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 PolyGamma

 http://functions.wolfram.com/06.15.06.0044.01

 Input Form

 PolyGamma[-n, z] \[Proportional] z^n Sum[(1/(z^(2 k) ((2 k)! Gamma[1 - 2 k + n]))) ((Log[z] - EulerGamma - PolyGamma[1 - 2 k + n]) BernoulliB[2 k] + (2 k - n) (((-1)^k (2 k)! Zeta[2 k + 1])/(2^(2 k + 1) Pi^(2 k) z))), {k, 1, Floor[n/2]}] + z^n Sum[(((-1)^(n - 1) (2 k - n - 1)!)/(z^(2 k) (2 k)!)) BernoulliB[2 k], {k, Floor[n/2] + 1, Infinity}] + (z^(n - 2)/2) Sum[((2 k - n + 1)/(z^(2 k) ((2 k + 1)! (k + 1) (n - 2 k - 1)!))) (2 (k + 1) Derivative[1][Zeta][-2 k - 1] - BernoulliB[2 k + 2] HarmonicNumber[2 k + 1]), {k, 1, Floor[(n - 1)/2]}] - (z^(n - 1) (2 z HarmonicNumber[n] - n Log[2 Pi] - 2 z Log[z]))/(2 n!) + (z^(-2 + n)/(2 (n - 1)!)) (-2 (1 - n) Log[Glaisher] + z (EulerGamma - Log[z] + PolyGamma[n])) /; (Abs[z] -> Infinity) && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", RowBox[List["n", "/", "2"]], "]"]]], RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["2", "k"]], "+", "n"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", "EulerGamma", "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", RowBox[List["2", "k"]], "+", "n"]], "]"]]]], ")"]], RowBox[List["BernoulliB", "[", RowBox[List["2", "k"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "n"]], ")"]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!"]], RowBox[List["Zeta", "[", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], SuperscriptBox["\[Pi]", RowBox[List["2", "k"]]], "z"]]]]]]], ")"]]]]]]]], "+", RowBox[List[SuperscriptBox["z", "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List[RowBox[List["Floor", "[", RowBox[List["n", "/", "2"]], "]"]], "+", "1"]]]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "n", "-", "1"]], ")"]], "!"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!"]]], RowBox[List["BernoulliB", "[", RowBox[List["2", "k"]], "]"]]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["n", "-", "2"]]], "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["Floor", "[", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "/", "2"]], "]"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "n", "+", "1"]], ")"]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "!"]], RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", "k"]], "-", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[RowBox[List[RowBox[List["Derivative", "[", "1", "]"]], "[", "Zeta", "]"]], "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "k"]], "-", "1"]], "]"]]]], "-", RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List[RowBox[List["2", "k"]], "+", "2"]], "]"]], RowBox[List["HarmonicNumber", "[", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], "]"]]]]]], ")"]]]]]]]], "-", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "z", " ", RowBox[List["HarmonicNumber", "[", "n", "]"]]]], "-", RowBox[List["n", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]], "-", RowBox[List["2", " ", "z", " ", RowBox[List["Log", "[", "z", "]"]]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["n", "!"]]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "+", "n"]]], RowBox[List["2", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "-", "n"]], ")"]], " ", RowBox[List["Log", "[", "Glaisher", "]"]]]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["EulerGamma", "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( z ) - z n - 1 2 n ! ( 2 z H HarmonicNumber n - n log ( 2 π ) - 2 z log ( z ) ) + z n k = 1 n 2 z - 2 k ( 2 k ) ! Γ ( 1 - 2 k + n ) ( ( - 1 ) k ( 2 k ) ! ( 2 k - n ) 2 2 k + 1 π 2 k z ζ ( 2 k + 1 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] + B TagBox["B", BernoulliB] 2 k ( log ( z ) - ψ TagBox["\[Psi]", PolyGamma] ( 1 - 2 k + n ) - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ) + z n k = n 2 + 1 ( - 1 ) n - 1 ( 2 k - n - 1 ) ! ( 2 k ) ! B TagBox["B", BernoulliB] 2 k z - 2 k + z n - 2 2 ( n - 1 ) ! ( z ( - log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - 2 ( 1 - n ) log ( A TagBox["A", Function[List[], Glaisher]] ) ) + 1 2 z n - 2 k = 1 n - 1 2 ( 2 k - n + 1 ) z - 2 k ( 2 k + 1 ) ! ( k + 1 ) ( n - 2 k - 1 ) ! ( 2 ( k + 1 ) ζ ( - 2 k - 1 ) - B TagBox["B", BernoulliB] 2 k + 2 H HarmonicNumber 2 k + 1 ) /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n + Condition Proportional PolyGamma -1 n z -1 z n -1 2 n -1 2 z HarmonicNumber n -1 n 2 -1 2 z z z n k 1 n 2 -1 z -2 k 2 k Gamma 1 -1 2 k n -1 -1 k 2 k 2 k -1 n 2 2 k 1 2 k z -1 Zeta 2 k 1 BernoulliB 2 k z -1 PolyGamma 1 -1 2 k n -1 z n k n 2 -1 1 -1 n -1 2 k -1 n -1 2 k -1 BernoulliB 2 k z -2 k z n -2 2 n -1 -1 z -1 z PolyGamma n -1 2 1 -1 n Glaisher 1 2 z n -2 k 1 n -1 2 -1 2 k -1 n 1 z -2 k 2 k 1 k 1 n -1 2 k -1 -1 2 k 1 D Zeta -2 k -1 -2 k -1 -1 BernoulliB 2 k 2 HarmonicNumber 2 k 1 Rule z n SuperPlus [/itex]

 Rule Form

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

 2007-05-02