Stirling number of the first kind: Series representations (formula 04.14.06.0004)

StirlingS1

$Mathematica Notation$

Integer Functions

StirlingS1[n,m]

Series representations

Asymptotic series expansions

http://functions.wolfram.com/04.14.06.0004.01

Input Form

StirlingS1[n, m] \[Proportional] (n - 1)! (Sum[(SeriesCoefficient[1/Gamma[t + 1], {t, 0, k}]/(m - k - 1)!) Log[n]^(m - k - 1), {k, 0, m - 1}] + O[Log[n]^(m - 2)/n^2]) /; (n -> Infinity) && Element[m, Integers] && m >= 0 && 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> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ∝ </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> t </mi> <mi> k </mi> </msup> <mtext> </mtext> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <semantics> <mfrac> <mn> 1 </mn> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox["1", RowBox[List["\[CapitalGamma]", "(", RowBox[List["t", "+", "1"]], ")"]]], Pochhammer] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <msubsup> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", StirlingS1] </annotation> </semantics> <mi> n </mi> <mrow> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ∝ </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <mfrac> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> t </mi> <mi> k </mi> </msup> <mtext> </mtext> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mtext> </mtext> <semantics> <mfrac> <mn> 1 </mn> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> t </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mfrac> <annotation encoding='Mathematica'> TagBox[FractionBox["1", RowBox[List["\[CapitalGamma]", "(", RowBox[List["t", "+", "1"]], ")"]]], Pochhammer] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mrow> <mi> m </mi> <mo> - </mo> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> log </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> ( </mo> <mi> n </mi> <mo> ) </mo> </mrow> <msup> <mi> n </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> m </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02