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21: 24.6 Explicit Formulas

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24.6.1 $$B_{2n}=\sum _{k=2}^{2n+1}\frac{{(-1)}^{k-1}}{k}\left(\genfrac{}{}{0pt}{}{2n+1}{k}\right)\sum _{j=1}^{k-1}{j}^{2n},$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.3 $$B_{2n}=\sum _{k=1}^n\frac{(k-1)!k!}{(2k+1)!}\sum _{j=1}^k{(-1)}^{j-1}\left(\genfrac{}{}{0pt}{}{2k}{k+j}\right)j^{2n}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $!$: factorial (as in $n!$), $j$: integer, $k$: integer and $n$: integer

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.4 $$E_{2n}=\sum _{k=1}^n\frac{1}{2^{k-1}}\sum _{j=1}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{2k}{k-j}\right)j^{2n},$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.5 $$E_{2n}=\frac{1}{2^{n-1}}\sum _{k=0}^{n-1}{(-1)}^{n-k}{(n-k)}^{2n}\sum _{j=0}^k\left(\genfrac{}{}{0pt}{}{2n-2j}{k-j}\right)2^j,$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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24.6.9 $$B_n=\sum _{k=0}^n\frac{1}{k+1}\sum _{j=0}^k{(-1)}^j\left(\genfrac{}{}{0pt}{}{k}{j}\right)j^n,$$

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Symbols:

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

Referenced by:

§24.6

Permalink:

http://dlmf.nist.gov/24.6.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.6 and Ch.24

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22: 24.20 Tables

… ►Abramowitz and Stegun (1964, Chapter 23) includes exact values of ${\sum }_{k=1}^m{k}^n$, $m=1(1)100$, $n=1(1)10$; ${\sum }_{k=1}^{\mathrm{\infty }}{k}^{-n}$, ${\sum }_{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}{k}^{-n}$, ${\sum }_{k=0}^{\mathrm{\infty }}{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 20D; ${\sum }_{k=0}^{\mathrm{\infty }}{(-1)}^k{(2k+1)}^{-n}$, $n=1,2,\mathrm{\dots }$, 18D. …

23: 29.13 Graphics

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Figure 29.13.5: ${\mathrm{𝑢𝐸}}_4^m(x,0.1)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.61244\mathrm{\dots }$. Magnify

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Figure 29.13.6: ${\mathrm{𝑢𝐸}}_4^m(x,0.9)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=2.57809\mathrm{\dots }$. Magnify

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Figure 29.13.7: ${\mathrm{𝑠𝐸}}_5^m(x,0.1)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.61244\mathrm{\dots }$. Magnify

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Figure 29.13.8: ${\mathrm{𝑠𝐸}}_5^m(x,0.9)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=2.57809\mathrm{\dots }$. Magnify

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Figure 29.13.9: ${\mathrm{𝑐𝐸}}_5^m(x,0.1)$ for $-2K\le x\le 2K$, $m=0,1,2$. $K=1.61244\mathrm{\dots }$. Magnify

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24: 22.4 Periods, Poles, and Zeros

… ►Figure 22.4.1 illustrates the locations in the $z$-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices $0$, $2K$, $2K+2\mathrm{i}{K}^{\prime }$, $2\mathrm{i}{K}^{\prime }$. … ►For the distribution of the $k$-zeros of the Jacobian elliptic functions see Walker (2009). … ►This half-period will be plus or minus a member of the triple $K,\mathrm{i}{K}^{\prime },K+\mathrm{i}{K}^{\prime }$; the other two members of this triple are quarter periods of $\mathrm{p}\mathrm{q}(z,k)$. … ►For example, $\mathrm{sn}(z+K,k)=\mathrm{cd}(z,k)$. (The modulus $k$ is suppressed throughout the table.) …

25: 33.19 Power-Series Expansions in $r$

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$k(k+2\mathrm{\ell }+1){\alpha }_k+2{\alpha }_{k-1}+ϵ{\alpha }_{k-2}=0,$ $k=2,3,\mathrm{\dots }$.

