26 Combinatorial Analysis Properties 26.7 Set Partitions: Bell Numbers 26.9 Integer Partitions: Restricted Number and Part Size

§26.8 Set Partitions: Stirling Numbers

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Keywords:: partitions
Referenced by:: §26.13, §26.14(iii)
Permalink:: http://dlmf.nist.gov/26.8
See also:: Annotations for Ch.26

§26.8(i) Definitions

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Defines:: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind and $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind
Keywords:: Stirling numbers (first and second kinds), definitions, of a set, partitions
Notes:: See Comtet (1974, pp. 206–207, 214), Riordan (1979, p. 195). Tables 26.8.1 and 26.8.2 are from Abramowitz and Stegun (1964, Tables 24.3 and 24.4).
Referenced by:: §24.15(iii), §26.17, §26.7(i)
Permalink:: http://dlmf.nist.gov/26.8.i
See also:: Annotations for §26.8 and Ch.26

$s\left(n,k\right)$ denotes the Stirling number of the first kind: $(-1)^{n-k}$ times the number of permutations of $\{1,2,\ldots,n\}$ with exactly $k$ cycles. See Table 26.8.1.

26.8.1	$\displaystyle s\left(n,n\right)$	$\displaystyle=1,$
$n\geq 0$ ,
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E1 Encodings: TeX, pMML, png See also: Annotations for §26.8(i), §26.8 and Ch.26
26.8.2	$\displaystyle s\left(1,k\right)$	$\displaystyle=\delta_{1,k},$
ⓘ Symbols: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind and $k$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E2 Encodings: TeX, pMML, png See also: Annotations for §26.8(i), §26.8 and Ch.26

26.8.3		$(-1)^{n-k}s\left(n,k\right)=\sum_{1\leq b_{1}<\cdots<b_{n-k}\leq n-1}b_{1}b_{2% }\cdots b_{n-k},$
$n>k\geq 1$ .
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E3 Encodings: TeX, pMML, png See also: Annotations for §26.8(i), §26.8 and Ch.26

$S\left(n,k\right)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\ldots,n\}$ into exactly $k$ nonempty subsets. See Table 26.8.2.

26.8.4		$S\left(n,n\right)=1,$
$n\geq 0$ ,
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E4 Encodings: TeX, pMML, png See also: Annotations for §26.8(i), §26.8 and Ch.26

26.8.5		$S\left(n,k\right)=\sum 1^{c_{1}}2^{c_{2}}\cdots k^{c_{k}},$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E5 Encodings: TeX, pMML, png See also: Annotations for §26.8(i), §26.8 and Ch.26

where the summation is over all nonnegative integers $c_{1},c_{2},\ldots,c_{k}$ such that $c_{1}+c_{2}+\cdots+c_{k}=n-k.$

26.8.6		$S\left(n,k\right)=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}\genfrac{(}{)}{0.0pt}{}{% k}{j}j^{n}.$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $!$ : factorial (as in $n!$ ), $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.4 Permalink: http://dlmf.nist.gov/26.8.E6 Encodings: TeX, pMML, png See also: Annotations for §26.8(i), §26.8 and Ch.26

Table 26.8.1: Stirling numbers of the first kind

s\left(n,k\right)

$n$	$k$
$n$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
0	$1$
1	$0$	$1$
2	$0$	$-1$	$1$
3	$0$	$2$	$-3$	$1$
4	$0$	$-6$	$11$	$-6$	$1$
5	$0$	$24$	$-50$	$35$	$-10$	$1$
6	$0$	$-120$	$274$	$-225$	$85$	$-15$	$1$
7	$0$	$720$	$-1764$	$1624$	$-735$	$175$	$-21$	$1$
8	$0$	$-5040$	$13068$	$-13132$	$6769$	$-1960$	$322$	$-28$	$1$
9	$0$	$40320$	$-1\,09584$	$1\,18124$	$-67284$	$22449$	$-4536$	$546$	$-36$	$1$
10	$0$	$-3\,62880$	$10\,26576$	$-11\,72700$	$7\,23680$	$-2\,69325$	$63273$	$-9450$	$870$	$-45$	$1$

ⓘ

Symbols:: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $k$ : nonnegative integer and $n$ : nonnegative integer
Keywords:: Stirling numbers (first and second kinds), tables
Referenced by:: §26.8(i), §26.8(i), Erratum (V1.0.7) for Table 26.8.1
Permalink:: http://dlmf.nist.gov/26.8.T1
Svante Janson (effective with 1.0.7):: The correct table entry for $s\left(10,6\right)$ is 63273. Originally it was given incorrectly given as 6327.
See also:: Annotations for §26.8(i), §26.8 and Ch.26

