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 PolyGamma

 http://functions.wolfram.com/06.15.06.0043.01

 Input Form

 PolyGamma[-n, z] \[Proportional] (z^(n - 1)/(n - 1)!) (EulerGamma (1 - z/n) - Log[z] + PolyGamma[n] + Sum[(Binomial[n - 1, k] (Sum[Binomial[k, j] PolyGamma[k - j + 1] Zeta[j - k, 1 + z] (-z)^j, {j, 0, k}] - PolyGamma[k + 1] Zeta[-k] - Derivative[1][Zeta][-k]))/z^k, {k, 0, n - 1}]) - (1 + z)^(-1 + n)/(2 n!) + (Log[1 + z] - 1/n) (BernoulliB[n, 1 + z]/n!) + (-1)^(-1 + n) Sum[(BernoulliB[k] (1 + z)^(-k + n))/ Pochhammer[k - n, 1 + n], {k, 1 + n, Infinity}] - (1/n!) Sum[BernoulliB[k] Binomial[n, k] (1 + z)^(n - k) Sum[(-1)^j/(k - j), {j, 0, k - 1}], {k, 2, n}] /; Abs[Arg[z]] <= Pi/2 && (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[FractionBox[SuperscriptBox["z", RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List["EulerGamma", RowBox[List["(", RowBox[List["1", "-", FractionBox["z", "n"]]], ")"]]]], "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]], SuperscriptBox["z", RowBox[List["-", "k"]]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "j", "+", "1"]], "]"]], RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["1", "+", "z"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "j"]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "k"]], "]"]]]], "-", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "k"]], "]"]]]], ")"]]]]]]]], ")"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List["2", RowBox[List["n", "!"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", FractionBox["1", "n"]]], ")"]], FractionBox[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", RowBox[List["1", "+", "z"]]]], "]"]], RowBox[List["n", "!"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["BernoulliB", "[", "k", "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "+", "n"]]], " "]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["k", "-", "n"]], ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List["n", "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "n"], RowBox[List[RowBox[List["BernoulliB", "[", "k", "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["n", "-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["k", "-", "j"]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]], "\[And]", 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 ) 1 n ! ( log ( z + 1 ) - 1 n ) B TagBox["B", BernoulliB] n ( z + 1 ) + ( - 1 ) n - 1 k = n + 1 B TagBox["B", BernoulliB] k ( z + 1 ) n - k ( k - n ) n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["k", "-", "n"]], ")"]], RowBox[List["n", "+", "1"]]], Pochhammer] - 1 n ! k = 2 n B TagBox["B", BernoulliB] k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( z + 1 ) n - k j = 0 k - 1 ( - 1 ) j k - j + z n - 1 ( n - 1 ) ! ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ( 1 - z n ) - log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) + k = 0 n - 1 ( 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]] z - k ( j = 0 k ( 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]] ψ TagBox["\[Psi]", PolyGamma] ( - j + k + 1 ) ζ ( j - k , z + 1 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[RowBox[List["z", "+", "1"]], Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$], Zeta[ZetaDump`e1\$, ZetaDump`e2\$]]]] ( - z ) j - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) ζ ( - k ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["-", "k"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] - ζ ( - k ) ) ) - ( z + 1 ) n - 1 2 n ! /; "\[LeftBracketingBar]" arg ( z ) "\[RightBracketingBar]" π 2 ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n + Condition Proportional PolyGamma -1 n z 1 n -1 z 1 -1 1 n -1 BernoulliB n z 1 -1 n -1 k n 1 BernoulliB k z 1 n -1 k Pochhammer k -1 n n 1 -1 -1 1 n -1 k 2 n BernoulliB k Binomial n k z 1 n -1 k j 0 k -1 -1 j k -1 j -1 z n -1 n -1 -1 1 -1 z n -1 -1 z PolyGamma n k 0 n -1 Binomial n -1 k z -1 k j 0 k Binomial k j PolyGamma -1 j k 1 Zeta j -1 k z 1 -1 z j -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k -1 z 1 n -1 2 n -1 z 2 -1 Rule z n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n_"]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["1", "-", FractionBox["z", "n"]]], ")"]]]], "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "j", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["1", "+", "z"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], "j"]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "k"]], "]"]]]], "-", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "k"]], "]"]]]], ")"]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], RowBox[List["2", " ", RowBox[List["n", "!"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["1", "+", "z"]], "]"]], "-", FractionBox["1", "n"]]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n", ",", RowBox[List["1", "+", "z"]]]], "]"]]]], RowBox[List["n", "!"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["BernoulliB", "[", "k", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List[RowBox[List["-", "k"]], "+", "n"]]]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["k", "-", "n"]], ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "2"]], "n"], RowBox[List[RowBox[List["BernoulliB", "[", "k", "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["n", "-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["k", "-", "j"]]]]]]]]], RowBox[List["n", "!"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", "z", "]"]], "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02