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variants of this functions
NorlundB






Mathematica Notation

Traditional Notation









Polynomials > NorlundB[n,α] > Differentiation > Low-order differentiation





http://functions.wolfram.com/05.16.20.0004.01









  


  










Input Form





D[NorlundB[n, z], {z, 2}] == (-1)^(n - 1) Sum[(-1)^i (1 + i) (2 + i) Sum[StirlingS1[n + 1, k] Sum[(-1)^j j^(k - i - 3) Binomial[n, j] Subscript[p, j, n] z^(i - 1), {j, 1, n}], {k, 1, i + 2}], {i, 0, n - 2}] /; Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/k) Sum[(j v - k + v) Subscript[a, v] Subscript[p, j, k - v], {v, 1, k}] && Subscript[a, k] == 1/(k + 1)! && Element[k, Integers] && k >= 0










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "2"]], "}"]]]]], RowBox[List["NorlundB", "[", RowBox[List["n_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["n", "-", "2"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "i"], " ", RowBox[List["(", RowBox[List["1", "+", "i"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "i"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], RowBox[List["i", "+", "2"]]], RowBox[List[RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["n", "+", "1"]], ",", "k"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["j", RowBox[List["k", "-", "i", "-", "3"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", "n"]]], " ", SuperscriptBox["z", RowBox[List["i", "-", "1"]]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["v", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "v"]], "-", "k", "+", "v"]], ")"]], " ", SubscriptBox["a", "v"], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "v"]]]]]]]]], "k"]]], "&&", RowBox[List[SubscriptBox["a", "k"], "\[Equal]", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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