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 PolyGamma

 http://functions.wolfram.com/06.15.03.0063.01

 Input Form

 PolyGamma[-2 n, 2/3] == (1/(-1 + 2 n)!) ((3/2)^(1 - 2 n) EulerGamma - (2^(-1 + 2 n) EulerGamma)/(9^n n) + (3/2)^(1 - 2 n) PolyGamma[2 n] + I (4/9)^n Pi - (I 2^(-1 + 2 n) Pi)/(9^n n) + 3^(1 - 2 n) 4^(-1 + n) Log[(4 Pi^2)/3] - (BernoulliB[2 n] (Sqrt[3] (-1 + 9^n) Pi + Log[729]))/ (9^n (8 n)) - (-(1/4))^n 3^(1/2 - 2 n) Pi^(1 - 2 n) PolyGamma[-1 + 2 n, 1/3] - ((1/2) (-3 + 9^n) Derivative[1][Zeta][1 - 2 n])/9^n - (3/2)^(1 - 2 n) Sum[((-1)^j Binomial[-1 + 2 n, j])/(-1 - j + 2 n), {j, 0, -2 + 2 n}] - (3/2)^(1 - 2 n) Sum[(-1)^j Binomial[-1 + 2 n, j] Sum[(Binomial[-1 - j + 2 n, k] k! PolyLog[1 + k, E^((2 I Pi)/3)])/((4 I Pi)/3)^k, {k, 0, -1 - j + 2 n}], {j, 0, -2 + 2 n}] - (3/2)^(1 - 2 n) (-1 + 2 n)! Sum[Zeta[1 + j]/((-((4 I Pi)/3))^j (-1 - j + 2 n)!), {j, 1, -1 + 2 n}] - (3/2)^(1 - 2 n) Sum[(-1)^k (3/2)^k Binomial[-1 + 2 n, k] (Sum[(2/3)^j Binomial[k, j] PolyGamma[1 - j + k] Zeta[j - k, 1/3], {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, -1 + 2 n}]) /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], ",", FractionBox["2", "3"]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " "]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", "EulerGamma"]], "-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["9", RowBox[List["-", "n"]]], " ", "EulerGamma"]], "n"], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", 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RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "j", "+", "k"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", FractionBox["1", "3"]]], "]"]]]]]], "-", 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", ">", "0"]]]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - 2 n ) ( 2 3 ) 1 ( 2 n - 1 ) ! ( - 3 1 2 - 2 n π 1 - 2 n ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( 1 3 ) ( - 1 4 ) n + 3 1 - 2 n 4 n - 1 log ( 4 π 2 3 ) + ( 3 2 ) 1 - 2 n ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) - ( 3 2 ) 1 - 2 n j = 0 2 n - 2 ( - 1 ) j ( 2 n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] - j + 2 n - 1 - ( 3 2 ) 1 - 2 n ( 2 n - 1 ) ! j = 1 2 n - 1 ( - 1 3 ( 4 π ) ) - j ζ ( j + 1 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["j", "+", "1"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] ( - j + 2 n - 1 ) ! - ( 3 2 ) 1 - 2 n j = 0 2 n - 2 ( - 1 ) j ( 2 n - 1 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", " ", "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 - j + 2 n - 1 ( 4 π 3 ) - k ( - j + 2 n - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["-", "j"]], "+", 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]] k ! Li PolyLog k + 1 ( 2 π 3 ) - ( 3 2 ) 1 - 2 n k = 0 2 n - 1 ( - 1 ) k ( 3 2 ) 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]] j = 0 k ( 2 3 ) 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] ( - j + k + 1 ) ζ ( j - k , 1 3 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[FractionBox["1", "3"], Zeta, Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$], Zeta[ZetaDump`e1\$, ZetaDump`e2\$]]]] - 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 ) - 1 2 9 - n ( - 3 + 9 n ) ζ ( 1 - 2 n ) - 2 2 n - 1 9 - n π n - 2 2 n - 1 9 - n TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] n - 9 - n B TagBox["B", BernoulliB] 2 n ( 3 π ( - 1 + 9 n ) + log ( 729 ) ) 8 n + ( 4 9 ) n π + ( 3 2 ) 1 - 2 n TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -2 n 2 3 1 2 n -1 -1 -1 3 1 2 -1 2 n 1 -1 2 n PolyGamma 2 n -1 1 3 -1 1 4 n 3 1 -1 2 n 4 n -1 4 2 3 -1 3 2 1 -1 2 n PolyGamma 2 n -1 3 2 1 -1 2 n j 0 2 n -2 -1 j Binomial 2 n -1 j -1 j 2 n -1 -1 -1 3 2 1 -1 2 n 2 n -1 j 1 2 n -1 -1 1 3 4 -1 j Zeta j 1 -1 j 2 n -1 -1 -1 3 2 1 -1 2 n j 0 2 n -2 -1 j Binomial 2 n -1 j k 0 -1 j 2 n -1 4 3 -1 -1 k Binomial -1 j 2 n -1 k k PolyLog k 1 2 3 -1 -1 3 2 1 -1 2 n k 0 2 n -1 -1 k 3 2 k Binomial 2 n -1 k j 0 k 2 3 j Binomial k j PolyGamma -1 j k 1 Zeta j -1 k 1 3 -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k -1 1 2 9 -1 n -3 9 n D Zeta 1 -1 2 n 1 -1 2 n -1 2 2 n -1 9 -1 n n -1 -1 2 2 n -1 9 -1 n n -1 -1 9 -1 n BernoulliB 2 n 3 1 2 -1 9 n 729 8 n -1 4 9 n 3 2 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_"]], ",", FractionBox["2", "3"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", "EulerGamma"]], "-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["9", RowBox[List["-", "n"]]], " ", "EulerGamma"]], "n"], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", FractionBox["4", "9"], ")"]], "n"], " ", "\[Pi]"]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02