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 Zeta

 http://functions.wolfram.com/10.02.20.0029.01

 Input Form

 Derivative[1, 0][Zeta][-n, a] == n! PolyGamma[-n - 1, a - 1] - (a - 1)^n (EulerGamma - (EulerGamma (a - 1))/(1 + n) - Log[a - 1] + PolyGamma[1 + n] + Sum[(Binomial[n, k]/(a - 1)^k) (Sum[Binomial[k, j] PolyGamma[k - j + 1] (Zeta[j - k, a + Max[Floor[1 - Re[a]], 0]] + UnitStep[Floor[-Re[a]]] Sum[(a + i)^(k - j), {i, 0, Floor[-Re[a]]}]) (1 - a)^j, {j, 0, k}] - PolyGamma[k + 1] Zeta[-k] - Derivative[1][Zeta][-k]), {k, 0, n}]) + UnitStep[Floor[-Re[a]]] Sum[(a + i)^n Log[a + i] - (1/2) ((a + i)^2)^(n/2) Log[(a + i)^2], {i, 0, Floor[-Re[a]]}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "n"]], ",", "a"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", "1"]], ",", RowBox[List["a", "-", "1"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["EulerGamma", "-", FractionBox[RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]]]], RowBox[List["1", "+", "n"]]], "-", RowBox[List["Log", "[", RowBox[List["a", "-", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " "]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], "k"]], RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "j", "+", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["a", "+", RowBox[List["Max", "[", RowBox[List[RowBox[List["Floor", "[", RowBox[List["1", "-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], ",", "0"]], "]"]]]]]], "]"]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "i"]], ")"]], RowBox[List["k", "-", "j"]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], "j"]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "k"]], "]"]]]], "-", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "k"]], "]"]]]], ")"]]]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "]"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "i"]], ")"]], "n"], RowBox[List["Log", "[", RowBox[List["a", "+", "i"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "i"]], ")"]], "2"], ")"]], RowBox[List["n", "/", "2"]]], RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "i"]], ")"]], "2"], "]"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 ζ ( 1 , 0 ) TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative] ( - n , a ) n ! ψ TagBox["\[Psi]", PolyGamma] ( - n - 1 ) ( a - 1 ) - ( TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] - TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] ( a - 1 ) n + 1 - log ( a - 1 ) + ψ TagBox["\[Psi]", PolyGamma] ( n + 1 ) + k = 0 n 1 ( a - 1 ) k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( j = 0 k ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["j", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ψ TagBox["\[Psi]", PolyGamma] ( - j + k + 1 ) ( θ UnitStep ( - Re ( a ) ) i = 0 - Re ( a ) ( a + i ) k - j + ζ ( j - k , a + max ( 1 - Re ( a ) , 0 ) ) TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox[RowBox[List["j", "-", "k"]], Zeta, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["a", "+", RowBox[List["max", "(", RowBox[List[RowBox[List["\[LeftFloor]", RowBox[List["1", "-", RowBox[List["Re", "(", "a", ")"]]]], "\[RightFloor]"]], ",", "0"]], ")"]]]], Zeta, Rule[Editable, True], Rule[Selectable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1\$, ZetaDump`e2\$], Zeta[ZetaDump`e1\$, ZetaDump`e2\$]]], Rule[Editable, False], Rule[Selectable, False]] ) ( 1 - a ) j - ψ TagBox["\[Psi]", PolyGamma] ( k + 1 ) ζ ( - k ) TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["-", "k"]], Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]], Rule[Editable, False], Rule[Selectable, False]] - ζ ( - k ) ) ) ( a - 1 ) n + θ UnitStep ( - Re ( a ) ) i = 0 - Re ( a ) ( ( a + i ) n log ( a + i ) - 1 2 ( ( a + i ) 2 ) n / 2 log ( ( a + i ) 2 ) ) /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] Condition 1 0 Zeta -1 n a n PolyGamma -1 n -1 a -1 -1 -1 a -1 n 1 -1 -1 a -1 PolyGamma n 1 k 0 n 1 a -1 k -1 Binomial n k j 0 k Binomial k j PolyGamma -1 j k 1 UnitStep -1 a i 0 -1 a a i k -1 j Zeta j -1 k a 1 -1 a 0 1 -1 a j -1 PolyGamma k 1 Zeta -1 k -1 D Zeta -1 k -1 k a -1 n UnitStep -1 a i 0 -1 a a i n a i -1 1 2 a i 2 n 2 -1 a i 2 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "n_"]], ",", "a_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["PolyGamma", "[", RowBox[List[RowBox[List[RowBox[List["-", "n"]], "-", "1"]], ",", RowBox[List["a", "-", "1"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["EulerGamma", "-", FractionBox[RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]]]], RowBox[List["1", "+", "n"]]], "-", RowBox[List["Log", "[", RowBox[List["a", "-", "1"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", "j", "+", "1"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", RowBox[List[RowBox[List["j", "-", "k"]], ",", RowBox[List["a", "+", RowBox[List["Max", "[", RowBox[List[RowBox[List["Floor", "[", RowBox[List["1", "-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], ",", "0"]], "]"]]]]]], "]"]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "i"]], ")"]], RowBox[List["k", "-", "j"]]]]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "a"]], ")"]], "j"]]]]], "-", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["-", "k"]], "]"]]]], "-", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["-", "k"]], "]"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], "k"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "a", "]"]]]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "i"]], ")"]], "n"], " ", RowBox[List["Log", "[", RowBox[List["a", "+", "i"]], "]"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "i"]], ")"]], "2"], ")"]], RowBox[List["n", "/", "2"]]], " ", RowBox[List["Log", "[", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "i"]], ")"]], "2"], "]"]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02