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http://functions.wolfram.com/06.01.06.0002.02
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n! == Sum[(Derivative[k][Gamma][1 + Subscript[n, 0]]/k!)
(n - Subscript[n, 0])^k, {k, 0, Infinity}] /;
(n -> Subscript[n, 0]) && !(Element[-Subscript[n, 0], Integers] &&
-Subscript[n, 0] > 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["n", "!"]], "\[Equal]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "+", SubscriptBox["n", "0"]]], "]"]], RowBox[List[" ", RowBox[List["k", "!"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", SubscriptBox["n", "0"]]], ")"]], "k"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "->", SubscriptBox["n", "0"]]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["n", "0"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", SubscriptBox["n", "0"]]], ">", "0"]]]], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⩵ </mo> <mtext> </mtext> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mi> Γ </mi> <semantics> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "k", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ∉ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <factorial /> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> D </ci> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <list> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </list> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["n_", "!"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "+", SubscriptBox["nn", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", SubscriptBox["nn", "0"]]], ")"]], "k"]]], RowBox[List["k", "!"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", SubscriptBox["nn", "0"]]], ")"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SubscriptBox["nn", "0"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", SubscriptBox["nn", "0"]]], ">", "0"]]]], ")"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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