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 PolyGamma

 http://functions.wolfram.com/06.15.03.0062.01

 Input Form

 PolyGamma[-2 n, 1/2] == (2^(-1 - 2 n) (EulerGamma (-1 + 4 n) + I Pi (1 - 2 n) + 4 n Log[Pi] + 4 n PolyGamma[2 n]))/(n (-1 + 2 n)!) - (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)!) - (2^(1 - 2 n)/(-1 + 2 n)!) Sum[((-1)^j Binomial[-1 + 2 n, j])/(-1 - j + 2 n), {j, 0, -2 + 2 n}] - 2^(1 - 2 n) Sum[Zeta[1 + j]/((I Pi)^j (-1 - j + 2 n)!), {j, 1, -1 + 2 n}] - (2^(1 - 2 n)/(-1 + 2 n)!) Sum[(-2)^k Binomial[-1 + 2 n, 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}] + (2^(1 - 2 n)/(-1 + 2 n)!) Sum[(-1)^j Binomial[-1 + 2 n, j] (Log[2] + Sum[((1 - 2^(-k)) Binomial[-1 - j + 2 n, k] k! Zeta[1 + k])/ ((-I) Pi)^k, {k, 1, -1 - 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 ) ( 1 2 ) - log ( 2 ) 2 2 n n ( 2 n - 1 ) ! B TagBox["B", BernoulliB] 2 n + 2 - 2 n - 1 ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ( 4 n - 1 ) + 4 n log ( π ) + 4 n ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) + ( 1 - 2 n ) π ) n ( 2 n - 1 ) ! - 2 1 - 2 n ( 2 n - 1 ) ! j = 0 2 n - 2 ( - 1 ) j - 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]] - 2 1 - 2 n j = 1 2 n - 1 ( 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 ) ! + 2 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]] ( log ( 2 ) + k = 1 - j + 2 n - 1 ( 1 - 2 - k ) ( - 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 ! ζ ( k + 1 ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["k", "+", "1"]], Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]]] ) - 2 1 - 2 n ( 2 n - 1 ) ! k = 0 2 n - 1 ( - 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 - 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 2 -1 2 2 2 n n 2 n -1 -1 BernoulliB 2 n 2 -2 n -1 4 n -1 4 n 4 n PolyGamma 2 n 1 -1 2 n n 2 n -1 -1 -1 2 1 -1 2 n 2 n -1 -1 j 0 2 n -2 -1 j -1 j 2 n -1 -1 Binomial 2 n -1 j -1 2 1 -1 2 n j 1 2 n -1 -1 j Zeta j 1 -1 j 2 n -1 -1 2 1 -1 2 n 2 n -1 -1 j 0 2 n -2 -1 j Binomial 2 n -1 j 2 k 1 -1 j 2 n -1 1 -1 2 -1 k -1 -1 k Binomial -1 j 2 n -1 k k Zeta k 1 -1 2 1 -1 2 n 2 n -1 -1 k 0 2 n -1 -2 k Binomial 2 n -1 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_"]], ",", FractionBox["1", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "n"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]], ")"]]]], "+", RowBox[List["4", " ", "n", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]], "+", RowBox[List["4", " ", "n", " ", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]]]]]], ")"]]]], RowBox[List["n", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]], "-", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02