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 PolyGamma

 http://functions.wolfram.com/06.15.03.0060.01

 Input Form

 PolyGamma[-2 n, 1/4] == (1/(-1 + 2 n)!) ((-1)^n 2^(1 - 6 n) Pi^(1 - 2 n) PolyGamma[-1 + 2 n, 1/4] - ((-2 + 4^n) Derivative[1][Zeta][1 - 2 n])/16^n + ((4^(-1 - 2 n) ((-4^n + 16^n) Pi + (-4 + 4^n) Log[4]))/n) BernoulliB[2 n] + (1/n) (2^(-1 - 4 n) (-EulerGamma + I Pi - 2 I n Pi + n Log[16 Pi^8] + 8 n (EulerGamma + PolyGamma[2 n]))) - 4^(1 - 2 n) Sum[(-4)^k Binomial[-1 + 2 n, k] (Sum[(Binomial[k, j] PolyGamma[1 - j + k] Zeta[j - k, 3/4])/4^j, {j, 0, k}] - PolyGamma[1 + k] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, -1 + 2 n}] - 4^(1 - 2 n) Sum[((-1)^j Binomial[-1 + 2 n, j])/ (-1 - j + 2 n), {j, 0, -2 + 2 n}] - 4^(1 - 2 n) Sum[(-1)^j Binomial[-1 + 2 n, j] Sum[(Binomial[-1 - j + 2 n, k] k! PolyLog[1 + k, I])/(-((Pi I)/2))^k, {k, 0, -1 - j + 2 n}], {j, 0, -2 + 2 n}] - (2 I Pi)^(1 - 2 n) Sum[((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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], ",", FractionBox["1", "4"]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["6", " ", "n"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", FractionBox["1", "4"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["16", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["4", "n"]]], ")"]], RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", 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">", "0"]]]]]]]]

 MathML Form

 ψ TagBox["\[Psi]", PolyGamma] ( - 2 n ) ( 1 4 ) 1 ( 2 n - 1 ) ! ( ( - 1 ) n 2 1 - 6 n π 1 - 2 n ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( 1 4 ) - j = 0 2 n - 2 ( 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]] ( 2 n - j - 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 π ) 1 - 2 n + 4 - 2 n - 1 ( π ( - 4 n + 16 n ) + ( - 4 + 4 n ) log ( 4 ) ) n B TagBox["B", BernoulliB] 2 n + 2 - 4 n - 1 ( - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + π - 2 n π + n log ( 16 π 8 ) + 8 n ( ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ) ) n - 4 1 - 2 n j = 0 2 n - 2 ( - 1 ) j 2 n - j - 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]] - 4 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 2 n - j - 1 ( - π 2 ) - k ( 2 n - j - 1 k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List[RowBox[List["2", "n"]], "-", "j", "-", "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 ( ) - 4 1 - 2 n k = 0 2 n - 1 ( - 4 ) 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 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] ( k - j + 1 ) ζ ( j - k , 3 4 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[FractionBox["3", "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 ) ) - 16 - n ( - 2 + 4 n ) ζ ( 1 - 2 n ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -2 n 1 4 1 2 n -1 -1 -1 n 2 1 -1 6 n 1 -1 2 n PolyGamma 2 n -1 1 4 -1 j 0 2 n -2 2 -1 j Binomial 2 n -1 j 2 n -1 j -1 Zeta 2 n -1 j 2 1 -1 2 n 4 -2 n -1 -1 4 n 16 n -4 4 n 4 n -1 BernoulliB 2 n 2 -4 n -1 -1 -1 2 n n 16 8 8 n PolyGamma 2 n n -1 -1 4 1 -1 2 n j 0 2 n -2 -1 j 2 n -1 j -1 -1 Binomial 2 n -1 j -1 4 1 -1 2 n j 0 2 n -2 -1 j Binomial 2 n -1 j k 0 2 n -1 j -1 -1 2 -1 -1 k Binomial 2 n -1 j -1 k k PolyLog k 1 -1 4 1 -1 2 n k 0 2 n -1 -4 k Binomial 2 n -1 k j 0 k 4 -1 j Binomial k j PolyGamma k -1 j 1 Zeta j -1 k 3 4 -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k -1 16 -1 n -2 4 n 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_"]], ",", FractionBox["1", "4"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["1", "-", RowBox[List["6", " ", "n"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", FractionBox["1", "4"]]], "]"]]]], "-", RowBox[List[SuperscriptBox["16", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["4", "n"]]], ")"]], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]], "]"]]]], "+", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02