binomial expansion

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21: 26.8 Set Partitions: Stirling Numbers

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26.8.16 $$-s(n,n-1)=S(n,n-1)=\left(\genfrac{}{}{0.0pt}{}{n}{2}\right),$$

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Symbols:

$s(n,k)$: Stirling number of the first kind, $S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient and $n$: nonnegative integer

A&S Ref:

24.1.3

Permalink:

http://dlmf.nist.gov/26.8.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(iii), §26.8 and Ch.26

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26.8.19 $$\left(\genfrac{}{}{0.0pt}{}{k}{h}\right)s(n,k)=\sum _{j=k-h}^{n-h}\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)s(n-j,h)s(j,k-h),$$ $n\ge k\ge h$,

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Symbols:

$s(n,k)$: Stirling number of the first kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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:

pMML, png, TeX

See also:

Annotations for §26.8(iv), §26.8 and Ch.26

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26.8.23 $$\left(\genfrac{}{}{0.0pt}{}{k}{h}\right)S(n,k)=\sum _{j=k-h}^{n-h}\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)S(n-j,h)S(j,k-h),$$ $n\ge k\ge h$,

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Symbols:

$S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: 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:

pMML, png, TeX

See also:

Annotations for §26.8(iv), §26.8 and Ch.26

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26.8.25 $$S(n+1,k+1)=\sum _{j=k}^n\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)S(j,k),$$

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Symbols:

$S(n,k)$: Stirling number of the second kind, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(iv), §26.8 and Ch.26

… ►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. …

22: 24.16 Generalizations

… ►For extensions of $B_n^{(\mathrm{\ell })}\left(x\right)$ to complex values of $x$, $n$, and $\mathrm{\ell }$, and also for uniform asymptotic expansions for large $x$ and large $n$, see Temme (1995b) and López and Temme (1999b, 2010b). … ►

24.16.11 $$B_{n,\chi }(x)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_{k,\chi }{x}^{n-k}.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer, $x$: real or complex, $\chi (a)$: Dirichlet character and $B_{n,\chi }$: Generalized Bernoulli numbers

Permalink:

http://dlmf.nist.gov/24.16.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.16(ii), §24.16 and Ch.24

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