Stirling number of the first kind: Identities (formula 04.14.17.0003)

StirlingS1

$Mathematica Notation$

Integer Functions

StirlingS1[n,m]

Identities

Identities involving determinants

http://functions.wolfram.com/04.14.17.0003.01

Input Form

Det[Table[((x k)! StirlingS2[x k + l, x k])/(x k + l)!, {k, 0, n}, {l, 0, n}]] == (-(x/2))^Binomial[n + 1, 2] /; Element[n, Integers] && n > 0

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mrow> <mo> ❘ </mo> <msub> <mrow> <mo> ( </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> 𝒮 </mi> <mrow> <mi> l </mi> <mo> + </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> </mrow> <mrow> <mo> ( </mo> <mrow> <mi> x </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </msubsup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> l </mi> <mo> + </mo> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> ≤ </mo> <mi> l </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> </mtd> </mtr> </mtable> </msub> <mo> ❘ </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[LeftBracketingBar]", SubscriptBox[RowBox[List["(", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["x", " ", "k"]], ")"]], "!"]], " ", SubsuperscriptBox[TagBox["\[ScriptCapitalS]", StirlingS2], RowBox[List["l", "+", RowBox[List["k", " ", "x"]]]], RowBox[List["(", RowBox[List["x", " ", "k"]], ")"]]]]], RowBox[List[RowBox[List["(", RowBox[List["l", "+", RowBox[List["k", " ", "x"]]]], ")"]], "!"]]], ")"]], GridBox[List[List[RowBox[List["0", "\[LessEqual]", "k", "\[LessEqual]", "n"]]], List[RowBox[List["0", "\[LessEqual]", "l", "\[LessEqual]", "n"]]]]]], "\[RightBracketingBar]"]], List[Det]] </annotation> </semantics> <mo> ⩵ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> x </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn> 2 </mn> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["n", "+", "1"]], Identity, Rule[Editable, True]]], List[TagBox["2", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </msup> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <determinant /> <apply> <ci> Subscript </ci> <apply> <times /> <apply> <factorial /> <apply> <times /> <ci> x </ci> <ci> k </ci> </apply> </apply> <apply> <ci> StirlingS2 </ci> <apply> <plus /> <ci> l </ci> <apply> <times /> <ci> k </ci> <ci> x </ci> </apply> </apply> <apply> <times /> <ci> x </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> l </ci> <apply> <times /> <ci> k </ci> <ci> x </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <list> <list> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> k </ci> <ci> n </ci> </apply> </list> <list> <apply> <leq /> <cn type='integer'> 0 </cn> <ci> l </ci> <ci> n </ci> </apply> </list> </list> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2001-10-29