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variants of this functions
Zeta






Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > Zeta[s] > Summation > Infinite summation





http://functions.wolfram.com/10.01.23.0009.01









  


  










Input Form





Sum[(z^k/(n + k)) Zeta[2 + k], {k, 0, Infinity}] == (-z^(-n)) (PolyGamma[n - 1] Zeta[2 - n] + Derivative[1][Zeta][2 - n]) + ((n - 2)! Sum[(z^j/(j! (n - j - 2)!)) (PolyGamma[n - j - 1] Zeta[2 + j - n, 1 - z] + Derivative[1, 0][Zeta][2 + j - n, 1 - z]), {j, 0, n - 2}])/z^n /; Element[n, Integers] && n > 1










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["z_", "k"], " ", RowBox[List["Zeta", "[", RowBox[List["2", "+", "k"]], "]"]]]], RowBox[List["n_", "+", "k"]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["z", RowBox[List["-", "n"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["n", "-", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List["2", "-", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["Zeta", "\[Prime]", Rule[MultilineFunction, None]], "[", RowBox[List["2", "-", "n"]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "2"]], ")"]], "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["n", "-", "2"]]], FractionBox[RowBox[List[SuperscriptBox["z", "j"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["n", "-", "j", "-", "1"]], "]"]], " ", RowBox[List["Zeta", "[", RowBox[List[RowBox[List["2", "+", "j", "-", "n"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["Zeta", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["2", "+", "j", "-", "n"]], ",", RowBox[List["1", "-", "z"]]]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j", "-", "2"]], ")"]], "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "1"]]]]]]]]]]










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





2007-05-02





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