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 PolyGamma

 http://functions.wolfram.com/06.15.03.0053.01

 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)) /; Element[n, Integers] && n > 0 && Element[p, Integers] && 0 < p < q && Element[q, Integers] && q > 1

 Standard Form

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 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - 2 n ) ( - p q ) 1 ( 2 n - 1 ) ! ( ( - 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]] 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] ( - j + k + 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 ) ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + p TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] 0 < p < q q TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] q > 1 Condition PolyGamma -2 n -1 p q -1 1 2 n -1 -1 -1 n 1 2 -2 n 1 -1 2 n q -2 n j 1 q -1 PolyGamma 2 n -1 j q -1 2 j q -1 p q -1 -1 n 1 2 1 -1 2 n -2 n 2 n -1 q -2 n j 1 q -1 2 j q -1 p q -1 1 0 Zeta 2 n j q -1 D Zeta 1 -1 2 n 1 -1 2 n q -2 n -1 PolyGamma 2 n -1 2 q -2 n 2 n -1 BernoulliB 2 n PolyGamma 2 n -1 2 q 2 n -1 BernoulliB 2 n q -1 p q -1 -1 p q -1 2 n -1 p 2 n q -1 -1 -1 p q -1 PolyGamma 2 n k 0 2 n -1 -1 p q -1 -1 k Binomial 2 n -1 k j 0 k p q -1 j Binomial k j PolyGamma -1 j k 1 2 Zeta j -1 k 1 -1 p q -1 -1 Zeta j -1 k 1 -1 p q -1 -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k n SuperPlus p 0 p q q q 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n_"]], ",", RowBox[List["-", FractionBox["p_", "q_"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[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"]]], ")"]], 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02