q-multinomial%20coefficient

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21: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►The coefficients are given by … ►and the coefficients ${\gamma }_s$ are defined by … ►and the coefficients ${\widetilde{𝒜}}_s(t)$ and ${\widetilde{ℬ}}_s(t)$ are given by … ►The coefficients $A_s(\zeta )$ and $B_s(\zeta )$ are given by …The coefficients $C_s(\zeta )$ and $D_s(\zeta )$ in (12.10.36) and (12.10.38) are given by …

22: 3.8 Nonlinear Equations

… ►However, when the coefficients are all real, complex arithmetic can be avoided by the following iterative process. … ►Thus if $f$ is the polynomial (3.8.8) and $\alpha$ is the coefficient $a_j$, say, then … ►

3.8.15 $$p(x)=(x-1)(x-2)\mathrm{\cdots }(x-20)$$

ⓘ

Permalink:

http://dlmf.nist.gov/3.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

23: 36 Integrals with Coalescing Saddles

…

24: Gergő Nemes

… ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

25: Wolter Groenevelt

… ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. …

26: 25.6 Integer Arguments

… ►

25.6.3 $$\zeta \left(-n\right)=-\frac{B_{n+1}}{n+1},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function and $n$: nonnegative integer

Source:

Apostol (1976, (20), p. 266); with $n\ne 0$

A&S Ref:

23.2.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

… ►

25.6.8 $$\zeta \left(2\right)=3\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^2\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

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Symbols:

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

Keywords:

infinite series

Source:

van der Poorten (1980, (2), p. 271)

Permalink:

http://dlmf.nist.gov/25.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

►

25.6.9 $$\zeta \left(3\right)=\frac{5}{2}\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^{k-1}}{k^3\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

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Symbols:

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

Keywords:

infinite series

Source:

van der Poorten (1980, (3), p. 271)

Permalink:

http://dlmf.nist.gov/25.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

►

25.6.10 $$\zeta \left(4\right)=\frac{36}{17}\sum _{k=1}^{\mathrm{\infty }}\frac{1}{k^4\left(\genfrac{}{}{0.0pt}{}{2k}{k}\right)}.$$

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Symbols:

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

Keywords:

infinite series

Source:

van der Poorten (1980, (5), p. 274)

Permalink:

http://dlmf.nist.gov/25.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(i), §25.6 and Ch.25

… ►

25.6.13 $${(-1)}^k{\zeta }^{(k)}\left(-2n\right)=\frac{2{(-1)}^n}{{(2\pi )}^{2n+1}}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\mathrm{\Im }\left(c^{k-m}\right){\mathrm{\Gamma }}^{(r)}\left(2n+1\right){\zeta }^{(m-r)}\left(2n+1\right),$$

ⓘ

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, $\mathrm{\Im }$: imaginary part, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $c$

Source:

Apostol (1985a, p. 225)

Permalink:

http://dlmf.nist.gov/25.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §25.6(ii), §25.6 and Ch.25

…

27: 26.14 Permutations: Order Notation

… ►

26.14.5

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Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(ii), §26.14 and Ch.26

… ►

26.14.6 $n\ge 1$,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iii), §26.14 and Ch.26

►

26.14.7

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Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iii), §26.14 and Ch.26

… ►

26.14.12 $n\ge m$, $n\ge 1$.

ⓘ

Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iii), §26.14 and Ch.26

… ►

26.14.16 $n\ge 1$.

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Symbols:

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

Permalink:

http://dlmf.nist.gov/26.14.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §26.14(iv), §26.14 and Ch.26

28: 33.24 Tables

… ►

Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

…

29: 26.10 Integer Partitions: Other Restrictions

… ►

Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from $A_{j,k}$.

► ►►►

	$p(𝒟,n)$	$p(𝒟2,n)$	$p(𝒟2,\in T,n)$	$p({𝒟}^{\prime }3,n)$
…
$20$	$64$	$31$	$20$	$18$

►

… ►

26.10.3 $$(1-x)\sum _{m,n=0}^{\mathrm{\infty }}{p}_m(\le k,𝒟,n)x^m{q}^n=\sum _{m=0}^k{\left[\genfrac{}{}{0.0pt}{}{k}{m}\right]}_q{q}^{m(m+1)/2}{x}^m=\prod _{j=1}^k(1+x{q}^j),$$ ,

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Symbols:

${\left[\genfrac{}{}{0.0pt}{}{n}{m}\right]}_q$: $q$-binomial coefficient (or Gaussian polynomial), $p_k(\le m,n)$: number of partitions of $n$ into at most $k$ parts, each less than or equal to $m$ , $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $q$: parameter

Referenced by:

§26.10(ii)

Permalink:

http://dlmf.nist.gov/26.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

… ►

26.10.17 $$p(𝒟,n)=\pi \sum _{k=1}^{\mathrm{\infty }}\frac{A_{2k-1}(n)}{(2k-1)\sqrt{24n+1}}{I}_1\left(\frac{\pi }{2k-1}\sqrt{\frac{24n+1}{72}}\right),$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer, $n$: nonnegative integer and $A_k(n)$: coefficient

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(vi), §26.10 and Ch.26

►where $I_1\left(x\right)$ is the modified Bessel function (§10.25(ii)), and ►

26.10.18

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\left(m,n\right)$: greatest common divisor (gcd), $\mathrm{i}$: imaginary unit, $h$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $A_k(n)$: coefficient and $f(h,k)$: function

Permalink:

http://dlmf.nist.gov/26.10.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(vi), §26.10 and Ch.26

…

30: 24.2 Definitions and Generating Functions

… ►

24.2.5 $$B_n\left(x\right)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)B_k{x}^{n-k}.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

Permalink:

http://dlmf.nist.gov/24.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(i), §24.2 and Ch.24

… ►

24.2.10 $$E_n\left(x\right)=\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{E_k}{2^k}{(x-\frac{1}{2})}^{n-k}.$$

ⓘ

Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $k$: integer, $n$: integer and $x$: real or complex

A&S Ref:

23.1.7

Permalink:

http://dlmf.nist.gov/24.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.2(ii), §24.2 and Ch.24

… ►

Table 24.2.4: Euler numbers $E_n$.

► ►►►

$n$	$E_n$
…
$20$	$\mathrm{37037\hspace{0.33em}11882\hspace{0.33em}37525}$
…

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Table 24.2.5: Coefficients $b_{n,k}$ of the Bernoulli polynomials $B_n\left(x\right)={\sum }_{k=0}^n{b}_{n,k}{x}^k$.

► ►►

	$k$
…

►

►

Table 24.2.6: Coefficients $e_{n,k}$ of the Euler polynomials $E_n\left(x\right)={\sum }_{k=0}^n{e}_{n,k}{x}^k$.

► ►►

	$k$
…

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