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http://functions.wolfram.com/06.03.04.0005.01
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Residue[Binomial[n, k], {n, -j}] == (-1)^j/(j! k! (-j - k)!) /;
Element[j, Integers] && j > 0 && !Element[k, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], ",", RowBox[List["{", RowBox[List["n", ",", " ", RowBox[List["-", "j"]]]], "}"]]]], "]"]], " ", "\[Equal]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List[RowBox[List["j", "!"]], RowBox[List["k", "!"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "-", "k"]], ")"]], "!"]]]]]]], " ", "/;", RowBox[List[RowBox[List["j", "\[Element]", "Integers"]], "\[And]", RowBox[List["j", ">", "0"]], "\[And]", RowBox[List["Not", "[", RowBox[List["k", "\[Element]", "Integers"]], "]"]]]]]]]]
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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> <mrow> <msub> <mi> res </mi> <mi> n </mi> </msub> <mo> ( </mo> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <semantics> <mi> n </mi> <annotation encoding='Mathematica'> TagBox["n", Identity, Rule[Editable, True]] </annotation> </semantics> </mtd> </mtr> <mtr> <mtd> <semantics> <mi> k </mi> <annotation encoding='Mathematica'> TagBox["k", Identity, Rule[Editable, True]] </annotation> </semantics> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mrow> <mrow> <mi> j </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> j </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> j </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> <mo> ∧ </mo> <mrow> <mi> k </mi> <mo> ∉ </mo> <mi> ℤ </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> n </ci> </apply> <list> <list> <apply> <ident /> <ci> n </ci> </apply> </list> <list> <apply> <ident /> <ci> k </ci> </apply> </list> </list> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> j </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> j </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> <apply> <notin /> <ci> k </ci> <ci> ℤ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Residue", "[", RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n_", ",", "k_"]], "]"]], ",", RowBox[List["{", RowBox[List["n_", ",", RowBox[List["-", "j_"]]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "j"]], "-", "k"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["j", "\[Element]", "Integers"]], "&&", RowBox[List["j", ">", "0"]], "&&", RowBox[List["!", RowBox[List["k", "\[Element]", "Integers"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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