(1 - 2 n) Sum[((2 Pi I p)/q)^j Binomial[-1 + 2 n, j] (-1 - j + 2 n)! Zeta[-j + 2 n], {j, 0, -2 + 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["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], ",", FractionBox["p", "q"]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["p", "q"], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox[RowBox[List["EulerGamma", " ", "p"]], RowBox[List["2", " ", "n", " ", "q"]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List["Log", "[", FractionBox["p", "q"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", 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RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]], "]"]]]]]], ")"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", FractionBox["p", "q"], ")"]], RowBox[List["2", " ", "n"]]]]], RowBox[List["2", " ", "n"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["p", "q"], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "EulerGamma"]], "+", " ", RowBox[List["Log", "[", FractionBox["p", RowBox[List["\[Pi]", " ", "q"]]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", FractionBox[RowBox[List["p", " ", "\[Pi]"]], "q"], "]"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", 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RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "\[Pi]"]], "q"]]]], "]"]]]]]]]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "j"]], "+", RowBox[List["2", " ", "n"]]]], "]"]]]]]]]]]], ")"]]]]]], "/;", 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

 ψ TagBox["\[Psi]", PolyGamma] ( - 2 n ) ( p q ) 1 ( 2 n - 1 ) ! ( - ( 2 n - 1 ) π 2 n ( p q ) 2 n + ( - p q ) 2 n - 1 ( log ( p π q ) + π + log ( sin ( p π q ) ) - 2 ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) - 2 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + j = 0 2 n - 2 ( - 1 ) j 2 n - j - 1 ( 2 n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] + j = 0 2 n - 2 ( - 1 ) j ( 2 n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] k = 0 2 n - j - 1 ( - 2 π p q ) - k ( 2 n - j - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", "n"]], "-", "j", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] k ! Li PolyLog k + 1 ( 2 p π q ) ) - ( 2 π ) 1 - 2 n j = 0 2 n - 2 ( 2 π p q ) j ( 2 n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 2 n - j - 1 ) ! ζ ( 2 n - j ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "j"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] + ( ( - 1 ) n + 1 2 - 2 n π 1 - 2 n q - 2 n j = 1 q - 1 ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( j q ) sin ( 2 j π ( q - p ) q ) + ( - 1 ) n + 1 2 1 - 2 n π - 2 n ( 2 n - 1 ) ! q - 2 n j = 1 q - 1 cos ( 2 j π ( q - p ) q ) ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( 2 n , j q ) + ζ ( 1 - 2 n ) q - 2 n - ( ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) - log ( 2 π ) ) q - 2 n 2 n B TagBox["B", BernoulliB] 2 n + ( ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) - log ( 2 π q ) 2 n B TagBox["B", BernoulliB] 2 n ( q - p q ) - ( p q ) 2 n - 1 ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] p 2 n q - π - log ( p q ) + ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) + k = 0 2 n - 1 ( - p q ) - k ( 2 n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( j = 0 k ( p q ) j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ψ TagBox["\[Psi]", PolyGamma] ( k - j + 1 ) ( 2 ζ ( j - k , 1 - p q ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["p", "q"]]], Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$], Zeta[ZetaDump`e1\$, ZetaDump`e2\$]]]] - ζ ( j - k , 1 - p q ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[RowBox[List["1", "-", FractionBox["p", "q"]]], Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$], Zeta[ZetaDump`e1\$, ZetaDump`e2\$]]]] ) - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) ζ ( - k ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["-", "k"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] - ζ ( - k ) ) ) ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + p TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] 0 < p < q q TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] q > 1 FormBox RowBox RowBox RowBox SuperscriptBox TagBox ψ PolyGamma RowBox ( RowBox RowBox - 2 n ) ( FractionBox p q ) RowBox FractionBox 1 RowBox RowBox ( RowBox RowBox 2 n - 1 ) ! RowBox ( RowBox RowBox RowBox - FractionBox RowBox RowBox ( RowBox RowBox 2 n - 1 ) π RowBox 2 n SuperscriptBox RowBox ( FractionBox p q ) RowBox 2 n + RowBox SuperscriptBox RowBox ( RowBox - FractionBox p q ) RowBox RowBox 2 n - 1 RowBox ( RowBox RowBox log ( FractionBox p RowBox π q ) + RowBox π + RowBox log ( RowBox sin ( FractionBox RowBox p π q ) ) - RowBox 2 RowBox TagBox ψ PolyGamma ( RowBox 2 n ) - RowBox 2 TagBox Function + RowBox UnderoverscriptBox RowBox j = 0 RowBox RowBox 2 n - 2 RowBox FractionBox SuperscriptBox RowBox ( RowBox - 1 ) j RowBox RowBox 2 n - j - 1 TagBox RowBox ( GridBox TagBox RowBox RowBox 2 n - 1 Rule Editable TagBox j Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable + RowBox UnderoverscriptBox RowBox j = 0 RowBox RowBox 2 n - 2 RowBox SuperscriptBox RowBox ( RowBox - 1 ) j TagBox RowBox ( GridBox TagBox RowBox RowBox 2 n - 1 Rule Editable TagBox j Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox UnderoverscriptBox RowBox k = 0 RowBox RowBox 2 n - j - 1 RowBox SuperscriptBox RowBox ( RowBox - FractionBox RowBox 2 π p q ) RowBox - k TagBox RowBox ( GridBox TagBox RowBox RowBox 2 n - j - 1 Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox k ! RowBox SubscriptBox InterpretationBox Li PolyLog Rule Editable Rule Selectable RowBox k + 1 ( SuperscriptBox FractionBox RowBox 2 p π q ) ) - RowBox SuperscriptBox RowBox ( RowBox 2 π ) RowBox 1 - RowBox 2 n RowBox UnderoverscriptBox RowBox j = 0 RowBox RowBox 2 n - 2 RowBox SuperscriptBox RowBox ( FractionBox RowBox 2 π p q ) j TagBox RowBox ( GridBox TagBox RowBox RowBox 2 n - 1 Rule Editable TagBox j Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox RowBox ( RowBox RowBox 2 n - j - 1 ) ! TagBox RowBox ζ ( TagBox RowBox RowBox 2 n - j Zeta Rule Editable ) InterpretTemplate Function BoxForm`e\$ Zeta BoxForm`e\$ + RowBox ( RowBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox n + 1 SuperscriptBox 2 RowBox RowBox - 2 n SuperscriptBox π RowBox 1 - RowBox 2 n SuperscriptBox q RowBox RowBox - 2 n RowBox UnderoverscriptBox RowBox j = 1 RowBox q - 1 RowBox RowBox SuperscriptBox TagBox ψ PolyGamma RowBox ( RowBox RowBox 2 n - 1 ) ( FractionBox j q ) RowBox sin ( FractionBox RowBox 2 j π RowBox ( RowBox q - p ) q ) + RowBox SuperscriptBox RowBox ( RowBox - 1 ) RowBox n + 1 SuperscriptBox 2 RowBox 1 - RowBox 2 n SuperscriptBox π RowBox RowBox - 2 n RowBox RowBox ( RowBox RowBox 2 n - 1 ) ! SuperscriptBox q RowBox RowBox - 2 n RowBox UnderoverscriptBox RowBox j = 1 RowBox q - 1 RowBox RowBox cos ( FractionBox RowBox 2 j π RowBox ( RowBox q - p ) q ) RowBox SuperscriptBox ζ TagBox RowBox ( RowBox 1 , 0 ) Derivative ( RowBox RowBox 2 n , FractionBox j q ) + RowBox RowBox SuperscriptBox ζ ( RowBox 1 - RowBox 2 n ) SuperscriptBox q RowBox RowBox - 2 n - RowBox FractionBox RowBox RowBox ( RowBox RowBox TagBox ψ PolyGamma ( RowBox 2 n ) - RowBox log ( RowBox 2 π ) ) SuperscriptBox q RowBox RowBox - 2 n RowBox 2 n SubscriptBox TagBox B BernoulliB RowBox 2 n + RowBox FractionBox ErrorBox RowBox ( RowBox RowBox TagBox ψ PolyGamma ( RowBox 2 n ) - RowBox log ( RowBox 2 π q ) RowBox 2 n RowBox SubscriptBox TagBox B BernoulliB RowBox 2 n ( FractionBox RowBox q - p q ) - RowBox SuperscriptBox RowBox ( FractionBox p q ) RowBox RowBox 2 n - 1 RowBox ( RowBox TagBox Function + FractionBox RowBox TagBox Function p RowBox 2 n q - RowBox π - RowBox log ( FractionBox p q ) + RowBox TagBox ψ PolyGamma ( RowBox 2 n ) + RowBox UnderoverscriptBox RowBox k = 0 RowBox RowBox 2 n - 1 RowBox SuperscriptBox RowBox ( RowBox - FractionBox p q ) RowBox - k TagBox RowBox ( GridBox TagBox RowBox RowBox 2 n - 1 Rule Editable TagBox k Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox ( RowBox RowBox