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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

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 MathML Form

 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"]]], 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02