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 PolyGamma

 http://functions.wolfram.com/06.15.03.0061.01

 Input Form

 PolyGamma[-2 n, 1/3] == (1/(3^(4 n) Pi^(2 n) (8 n (-1 + 2 n)!))) (4 EulerGamma (6 n - 1) (3 Pi)^(2 n) - 4 Pi I (2 n - 1) (3 Pi)^(2 n) + 3 n (3 Pi)^(2 n) Log[(256 Pi^8)/81] + (3 Pi)^(2 n) BernoulliB[2 n] (Sqrt[3] (-1 + 9^n) Pi - 6 Log[3]) + 24 n (3 Pi)^(2 n) PolyGamma[2 n] + (-1)^n 2^(3 - 2 n) 3^(1/2 + 2 n) n Pi PolyGamma[-1 + 2 n, 1/3] + 4 (3 - 9^n) n (3 Pi)^(2 n) Derivative[1][Zeta][1 - 2 n]) - (3^(1 - 2 n)/(-1 + 2 n)!) (Sum[((-1)^j Binomial[-1 + 2 n, j])/ (-1 - j + 2 n), {j, 0, -2 + 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)])/(-((2 Pi I)/3))^k, {k, 0, -1 - j + 2 n}], {j, 0, -2 + 2 n}] + Sum[(-1)^k Binomial[-1 + 2 n, k] 3^k (Sum[(Binomial[k, j] PolyGamma[1 - j + k] Zeta[j - k, 2/3])/3^j, {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, -1 + 2 n}]) - 3^(1 - 2 n) Sum[Zeta[1 + j]/(((2 Pi I)/3)^j (-1 - j + 2 n)!), {j, 1, -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["1", "3"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["3", RowBox[List[RowBox[List["-", "4"]], "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "2"]], " ", "n"]]], " "]], RowBox[List["8", " ", "n", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", "n"]], "-", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["3", "\[Pi]"]], ")"]], RowBox[List["2", " ", "n"]]]]], "-", RowBox[List["4", "\[Pi]", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["3", "\[Pi]"]], ")"]], RowBox[List["2", " ", "n"]]]]], 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 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - 2 n ) ( 1 3 ) - 3 1 - 2 n j = 1 2 n - 1 ( 2 π 3 ) - 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 1 - 2 n ( 2 n - 1 ) ! ( 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 + 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 ( - 1 3 ( 2 π ) ) - 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 ) + 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]] 3 k ( j = 0 k 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 , 2 3 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[FractionBox["2", "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 ) ) ) + 3 - 4 n π - 2 n 8 n ( 2 n - 1 ) ! ( - 4 π ( 2 n - 1 ) ( 3 π ) 2 n + B TagBox["B", BernoulliB] 2 n ( 3 ( - 1 + 9 n ) π - 6 log ( 3 ) ) ( 3 π ) 2 n + 3 n log ( 256 π 8 81 ) ( 3 π ) 2 n + 24 n ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) ( 3 π ) 2 n + 4 ( 3 - 9 n ) n ζ ( 1 - 2 n ) ( 3 π ) 2 n + 4 ( 6 n - 1 ) TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ( 3 π ) 2 n + ( - 1 ) n 2 3 - 2 n 3 2 n + 1 2 n π ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( 1 3 ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -2 n 1 3 -1 3 1 -1 2 n j 1 2 n -1 2 3 -1 -1 j Zeta j 1 -1 j 2 n -1 -1 -1 3 1 -1 2 n 2 n -1 -1 j 0 2 n -2 -1 j Binomial 2 n -1 j -1 j 2 n -1 -1 j 0 2 n -2 -1 j Binomial 2 n -1 j k 0 -1 j 2 n -1 -1 1 3 2 -1 k Binomial -1 j 2 n -1 k k PolyLog k 1 2 3 -1 k 0 2 n -1 -1 k Binomial 2 n -1 k 3 k j 0 k 3 -1 j Binomial k j PolyGamma -1 j k 1 Zeta j -1 k 2 3 -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k 3 -4 n -2 n 8 n 2 n -1 -1 -4 2 n -1 3 2 n BernoulliB 2 n 3 1 2 -1 9 n -1 6 3 3 2 n 3 n 256 8 81 -1 3 2 n 24 n PolyGamma 2 n 3 2 n 4 3 -1 9 n n D Zeta 1 -1 2 n 1 -1 2 n 3 2 n 4 6 n -1 3 2 n -1 n 2 3 -1 2 n 3 2 n 1 2 n PolyGamma 2 n -1 1 3 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n_"]], ",", FractionBox["1", "3"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["3", RowBox[List[RowBox[List["-", "4"]], " ", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "2"]], " ", "n"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["6", " ", "n"]], "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["3", " ", "\[Pi]"]], ")"]], RowBox[List["2", " ", "n"]]]]], "-", RowBox[List["4", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["3", " ", "\[Pi]"]], ")"]], RowBox[List["2", " ", "n"]]]]], "+", RowBox[List["3", " ", "n", " ", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02

© 1998-2013 Wolfram Research, Inc.