Table 26.8.2: Stirling numbers of the second kind

S\left(n,k\right)

$n$	$k$
$n$	0	1	2	3	4	5	6	7	8	9	10
0	1
1	0	1
2	0	1	1
3	0	1	3	1
4	0	1	7	6	1
5	0	1	15	25	10	1
6	0	1	31	90	65	15	1
7	0	1	63	301	350	140	21	1
8	0	1	127	966	1701	1050	266	28	1
9	0	1	255	3025	7770	6951	2646	462	36	1
10	0	1	511	9330	34105	42525	22827	5880	750	45	1

ⓘ

Symbols:: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $k$ : nonnegative integer and $n$ : nonnegative integer
Keywords:: Stirling numbers (first and second kinds), tables
Referenced by:: §26.8(i), §26.8(i)
Permalink:: http://dlmf.nist.gov/26.8.T2
See also:: Annotations for §26.8(i), §26.8 and Ch.26

§26.8(ii) Generating Functions

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Keywords:: Stirling numbers (first and second kinds), generating functions
Notes:: See Comtet (1974, pp. 206–213).
Permalink:: http://dlmf.nist.gov/26.8.ii
See also:: Annotations for §26.8 and Ch.26

26.8.7		$\sum_{k=0}^{n}s\left(n,k\right)x^{k}={\left(x-n+1\right)_{n}},$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 (in slightly different form) Permalink: http://dlmf.nist.gov/26.8.E7 Encodings: TeX, pMML, png Addition (effective with 1.1.2): The Pochhammer symbol now links to its definition. See also: Annotations for §26.8(ii), §26.8 and Ch.26

where ${\left(x\right)_{n}}$ is the Pochhammer symbol: $x(x+1)\cdots(x+n-1)$ .

26.8.8		$\sum_{n=0}^{\infty}s\left(n,k\right)\frac{x^{n}}{n!}=\frac{(\ln\left(1+x\right% ))^{k}}{k!},$
$\|x\|<1$ ,
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E8 Encodings: TeX, pMML, png See also: Annotations for §26.8(ii), §26.8 and Ch.26

26.8.9		$\sum_{n,k=0}^{\infty}s\left(n,k\right)\frac{x^{n}}{n!}y^{k}=(1+x)^{y},$
$\|x\|<1$ .
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ), $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E9 Encodings: TeX, pMML, png See also: Annotations for §26.8(ii), §26.8 and Ch.26

26.8.10		$\sum_{k=1}^{n}S\left(n,k\right){\left(x-k+1\right)_{k}}=x^{n},$
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.4 Referenced by: §26.15 Permalink: http://dlmf.nist.gov/26.8.E10 Encodings: TeX, pMML, png Addition (effective with 1.1.2): The Pochhammer symbol now links to its definition. See also: Annotations for §26.8(ii), §26.8 and Ch.26

26.8.11		$\sum_{n=0}^{\infty}S\left(n,k\right)\,x^{n}=\frac{x^{k}}{(1-x)(1-2x)\cdots(1-% kx)},$
$\|x\|<1/k$ ,
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.4 Permalink: http://dlmf.nist.gov/26.8.E11 Encodings: TeX, pMML, png See also: Annotations for §26.8(ii), §26.8 and Ch.26

26.8.12		$\sum_{n=0}^{\infty}S\left(n,k\right)\frac{x^{n}}{n!}=\frac{({\mathrm{e}}^{x}-1% )^{k}}{k!},$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.4 Permalink: http://dlmf.nist.gov/26.8.E12 Encodings: TeX, pMML, png See also: Annotations for §26.8(ii), §26.8 and Ch.26

26.8.13		$\sum_{n,k=0}^{\infty}S\left(n,k\right)\frac{x^{n}}{n!}y^{k}=\exp\left(y({% \mathrm{e}}^{x}-1)\right).$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $!$ : factorial (as in $n!$ ), $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E13 Encodings: TeX, pMML, png See also: Annotations for §26.8(ii), §26.8 and Ch.26

§26.8(iii) Special Values

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Keywords:: Stirling numbers (first and second kinds), special values
Notes:: See Graham et al. (1994, p. 264).
Permalink:: http://dlmf.nist.gov/26.8.iii
See also:: Annotations for §26.8 and Ch.26

For $n\geq 1$ ,

26.8.14	$\displaystyle s\left(n,0\right)$	$\displaystyle=0,$
26.8.14	$\displaystyle s\left(n,1\right)$	$\displaystyle=(-1)^{n-1}(n-1)!,$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ) and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §26.8(iii), §26.8 and Ch.26

