binomials

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21: 10.29 Recurrence Relations and Derivatives

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10.29.5 $${𝒵}_{\nu }^{(k)}\left(z\right)=\frac{1}{2^k}\left({𝒵}_{\nu -k}\left(z\right)+\left(\genfrac{}{}{0.0pt}{}{k}{1}\right){𝒵}_{\nu -k+2}\left(z\right)+\left(\genfrac{}{}{0.0pt}{}{k}{2}\right){𝒵}_{\nu -k+4}\left(z\right)+\mathrm{\cdots }+{𝒵}_{\nu +k}\left(z\right)\right).$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.29

Referenced by:

§10.29(ii)

Permalink:

http://dlmf.nist.gov/10.29.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.29(ii), §10.29 and Ch.10

22: 25.4 Reflection Formulas

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25.4.5 $${(-1)}^k{\zeta }^{(k)}\left(1-s\right)=\frac{2}{{(2\pi )}^s}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\left(\mathrm{\Re }\left(c^{k-m}\right)\cos \left(\frac{1}{2}\pi s\right)+\mathrm{\Im }\left(c^{k-m}\right)\sin \left(\frac{1}{2}\pi s\right)\right){\mathrm{\Gamma }}^{(r)}\left(s\right){\zeta }^{(m-r)}\left(s\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $\sin z$: sine function, $k$: nonnegative integer, $m$: nonnegative integer, $s$: complex variable and $c$

Keywords:

derivative

Source:

Apostol (1985a, (6), p. 224)

Permalink:

http://dlmf.nist.gov/25.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

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23: 26.11 Integer Partitions: Compositions

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26.11.3 $$c_m\left(n\right)=\left(\genfrac{}{}{0.0pt}{}{n-1}{m-1}\right),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $c_m\left(n\right)$: number of compositions of $n$ into exactly $m$ parts, $m$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

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24: 26.18 Counting Techniques

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26.18.4 $$k^n+\sum _{t=1}^n{(-1)}^t\left(\genfrac{}{}{0.0pt}{}{k}{t}\right){(k-t)}^n.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §26.18, §26.18 and Ch.26

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25: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

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26.4.1 $$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2}{n_1,n_2}\right)=\left(\genfrac{}{}{0.0pt}{}{n_1+n_2}{n_1}\right)=\left(\genfrac{}{}{0.0pt}{}{n_1+n_2}{n_2}\right),$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)$: multinomial coefficient and $n$: nonnegative integer

A&S Ref:

24.1.2

Permalink:

http://dlmf.nist.gov/26.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.4(i), §26.4 and Ch.26

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26.4.2 $$\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)=\frac{(n_1+n_2+\mathrm{\cdots }+n_k)!}{n_1!n_2!\mathrm{\cdots }{n}_k!}=\prod _{j=1}^{k-1}\left(\genfrac{}{}{0.0pt}{}{n_j+n_{j+1}+\mathrm{\cdots }+n_k}{n_j}\right).$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $\left(\genfrac{}{}{0.0pt}{}{n_1+n_2+\mathrm{\cdots }+n_k}{n_1,n_2,\mathrm{\dots },n_k}\right)$: multinomial coefficient, $j$: nonnegative integer, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

24.1.2

Referenced by:

§26.4(i)

Permalink:

http://dlmf.nist.gov/26.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §26.4(i), §26.4 and Ch.26

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26: 26.15 Permutations: Matrix Notation

… ►For the problem of derangements, $r_j(B)=\left(\genfrac{}{}{0.0pt}{}{n}{j}\right)$. … ►

26.15.9 $$r_k(B)=\frac{2n}{2n-k}\left(\genfrac{}{}{0.0pt}{}{2n-k}{k}\right).$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $r_j(B)$: number of rook positions

Permalink:

http://dlmf.nist.gov/26.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15, §26.15 and Ch.26

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26.15.10 $$2(n!)N_0(B)=2(n!)\sum _{k=0}^n{(-1)}^k\frac{2n}{2n-k}\left(\genfrac{}{}{0.0pt}{}{2n-k}{k}\right)(n-k)!.$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $N(x,B)$: generating function

Permalink:

http://dlmf.nist.gov/26.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §26.15, §26.15 and Ch.26

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27: 5.11 Asymptotic Expansions

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$G_2(a,b)={\displaystyle \frac{1}{12}}\left({\displaystyle \genfrac{}{}{0.0pt}{}{a-b}{2}}\right)(3{(a+b-1)}^2-(a-b+1)),$

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$H_1(a,b)=-{\displaystyle \frac{1}{12}}\left({\displaystyle \genfrac{}{}{0.0pt}{}{a-b}{2}}\right)(a-b+1),$

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$H_2(a,b)={\displaystyle \frac{1}{240}}\left({\displaystyle \genfrac{}{}{0.0pt}{}{a-b}{4}}\right)(2(a-b+1)+5{(a-b+1)}^2).$

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5.11.17 $$G_k(a,b)=\left(\genfrac{}{}{0.0pt}{}{a-b}{k}\right)B_k^{(a-b+1)}\left(a\right),$$

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Defines:

$G_k(a,b)$: coefficients (locally)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $k$: nonnegative integer, $a$: real or complex variable and $b$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

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5.11.18 $$H_k(a,b)=\left(\genfrac{}{}{0.0pt}{}{a-b}{2k}\right)B_{2k}^{(a-b+1)}\left(\frac{a-b+1}{2}\right).$$

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Defines:

$H_k(a,b)$: coefficients (locally)

Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $B_n^{(\mathrm{\ell })}\left(x\right)$: generalized Bernoulli polynomials, $k$: nonnegative integer, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.11(iii)

Permalink:

http://dlmf.nist.gov/5.11.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §5.11(iii), §5.11 and Ch.5

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28: 26.7 Set Partitions: Bell Numbers

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26.7.6 $$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(k\right).$$

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Symbols:

$B\left(n\right)$: Bell number, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

Erratum (V1.0.1) for Equation (26.7.6)

Permalink:

http://dlmf.nist.gov/26.7.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.1):

Originally appeared with $B\left(n\right)$ in the summation, instead of $B\left(k\right)$:

$$B\left(n+1\right)=\sum _{k=0}^n\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)B\left(n\right)$$

Reported 2010-11-07 by Layne Watson

See also:

Annotations for §26.7(iii), §26.7 and Ch.26

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29: 4.6 Power Series

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Binomial Expansion

… ►Note that (4.6.7) is a generalization of the binomial expansion (1.2.2) with the binomial coefficients defined in (1.2.6).

30: 26.16 Multiset Permutations

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26.16.1 $${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q=\prod _{k=1}^{n-1}{\left[\genfrac{}{}{0.0pt}{}{a_k+a_{k+1}+\mathrm{\cdots }+a_n}{a_k}\right]}_q,$$

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), ${\left[\genfrac{}{}{0.0pt}{}{a_1+a_2+\mathrm{\cdots }+a_n}{a_1,a_2,\mathrm{\dots },a_n}\right]}_q$: $q$-multinomial coefficient, $k$: nonnegative integer, $n$: nonnegative integer, $a_j$: numbers and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §26.16 and Ch.26

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