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Subfactorial






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Subfactorial[n] > Differentiation > Symbolic differentiation





http://functions.wolfram.com/06.42.20.0006.01









  


  










Input Form





D[Subfactorial[z], {z, n}] == (Pi I)^n Subfactorial[z] + (n!/E) Sum[((Pi I)^(n - k)/(n - k)!) MeijerG[{{}, {Subscript[a, 1] + 1, Subscript[a, 2] + 1, \[Ellipsis], Subscript[a, k + 1] + 1}}, {{Subscript[a, 1], Subscript[a, 2], \[Ellipsis], Subscript[a, k + 1], z + 1}, {}}, -1], {k, 1, n}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == 0 && Element[n, Integers] && n >= 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_", ",", "n_"]], "}"]]]]], RowBox[List["Subfactorial", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], "n"], " ", RowBox[List["Subfactorial", "[", "z", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], RowBox[List["n", "-", "k"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "+", "1"]], ",", RowBox[List[SubscriptBox["a", "2"], "+", "1"]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["a", RowBox[List["k", "+", "1"]]], "+", "1"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["k", "+", "1"]]], ",", RowBox[List["z", "+", "1"]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]]]], "\[ExponentialE]"]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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