26.8.15		$s\left(n,2\right)=(-1)^{n}(n-1)!\left(1+\frac{1}{2}+\cdots+\frac{1}{n-1}\right),$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ) and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E15 Encodings: TeX, pMML, png See also: Annotations for §26.8(iii), §26.8 and Ch.26

26.8.16		$-s\left(n,n-1\right)=S\left(n,n-1\right)=\genfrac{(}{)}{0.0pt}{}{n}{2},$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E16 Encodings: TeX, pMML, png See also: Annotations for §26.8(iii), §26.8 and Ch.26

26.8.17	$\displaystyle S\left(n,0\right)$	$\displaystyle=0,$
	$\displaystyle S\left(n,1\right)$	$\displaystyle=1,$
	$\displaystyle S\left(n,2\right)$	$\displaystyle=2^{n-1}-1.$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind and $n$ : nonnegative integer A&S Ref: 24.1.4 Permalink: http://dlmf.nist.gov/26.8.E17 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §26.8(iii), §26.8 and Ch.26

§26.8(iv) Recurrence Relations

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Keywords:: Stirling numbers (first and second kinds), recurrence relations
Notes:: See Comtet (1974, pp. 208–214), Graham et al. (1994, p. 265), Riordan (1979, pp. 204–227).
Permalink:: http://dlmf.nist.gov/26.8.iv
See also:: Annotations for §26.8 and Ch.26

26.8.18		$s\left(n,k\right)=s\left(n-1,k-1\right)-(n-1)s\left(n-1,k\right),$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E18 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

26.8.19		$\genfrac{(}{)}{0.0pt}{}{k}{h}s\left(n,k\right)=\sum_{j=k-h}^{n-h}\genfrac{(}{)% }{0.0pt}{}{n}{j}s\left(n-j,h\right)s\left(j,k-h\right),$
$n\geq k\geq h$ ,
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $h$ : nonnegative integer, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E19 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

26.8.20		$s\left(n+1,k+1\right)=n!\sum_{j=k}^{n}\frac{(-1)^{n-j}}{j!}\,s\left(j,k\right),$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ), $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E20 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

26.8.21		$s\left(n+k+1,k\right)=-\sum_{j=0}^{k}(n+j)s\left(n+j,j\right).$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E21 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

26.8.22		$S\left(n,k\right)=kS\left(n-1,k\right)+S\left(n-1,k-1\right),$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.4 Permalink: http://dlmf.nist.gov/26.8.E22 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

26.8.23		$\genfrac{(}{)}{0.0pt}{}{k}{h}S\left(n,k\right)=\sum_{j=k-h}^{n-h}\genfrac{(}{)% }{0.0pt}{}{n}{j}S\left(n-j,h\right)S\left(j,k-h\right),$
$n\geq k\geq h$ ,
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $h$ : nonnegative integer, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.4 Permalink: http://dlmf.nist.gov/26.8.E23 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

26.8.24		$S\left(n,k\right)=\sum_{j=k}^{n}S\left(j-1,k-1\right)k^{n-j},$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E24 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

26.8.25		$S\left(n+1,k+1\right)=\sum_{j=k}^{n}\genfrac{(}{)}{0.0pt}{}{n}{j}S\left(j,k% \right),$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E25 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

26.8.26		$S\left(n+k+1,k\right)=\sum_{j=0}^{k}jS\left(n+j,j\right).$
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E26 Encodings: TeX, pMML, png See also: Annotations for §26.8(iv), §26.8 and Ch.26

§26.8(v) Identities

ⓘ

Keywords:: Stirling numbers (first and second kinds), identities
Notes:: See Comtet (1974, pp. 213–216), Graham et al. (1994, pp. 264–265). Equation (26.8.37) is in Riordan (1979, p. 203). It and (26.8.39) imply (26.8.31). Equations (26.8.34) and (26.8.35) follow from equation (34) in Riordan (1979, p. 218).
Permalink:: http://dlmf.nist.gov/26.8.v
See also:: Annotations for §26.8 and Ch.26

26.8.27		$s\left(n,n-k\right)=\sum_{j=0}^{k}(-1)^{j}\genfrac{(}{)}{0.0pt}{}{n-1+j}{k+j}% \,\genfrac{(}{)}{0.0pt}{}{n+k}{k-j}\*S\left(k+j,j\right),$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E27 Encodings: TeX, pMML, png See also: Annotations for §26.8(v), §26.8 and Ch.26

26.8.28		$\sum_{k=1}^{n}s\left(n,k\right)=0,$
$n>1$ ,
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E28 Encodings: TeX, pMML, png See also: Annotations for §26.8(v), §26.8 and Ch.26

