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 PolyGamma

 http://functions.wolfram.com/06.15.16.0021.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["-", "z"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "2"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "p"]], ")"]], "k"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "p", "+", "1"]], ")"]], "k"]]], RowBox[List["k", "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "k", "-", "2"]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["j", "!"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["j", "+", "k", "-", "n"]], ",", "1"]], "]"]]]]]]]]]], "+", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "p"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "p"]], "+", "z"]], "]"]], "-", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "p", "+", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "1"]], "+", "p", "-", "z"]], "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "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]", " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], ")"]]]], "n"], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], "\[Pi]", " ", "\[ImaginaryI]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "-", "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]", " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "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["2", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "-", "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["Re", "[", "z", "]"]], "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["-", "z"]]]], "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "p"]], ")"]], "k"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "p", "+", "1"]], ")"]], "k"]]], ")"]], " ", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02