UnderoverscriptBox RowBox j = 0 k RowBox SuperscriptBox RowBox ( FractionBox p q ) j TagBox RowBox ( GridBox TagBox k Rule Editable TagBox j Rule Editable ) InterpretTemplate Function Binomial Slot 1 Slot 2 Rule Editable RowBox TagBox ψ PolyGamma ( RowBox k - j + 1 ) RowBox ( RowBox RowBox 2 TagBox RowBox ζ ( RowBox TagBox RowBox j - k Zeta Rule Editable , TagBox RowBox 1 - FractionBox p q Zeta Rule Editable ) InterpretTemplate ZetaDump`e1\$ ZetaDump`e2\$ Zeta ZetaDump`e1\$ ZetaDump`e2\$ - TagBox RowBox ζ ( RowBox TagBox RowBox j - k Zeta Rule Editable , TagBox RowBox 1 - FractionBox p q Zeta Rule Editable ) InterpretTemplate ZetaDump`e1\$ ZetaDump`e2\$ Zeta ZetaDump`e1\$ ZetaDump`e2\$ ) - RowBox RowBox TagBox ψ PolyGamma ( RowBox k + 1 ) TagBox RowBox ζ ( TagBox RowBox - k Zeta Rule Editable ) InterpretTemplate Function BoxForm`e\$ Zeta BoxForm`e\$ - RowBox SuperscriptBox ζ ( RowBox - k ) ) ) ) ) /; RowBox RowBox n SuperscriptBox TagBox Function + RowBox p TagBox Function RowBox 0 < p < q RowBox q TagBox Function RowBox q > 1 TraditionalForm [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n_"]], ",", FractionBox["p_", "q_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", FractionBox["p", "q"], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox[RowBox[List["EulerGamma", " ", "p"]], RowBox[List["2", " ", "n", " ", "q"]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "-", RowBox[List["Log", "[", FractionBox["p", "q"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["p", "q"]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["p", "q"], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "j", "+", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["1", "-", FractionBox["p", "q"]]]]], "]"]]]], "-", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["1", "-", FractionBox["p", "q"]]]]], "]"]]]], ")"]]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "k"]], "]"]]]], "-", 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RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "+", "n"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["q", RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "q"]]], RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", FractionBox["j", "q"]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["2", " ", "j", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "p"]], "+", "q"]], ")"]]]], "q"], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "+", "n"]]], " ", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", SuperscriptBox["q", RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "q"]]], RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["2", " ", "j", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "p"]], "+", "q"]], ")"]]]], "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["1", "-", RowBox[List["2", " ", "n"]]]], "]"]]]]]], ")"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", FractionBox["p", "q"], ")"]], RowBox[List["2", " ", "n"]]]]], RowBox[List["2", " ", "n"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["p", "q"], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "EulerGamma"]], "+", RowBox[List["Log", "[", FractionBox["p", RowBox[List["\[Pi]", " ", "q"]]], "]"]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["Log", "[", RowBox[List["Sin", "[", FractionBox[RowBox[List["p", " ", "\[Pi]"]], "q"], "]"]], "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "n"]]]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", "j"]], "]"]]]], RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"]]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "\[Pi]"]], "q"]]]], "]"]]]]]]]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]"]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "p"]], "q"], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "j", "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["-", "j"]], "+", RowBox[List["2", " ", "n"]]]], "]"]]]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], "/;", 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