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 PolyGamma

 http://functions.wolfram.com/06.15.03.0055.01

 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]) BernoulliB[2 n])/ (4^(2 n + 1) n) - (3/4)^(-1 + 2 n) (EulerGamma + (3 EulerGamma)/(8 n) - I Pi + Log[4/3] + PolyGamma[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}])) /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], ",", RowBox[List["-", FractionBox["3", "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["-", 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", RowBox[List[RowBox[List["4", " ", "n"]], "-", "1"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", "n"], "-", SuperscriptBox["4", RowBox[List["2", " ", "n"]]]]], ")"]], " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], "-", "8"]], ")"]], " ", RowBox[List["Log", "[", "2", "]"]]]]]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "n"]], "]"]]]], RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "n"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "4"], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox[RowBox[List["3", " ", "EulerGamma"]], RowBox[List["8", " ", "n"]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["Log", "[", FractionBox["4", "3"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["4", "3"]]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "4"], ")"]], "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", "4"]]], "]"]]]]]], "-", 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 ) ( - 3 4 ) 1 ( 2 n - 1 ) ! ( π ( 4 n - 4 2 n ) + ( 2 2 n + 1 - 8 ) log ( 2 ) 4 2 n + 1 n B TagBox["B", BernoulliB] 2 n - 2 2 n - 1 - 1 2 4 n - 1 ζ ( 1 - 2 n ) - ( - 1 ) n 4 ( 8 π ) 2 n - 1 ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( 1 4 ) - ( 3 4 ) 2 n - 1 ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + 3 TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] 8 n - π + log ( 4 3 ) + ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) + 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 ) ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -2 n -1 3 4 1 2 n -1 -1 4 n -1 4 2 n 2 2 n 1 -8 2 4 2 n 1 n -1 BernoulliB 2 n -1 2 2 n -1 -1 2 4 n -1 -1 D Zeta 1 -1 2 n 1 -1 2 n -1 -1 n 4 8 2 n -1 -1 PolyGamma 2 n -1 1 4 -1 3 4 2 n -1 3 8 n -1 -1 4 3 PolyGamma 2 n 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 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n_"]], ",", RowBox[List["-", 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", RowBox[List[RowBox[List["4", " ", "n"]], "-", "1"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["4", "n"], "-", SuperscriptBox["4", RowBox[List["2", " ", "n"]]]]], ")"]], " ", "\[Pi]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], "-", "8"]], ")"]], " ", RowBox[List["Log", "[", "2", "]"]]]]]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "n"]], "]"]]]], RowBox[List[SuperscriptBox["4", RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "n"]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "4"], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox[RowBox[List["3", " ", "EulerGamma"]], RowBox[List["8", " ", "n"]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["Log", "[", FractionBox["4", "3"], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", " ", "n"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["4", "3"]]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "4"], ")"]], "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", "4"]]], "]"]]]]]], "-", 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["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02