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 PolyGamma

 http://functions.wolfram.com/06.15.03.0041.01

 Input Form

 PolyGamma[-n, z] == (z^(n - 1)/(n - 1)!) (EulerGamma - (EulerGamma z)/n - Log[z] + PolyGamma[n] + Sum[(Binomial[n - 1, k]/z^k) (Sum[Binomial[k, j] PolyGamma[k - j + 1] (Zeta[j - k, z + 1 + Max[Floor[-Re[z]], 0]] + Sum[(z + i + 1)^(k - j), {i, 0, Floor[-Re[z]] - 1}]) (-z)^j, {j, 0, k}] - PolyGamma[k + 1] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, n - 1}]) + (1/(n - 1)!) (Derivative[1, 0][Zeta][1 - n, z + 1 + Max[Floor[-Re[z]], 0]] - Sum[Log[z + i + 1]/(z + i + 1)^(1 - n), {i, 0, Floor[-Re[z]] - 1}]) /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "-", FractionBox[RowBox[List["EulerGamma", " ", "z"]], "n"], "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]], " "]], SuperscriptBox["z", "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["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["z", "+", "1", "+", RowBox[List["Max", "[", RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], ",", "0"]], "]"]]]]]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "i", "+", "1"]], ")"]], RowBox[List["k", "-", "j"]]]]]]], ")"]], 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[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", RowBox[List["1", ",", "0"]], "]"]], "[", "Zeta", "]"]], "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", RowBox[List["z", "+", "1", "+", RowBox[List["Max", "[", RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], ",", "0"]], "]"]]]]]], "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "-", "1"]]], FractionBox[RowBox[List["Log", "[", RowBox[List["z", "+", "i", "+", "1"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "i", "+", "1"]], ")"]], RowBox[List["1", "-", "n"]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( z ) 1 ( n - 1 ) ! ( ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( 1 - n , z + max ( - Re ( z ) , 0 ) + 1 ) - i = 0 - Re ( z ) - 1 log ( i + z + 1 ) ( i + z + 1 ) 1 - n ) + z n - 1 ( n - 1 ) ! ( - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] z n - log ( z ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) + k = 0 n - 1 1 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]] ( 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] ( k - j + 1 ) ( i = 0 - Re ( z ) - 1 ( i + z + 1 ) k - j + ζ ( j - k , z + max ( - Re ( z ) , 0 ) + 1 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[RowBox[List["z", "+", RowBox[List["max", "(", RowBox[List[RowBox[List["\[LeftFloor]", RowBox[List["-", RowBox[List["Re", "(", "z", ")"]]]], "\[RightFloor]"]], ",", "0"]], ")"]], "+", "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 ) ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -1 n z 1 n -1 -1 1 0 Zeta 1 -1 n z -1 z 0 1 -1 i 0 -1 z -1 i z 1 i z 1 1 -1 n -1 z n -1 n -1 -1 -1 z n -1 -1 z PolyGamma n k 0 n -1 1 z k -1 Binomial n -1 k j 0 k Binomial k j PolyGamma k -1 j 1 i 0 -1 z -1 i z 1 k -1 j Zeta j -1 k z -1 z 0 1 -1 z j -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k 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["EulerGamma", "-", FractionBox[RowBox[List["EulerGamma", " ", "z"]], "n"], "-", RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "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["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["z", "+", "1", "+", RowBox[List["Max", "[", RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], ",", "0"]], "]"]]]]]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "i", "+", "1"]], ")"]], RowBox[List["k", "-", "j"]]]]]]], ")"]], " ", 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"]], "]"]]]], ")"]]]], SuperscriptBox["z", "k"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", RowBox[List["z", "+", "1", "+", RowBox[List["Max", "[", RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], ",", "0"]], "]"]]]]]], "]"]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], "-", "1"]]], FractionBox[RowBox[List["Log", "[", RowBox[List["z", "+", "i", "+", "1"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "i", "+", "1"]], ")"]], RowBox[List["1", "-", "n"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02