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 PolyGamma

 http://functions.wolfram.com/06.15.03.0050.01

 Input Form

 PolyGamma[-n, -m] == (1/(n - 1)!) Derivative[1, 0][Zeta][1 - n, 1 - m] - (1/(n - 1)!) Sum[(1 + i - m)^(n - 1) (I Pi + (1 + (-1)^n) Log[m - 1 - i]), {i, 0, m - 2}] + ((-m)^(n - 1)/(n - 1)!) (EulerGamma + (EulerGamma m)/n - I Pi - Log[m] + PolyGamma[n] + Sum[((-1)^k Binomial[n - 1, k] (Sum[m^j Binomial[k, j] PolyGamma[1 - j + k] (Zeta[j - k] + Sum[(1 + i - m)^(-j + k), {i, 0, m - 2}]), {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]))/m^k, {k, 0, n - 1}]) /; Element[n, Integers] && n > 1 && Element[m, Integers] && m > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n"]], ",", RowBox[List["-", "m"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "-", "m"]]]], "]"]]]], "-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["m", "-", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "i", "-", "m"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["m", "-", "1", "-", "i"]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["n", "-", "1"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox[RowBox[List["EulerGamma", " ", "m"]], "n"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "-", RowBox[List["Log", "[", "m", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["m", RowBox[List["-", "k"]]], " ", 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[SuperscriptBox["m", "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "j", "+", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["j", "-", "k"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["m", "-", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "i", "-", "m"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]]]]]], ")"]]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "k"]], "]"]]]], "-", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "k"]], "]"]]]], ")"]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "1"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - n ) ( - m ) 1 ( n - 1 ) ! ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( 1 - n , 1 - m ) - 1 ( n - 1 ) ! i = 0 m - 2 ( i - m + 1 ) n - 1 ( π + ( 1 + ( - 1 ) n ) log ( m - i - 1 ) ) + ( - m ) n - 1 ( n - 1 ) ! ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] m n - π - log ( m ) + ψ TagBox["\[Psi]", PolyGamma] ( n ) + k = 0 n - 1 ( - 1 ) k m - 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 m 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]] ψ TagBox["\[Psi]", PolyGamma] ( k - j + 1 ) ( i = 0 m - 2 ( i - m + 1 ) k - j + ζ ( j - k ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] ) - 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 ) ) ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n > 1 m TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -1 n -1 m 1 n -1 -1 1 0 Zeta 1 -1 n 1 -1 m -1 1 n -1 -1 i 0 m -2 i -1 m 1 n -1 1 -1 n m -1 i -1 -1 m n -1 n -1 -1 m n -1 -1 -1 m PolyGamma n k 0 n -1 -1 k m -1 k Binomial n -1 k j 0 k m j Binomial k j PolyGamma k -1 j 1 i 0 m -2 i -1 m 1 k -1 j Zeta j -1 k -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k n n 1 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List["-", "n_"]], ",", RowBox[List["-", "m_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "-", "m"]]]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["m", "-", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "i", "-", "m"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["m", "-", "1", "-", "i"]], "]"]]]]]], ")"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox[RowBox[List["EulerGamma", " ", "m"]], "n"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "-", RowBox[List["Log", "[", "m", "]"]], "+", RowBox[List["PolyGamma", "[", "n", "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["m", RowBox[List["-", "k"]]], " ", 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[SuperscriptBox["m", "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "j", "+", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["j", "-", "k"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["m", "-", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "i", "-", "m"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]]]]]], ")"]]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], " ", 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"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "1"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2007-05-02