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 PolyGamma

 http://functions.wolfram.com/06.15.03.0064.01

 Input Form

 PolyGamma[-2 n, 3/4] == (1/(-1 + 2 n)!) (-(((-1)^n PolyGamma[2 n - 1, 1/4])/(4 (8 Pi)^(2 n - 1))) - ((2^(2 n - 1) - 1) Derivative[1][Zeta][-2 n + 1])/2^(4 n - 1) + (((4^n - 4^(2 n)) Pi + (2^(2 n + 1) - 8) Log[2])/(4^(2 n + 1) n)) BernoulliB[2 n] - (I 2^(-1 - 4 n) 3^(2 n) (-1 + 2 n) Pi)/n + ((2^(-1 - 4 n) 3^(-1 + 2 n))/n) (-3 EulerGamma + n Log[16 Pi^8] + 8 n (EulerGamma + PolyGamma[2 n])) - (3/4)^(-1 + 2 n) Sum[(-(4/3))^k Binomial[-1 + 2 n, k] (Sum[(3/4)^j Binomial[k, j] PolyGamma[1 - j + k] Zeta[j - k, 1/4], {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, -1 + 2 n}] + (3/4)^(-1 + 2 n) Sum[((-1)^(-1 + j) Binomial[-1 + 2 n, j])/(-1 - j + 2 n), {j, 0, -2 + 2 n}] - (3/4)^(-1 + 2 n) Sum[(-1)^j Binomial[-1 + 2 n, j] Sum[(Binomial[-1 - j + 2 n, k] k! PolyLog[1 + k, -I])/((-(1/2)) (3 I Pi))^k, {k, 0, -1 - j + 2 n}], {j, 0, -2 + 2 n}] - (2 I Pi)^(1 - 2 n) Sum[((3 I Pi)/2)^j Binomial[-1 + 2 n, j] (-1 - j + 2 n)! Zeta[-j + 2 n], {j, 0, -2 + 2 n}]) /; Element[n, Integers] && n > 0

 Standard Form

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 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - 2 n ) ( 3 4 ) 1 ( 2 n - 1 ) ! ( - ( - 1 ) n 4 ( 8 π ) 2 n - 1 ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( 1 4 ) + ( 3 4 ) 2 n - 1 j = 0 2 n - 2 ( - 1 ) j - 1 - j + 2 n - 1 ( 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]] - ( 3 4 ) 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 2 ( 3 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 ( - ) - ( 3 4 ) 2 n - 1 k = 0 2 n - 1 ( - 4 3 ) 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 ( 3 4 ) 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 4 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[FractionBox["1", "4"], 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 ) ) + ( π ( 4 n - 4 2 n ) + ( 2 2 n + 1 - 8 ) log ( 2 ) ) B TagBox["B", BernoulliB] 2 n 4 2 n + 1 n + ( 2 - 4 n - 1 3 2 n - 1 ) ( n log ( 16 π 8 ) + 8 n ( ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) - 3 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) n - ( 2 π ) 1 - 2 n j = 0 2 n - 2 ( 3 π 2 ) 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 ) ! ζ ( 2 n - j ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List[RowBox[List["2", " ", "n"]], "-", "j"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] - 2 2 n - 1 - 1 2 4 n - 1 ζ ( 1 - 2 n ) - 2 - 4 n - 1 3 2 n ( 2 n - 1 ) π n ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -2 n 3 4 1 2 n -1 -1 -1 -1 n 4 8 2 n -1 -1 PolyGamma 2 n -1 1 4 3 4 2 n -1 j 0 2 n -2 -1 j -1 -1 j 2 n -1 -1 Binomial 2 n -1 j -1 3 4 2 n -1 j 0 2 n -2 -1 j Binomial 2 n -1 j k 0 -1 j 2 n -1 -1 1 2 3 -1 k Binomial -1 j 2 n -1 k k PolyLog k 1 -1 -1 3 4 2 n -1 k 0 2 n -1 -1 4 3 k Binomial 2 n -1 k j 0 k 3 4 j Binomial k j PolyGamma -1 j k 1 Zeta j -1 k 1 4 -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k 4 n -1 4 2 n 2 2 n 1 -8 2 BernoulliB 2 n 4 2 n 1 n -1 2 -4 n -1 3 2 n -1 n 16 8 8 n PolyGamma 2 n -1 3 n -1 -1 2 1 -1 2 n j 0 2 n -2 3 2 -1 j Binomial 2 n -1 j -1 j 2 n -1 Zeta 2 n -1 j -1 2 2 n -1 -1 2 4 n -1 -1 D Zeta 1 -1 2 n 1 -1 2 n -1 2 -4 n -1 3 2 n 2 n -1 n -1 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n_"]], ",", FractionBox["3", "4"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]], ",", FractionBox["1", "4"]]], "]"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["8", " ", "\[Pi]"]], ")"]], RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]]]]]]], "-", 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"]], "]"]]]], SuperscriptBox["2", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02