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 PolyGamma

 http://functions.wolfram.com/06.15.03.0057.01

 Input Form

 PolyGamma[-2 n, -(1/2)] == -((BernoulliB[2 n] Log[2])/(2^(2 n) n (-1 + 2 n)!)) - ((2^(2 n - 1) - 1) Derivative[1][Zeta][-2 n + 1])/ (2^(2 n - 1) (-1 + 2 n)!) - (1/(2^(2 n - 1) (-1 + 2 n)!)) (EulerGamma + EulerGamma/(4 n) - I Pi + Log[2] + PolyGamma[2 n] + Sum[(-1)^k Binomial[-1 + 2 n, k] 2^k (Sum[((-1 + 2^(j - k)) Binomial[k, j] PolyGamma[1 - j + k] Zeta[j - k])/2^j, {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, -1 + 2 n}]) /; Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - 2 n ) ( - 1 2 ) - B TagBox["B", BernoulliB] 2 n log ( 2 ) 2 2 n n ( 2 n - 1 ) ! - 1 2 2 n - 1 ( 2 n - 1 ) ! ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] 4 n - π + log ( 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) + k = 0 2 n - 1 ( - 1 ) k ( 2 n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] 2 k j = 0 k 2 - j ( - 1 + 2 j - 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 ) 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 ) ) - 2 2 n - 1 - 1 2 2 n - 1 ( 2 n - 1 ) ! ζ ( 1 - 2 n ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -2 n -1 1 2 -1 BernoulliB 2 n 2 2 2 n n 2 n -1 -1 -1 1 2 2 n -1 2 n -1 -1 4 n -1 -1 2 PolyGamma 2 n k 0 2 n -1 -1 k Binomial 2 n -1 k 2 k j 0 k 2 -1 j -1 2 j -1 k Binomial k j PolyGamma -1 j k 1 Zeta j -1 k -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k -1 2 2 n -1 -1 2 2 n -1 2 n -1 -1 D Zeta 1 -1 2 n 1 -1 2 n n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n_"]], ",", RowBox[List["-", FractionBox["1", "2"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "n"]], "]"]], " ", RowBox[List["Log", "[", "2", "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "n"]]], " ", "n", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]], "-", "1"]], ")"]], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], "+", "1"]], "]"]]]], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]], "-", FractionBox[RowBox[List["EulerGamma", "+", FractionBox["EulerGamma", RowBox[List["4", " ", "n"]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["Log", "[", "2", "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", "k"]], "]"]], " ", SuperscriptBox["2", "k"], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox["2", RowBox[List["-", "j"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["2", RowBox[List["j", "-", "k"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "j", "+", "k"]], "]"]], " ", RowBox[List["Zeta", "[", 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[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02