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 PolyGamma

 http://functions.wolfram.com/06.15.03.0056.01

 Input Form

 PolyGamma[-2 n, -(2/3)] == (((-1)^(n - 1) 3^(1/2 - 2 n) Pi^(1 - 2 n))/ (4^n (-1 + 2 n)!)) PolyGamma[-1 + 2 n, 1/3] + ((3 - 9^n)/(9^n (2 (-1 + 2 n)!))) Derivative[1][Zeta][1 - 2 n] - ((Sqrt[3] (-1 + 9^n) Pi + 6 Log[3])/(9^n (8 n (-1 + 2 n)!))) BernoulliB[2 n] - (1/(-1 + 2 n)!) (2/3)^(-1 + 2 n) (EulerGamma + EulerGamma/(3 n) - I Pi + Log[3/2] + PolyGamma[2 n] + Sum[(-(3/2))^k Binomial[-1 + 2 n, k] (Sum[(2/3)^j Binomial[k, j] PolyGamma[1 - j + k] Zeta[j - k, 1/3], {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["2", "3"]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["3", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["4", RowBox[List["-", "n"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ",", FractionBox["1", "3"]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["9", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List["3", "-", SuperscriptBox["9", "n"]]], ")"]]]], RowBox[List["2", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]], "]"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["9", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["9", "n"]]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["6", RowBox[List["Log", "[", "3", "]"]]]]]], ")"]]]], RowBox[List["8", " ", "n", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]], RowBox[List["BernoulliB", "[", RowBox[List["2", "n"]], "]"]]]], "-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox["EulerGamma", RowBox[List["3", " ", "n"]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["Log", "[", FractionBox["3", "2"], "]"]], "+", 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["3", "2"]]], ")"]], "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["2", "3"], ")"]], "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", "3"]]], "]"]]]]]], "-", 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 ) ( - 2 3 ) - 1 ( 2 n - 1 ) ! ( 2 3 ) 2 n - 1 ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] + TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] 3 n - π + log ( 3 2 ) + ψ TagBox["\[Psi]", PolyGamma] ( 2 n ) + k = 0 2 n - 1 ( - 3 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 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 , 1 3 ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True]], ",", TagBox[FractionBox["1", "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 ) ) - 9 - n ( 3 π ( - 1 + 9 n ) + 6 log ( 3 ) ) 8 n ( 2 n - 1 ) ! B TagBox["B", BernoulliB] 2 n + ( - 1 ) n - 1 3 1 2 - 2 n 4 - n π 1 - 2 n ( 2 n - 1 ) ! ψ TagBox["\[Psi]", PolyGamma] ( 2 n - 1 ) ( 1 3 ) + 9 - n ( 3 - 9 n ) 2 ( 2 n - 1 ) ! ζ ( 1 - 2 n ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] + Condition PolyGamma -2 n -1 2 3 -1 1 2 n -1 -1 2 3 2 n -1 3 n -1 -1 3 2 PolyGamma 2 n k 0 2 n -1 -1 3 2 k Binomial 2 n -1 k j 0 k 2 3 j Binomial k j PolyGamma -1 j k 1 Zeta j -1 k 1 3 -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k -1 9 -1 n 3 1 2 -1 9 n 6 3 8 n 2 n -1 -1 BernoulliB 2 n -1 n -1 3 1 2 -1 2 n 4 -1 n 1 -1 2 n 2 n -1 -1 PolyGamma 2 n -1 1 3 9 -1 n 3 -1 9 n 2 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_"]], ",", RowBox[List["-", FractionBox["2", "3"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["3", RowBox[List[FractionBox["1", "2"], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["4", RowBox[List["-", "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", "3"]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["9", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List["3", "-", SuperscriptBox["9", "n"]]], ")"]]]], ")"]], " ", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["1", "-", RowBox[List["2", " ", "n"]]]], "]"]]]], RowBox[List["2", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["9", RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["3"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["9", "n"]]], ")"]], " ", "\[Pi]"]], "+", RowBox[List["6", " ", RowBox[List["Log", "[", "3", "]"]]]]]], ")"]]]], ")"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "n"]], "]"]]]], RowBox[List["8", " ", "n", " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]]]]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["2", "3"], ")"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", "n"]]]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", FractionBox["EulerGamma", RowBox[List["3", " ", "n"]]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "+", RowBox[List["Log", "[", FractionBox["3", "2"], "]"]], "+", 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["3", "2"]]], ")"]], "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["2", "3"], ")"]], "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", "3"]]], "]"]]]]]], "-", 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