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 PolyGamma

 http://functions.wolfram.com/06.15.16.0019.01

 Input Form

 PolyGamma[-n, -z] == (-1)^n PolyGamma[-n, z] + ((-z)^(-1 + n)/(n - 1)!) (EulerGamma + (I Pi z)/n + 2 I Pi Floor[3/4 - Arg[z]/(2 Pi)] + PolyGamma[n] + Log[-2 Pi I] - Log[-z] - Sum[(Binomial[-1 + n, k] k! PolyLog[1 + k, 1])/(2 Pi I z)^k, {k, 1, -1 + n}] + Sum[(-1)^k Binomial[-1 + n, k] Sum[(Binomial[k, j] j! PolyLog[1 + j, E^(2 I Pi z)])/(-2 Pi I z)^j, {j, 0, k}], {k, 0, -1 + n}]) /; Element[n, Integers] && n > 0 && (Inequality[0, Less, Arg[z], LessEqual, Pi] || Inequality[-1, Less, Re[z], LessEqual, 1])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "z"]], "n"], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], "\[Pi]", " ", "\[ImaginaryI]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", "1"]], "]"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["1", "+", "j"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]], "]"]]]]]]]]]]]], ")"]]]]]]]], "/;", " ", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "\[Or]", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["Re", "[", "z", "]"]], "\[LessEqual]", "1"]]]], ")"]]]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( - z ) ( - 1 ) n ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( z ) + ( - z ) n - 1 ( n - 1 ) ! ( π z n + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + 2 π 3 4 - arg ( z ) 2 π + log ( - 2 π ) - log ( - z ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) - k = 1 n - 1 ( 2 π z ) - k ( n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "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 ( 1 ) + k = 0 n - 1 ( - 1 ) k ( n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["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 ( - 2 π z ) - 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]] j ! Li PolyLog j + 1 ( 2 π z ) ) /; n + ( 0 < arg ( z ) π - 1 < Re ( z ) 1 ) Condition PolyGamma -1 n -1 z -1 n PolyGamma -1 n z -1 z n -1 n -1 -1 z n -1 2 3 4 -1 z 2 -1 -2 -1 -1 z PolyGamma n -1 k 1 n -1 2 z -1 k Binomial n -1 k k PolyLog k 1 1 k 0 n -1 -1 k Binomial n -1 k j 0 k -2 z -1 j Binomial k j j PolyLog j 1 2 z n SuperPlus Inequality 0 z Inequality -1 z 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n_"]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "z"]], "n"], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", "1"]], "]"]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "j"]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["1", "+", "j"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]], "]"]]]]]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "||", RowBox[List[RowBox[List["-", "1"]], "<", RowBox[List["Re", "[", "z", "]"]], "\[LessEqual]", "1"]]]], ")"]]]]]]]]]]

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

 2007-05-02