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 PolyGamma

 http://functions.wolfram.com/06.15.16.0020.01

 Input Form

 PolyGamma[-n, -z] == (-1)^n PolyGamma[-n, z] + (1/(n - 1)!) (-((Pi I (n - 1) (-z)^n)/n) - (2 Pi I)^(1 - n) Sum[Binomial[n - 1, j] (n - j - 1)! (-2 Pi I z)^j Zeta[n - j], {j, 0, n - 2}] + z^(n - 1) ((-1)^n Log[-(z/Pi)] + (-1)^n Log[Sin[Pi z]] - 2 (-1)^n EulerGamma - 2 (-1)^n PolyGamma[n] - Sum[((-1)^(j - 1) Binomial[n - 1, j])/(n - j - 1), {j, 0, n - 2}] + Sum[(-1)^j Binomial[n - 1, j] Sum[(Binomial[n - j - 1, k] k! PolyLog[1 + k, E^(-2 I Pi z)])/(2 Pi I z)^k, {k, 0, n - j - 1}], {j, 0, n - 2}])) /; Element[n, Integers] && n > 0 && Abs[Re[z]] < 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["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "n"]]], "n"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]], RowBox[List["1", "-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "2"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], ")"]], "j"], RowBox[List["Zeta", "[", RowBox[List["n", "-", "j"]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["Log", "[", RowBox[List["-", FractionBox["z", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], "EulerGamma"]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["PolyGamma", "[", "n", "]"]]]], "-", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "j"]], "]"]]]], RowBox[List["n", "-", "j", "-", "1"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "j"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "j", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "k"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "j", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]], "]"]]]]]]]]]]]], ")"]]]]]], ")"]]]]]]]], "/;", " ", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "<", "1"]]]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( - z ) ( - 1 ) n ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( z ) + 1 ( n - 1 ) ! ( z n - 1 ( ( - 1 ) n log ( - z π ) + ( - 1 ) n log ( sin ( π z ) ) - 2 ( - 1 ) n ψ TagBox["\[Psi]", PolyGamma] ( n ) - j = 0 n - 2 ( - 1 ) j - 1 n - j - 1 ( n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] + j = 0 n - 2 ( - 1 ) j ( n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] k = 0 n - j - 1 ( 2 π z ) - k ( n - j - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "j", "-", "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 ( - 2 π z ) - 2 ( - 1 ) n TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - ( 2 π ) 1 - n j = 0 n - 2 ( n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n - j - 1 ) ! ( - 2 π z ) j ζ ( n - j ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["n", "-", "j"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] - π ( n - 1 ) ( - z ) n n ) /; n + "\[LeftBracketingBar]" Re ( z ) "\[RightBracketingBar]" < 1 Condition PolyGamma -1 n -1 z -1 n PolyGamma -1 n z 1 n -1 -1 z n -1 -1 n -1 z -1 -1 n z -1 2 -1 n PolyGamma n -1 j 0 n -2 -1 j -1 n -1 j -1 -1 Binomial n -1 j j 0 n -2 -1 j Binomial n -1 j k 0 n -1 j -1 2 z -1 k Binomial n -1 j -1 k k PolyLog k 1 -2 z -1 2 -1 n -1 2 1 -1 n j 0 n -2 Binomial n -1 j n -1 j -1 -2 z j Zeta n -1 j -1 n -1 -1 z n n -1 n SuperPlus 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[RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "n"]]], "n"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]], RowBox[List["1", "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "2"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "j"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], ")"]], "j"], " ", RowBox[List["Zeta", "[", RowBox[List["n", "-", "j"]], "]"]]]]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Log", "[", RowBox[List["-", FractionBox["z", "\[Pi]"]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "EulerGamma"]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["PolyGamma", "[", "n", "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "j"]], "]"]]]], RowBox[List["n", "-", "j", "-", "1"]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "j"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "j", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "j", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List["k", "!"]], " ", RowBox[List["PolyLog", "[", RowBox[List[RowBox[List["1", "+", "k"]], ",", SuperscriptBox["\[ExponentialE]", RowBox[List[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["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "<", "1"]]]]]]]]]]

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

 2007-05-02

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