26.8.29		$\sum_{k=1}^{n}(-1)^{n-k}s\left(n,k\right)=n!,$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E29 Encodings: TeX, pMML, png See also: Annotations for §26.8(v), §26.8 and Ch.26

26.8.30		$\sum_{j=k}^{n}s\left(n+1,j+1\right)\,n^{j-k}=s\left(n,k\right).$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $j$ : nonnegative integer, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 Permalink: http://dlmf.nist.gov/26.8.E30 Encodings: TeX, pMML, png See also: Annotations for §26.8(v), §26.8 and Ch.26

26.8.31		$\frac{1}{k!}\,\frac{{\mathrm{d}}^{k}}{{\mathrm{d}x}^{k}}f(x)=\sum_{n=k}^{% \infty}\frac{s\left(n,k\right)}{n!}\,\Delta^{n}f(x),$
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $!$ : factorial (as in $n!$ ), $\Delta$ : forward difference operator, $x$ : real variable, $k$ : nonnegative integer and $n$ : nonnegative integer A&S Ref: 24.1.3 Referenced by: §26.8(v) Permalink: http://dlmf.nist.gov/26.8.E31 Encodings: TeX, pMML, png See also: Annotations for §26.8(v), §26.8 and Ch.26

when $f(x)$ is analytic for all $x$ , and the series converges, where

26.8.32		$\Delta f(x)=f(x+1)-f(x);$
ⓘ Symbols: $\Delta$ : forward difference operator and $x$ : real variable Permalink: http://dlmf.nist.gov/26.8.E32 Encodings: TeX, pMML, png See also: Annotations for §26.8(v), §26.8 and Ch.26

§26.8(vi) Relations to Bernoulli Numbers

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Permalink:: http://dlmf.nist.gov/26.8.vi
See also:: Annotations for §26.8 and Ch.26

See §24.15(iii).

§26.8(vii) Asymptotic Approximations

ⓘ

Keywords:: Stirling numbers (first and second kinds), asymptotic approximations, generalized, of a set, partitions
Notes:: See Jordan (1939, pp. 161–174) (as corrected here).
Referenced by:: Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/26.8.vii
Addition (effective with 1.0.11):: A reference to Temme (2015, Chapter 34) was added after (26.8.43).
See also:: Annotations for §26.8 and Ch.26

26.8.40		$s\left(n+1,k+1\right)\sim(-1)^{n-k}\frac{n!}{k!}(\gamma+\ln n)^{k},$
$n\to\infty$ ,
ⓘ Symbols: $\gamma$ : Euler’s constant, $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $\sim$ : asymptotic equality, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E40 Encodings: TeX, pMML, png See also: Annotations for §26.8(vii), §26.8 and Ch.26

uniformly for $k=o\left(\ln n\right)$ , where $\gamma$ is Euler’s constant (§5.2(ii)).

26.8.41		$s\left(n+k,k\right)\sim\frac{(-1)^{n}}{2^{n}n!}k^{2n},$
$k\to\infty$ ,
ⓘ Symbols: $s\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the first kind, $\sim$ : asymptotic equality, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E41 Encodings: TeX, pMML, png See also: Annotations for §26.8(vii), §26.8 and Ch.26

$n$ fixed.

26.8.42		$S\left(n,k\right)\sim\frac{k^{n}}{k!},$
$n\to\infty$ ,
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\sim$ : asymptotic equality, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/26.8.E42 Encodings: TeX, pMML, png See also: Annotations for §26.8(vii), §26.8 and Ch.26

$k$ fixed.

26.8.43		$S\left(n+k,k\right)\sim\frac{k^{2n}}{2^{n}n!},$
$k\to\infty$ ,
ⓘ Symbols: $S\left(\NVar{n},\NVar{k}\right)$ : Stirling number of the second kind, $\sim$ : asymptotic equality, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer and $n$ : nonnegative integer Referenced by: §26.8(vii) Permalink: http://dlmf.nist.gov/26.8.E43 Encodings: TeX, pMML, png See also: Annotations for §26.8(vii), §26.8 and Ch.26

uniformly for $n=o\left(k^{1/2}\right)$ .

For asymptotic approximations for $s\left(n+1,k+1\right)$ and $S\left(n,k\right)$ that apply uniformly for $1\leq k\leq n$ as $n\to\infty$ see Temme (1993) and Temme (2015, Chapter 34).

For other asymptotic approximations and also expansions see Moser and Wyman (1958a) for Stirling numbers of the first kind, and Moser and Wyman (1958b), Bleick and Wang (1974) for Stirling numbers of the second kind.

For asymptotic estimates for generalized Stirling numbers see Chelluri et al. (2000).

26.7 Set Partitions: Bell Numbers 26.9 Integer Partitions: Restricted Number and Part Size

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