… ►Here $\kappa$ is defined by (33.14.6), $A(ϵ,\mathrm{\ell })$ is defined by (33.14.11) or (33.14.12), ${\gamma }_0=1$, ${\gamma }_1=1$, and ►

33.19.4 $${\gamma }_k-{\gamma }_{k-1}+\frac{1}{4}(k-1)(k-2\mathrm{\ell }-2)ϵ{\gamma }_{k-2}=0,$$ $k=2,3,\mathrm{\dots }$.

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Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter and ${\gamma }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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33.19.6 $$k(k+2\mathrm{\ell }+1){\delta }_k+2{\delta }_{k-1}+ϵ{\delta }_{k-2}+2(2k+2\mathrm{\ell }+1)A(ϵ,\mathrm{\ell }){\alpha }_k=0,$$ $k=2,3,\mathrm{\dots }$,

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Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, ${\alpha }_k$: term and ${\delta }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

►with ${\beta }_0={\beta }_1=0$, and …

26: 19.7 Connection Formulas

… ►If $\mathrm{\Re }k>0$ then …where upper signs apply if $\mathrm{\Im }{k}^2>0$ and lower signs if . … ►

§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

►There are three relations connecting $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\omega }^2,k)$, where ${\omega }^2$ is a rational function of ${\alpha }^2$. If $k^2$ and ${\alpha }^2$ are real, then both integrals are circular cases or both are hyperbolic cases (see §19.2(ii)). …

27: 26.8 Set Partitions: Stirling Numbers

… ► $s(n,k)$ denotes the Stirling number of the first kind: ${(-1)}^{n-k}$ times the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. … … ► $S(n,k)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\mathrm{\dots },n\}$ into exactly $k$ nonempty subsets. …where the summation is over all nonnegative integers $c_1,c_2,\mathrm{\dots },c_k$ such that $c_1+c_2+\mathrm{\cdots }+c_k=n-k.$ … ► $k$ fixed. …

28: 25.8 Sums

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25.8.1 $$\sum _{k=2}^{\mathrm{\infty }}\left(\zeta \left(k\right)-1\right)=1.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (1.4), p. 132)

Permalink:

http://dlmf.nist.gov/25.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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25.8.2 $$\sum _{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)}{(k+1)!}\left(\zeta \left(s+k\right)-1\right)=\mathrm{\Gamma }\left(s-1\right),$$ $s\ne 1,0,-1,-2,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Landau (1953, (2), p. 273); with (5.2.4), (5.2.5)

Permalink:

http://dlmf.nist.gov/25.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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25.8.3 $$\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_k\zeta \left(s+k\right)}{k!2^{s+k}}=(1-2^{-s})\zeta \left(s\right),$$ $s\ne 1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\zeta \left(s\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Source:

Srivastava (1988, (2.6), p. 131); with $k=0$ term

Referenced by:

§25.11(iv), Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.8.E3

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{k=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s+k\right)\zeta \left(s+k\right)}{k!\mathrm{\Gamma }\left(s\right)2^{s+k}}$ more concisely as ${\sum }_{k=0}^{\mathrm{\infty }}\frac{{\left(s\right)}_k\zeta \left(s+k\right)}{k!2^{s+k}}$ using the Pochhammer symbol.

See also:

Annotations for §25.8 and Ch.25

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25.8.9 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2.$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $k$: nonnegative integer

Keywords:

infinite series

Source:

Srivastava and Choi (2001, (467), p. 212)

Permalink:

http://dlmf.nist.gov/25.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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25.8.10 $$\sum _{k=1}^{\mathrm{\infty }}\frac{\zeta \left(2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4{\pi }^2}\zeta \left(3\right).$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter and $k$: nonnegative integer

Keywords:

infinite series

Source:

Ewell (1990, (1), p. 219)

Permalink:

http://dlmf.nist.gov/25.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

…

29: 26.9 Integer Partitions: Restricted Number and Part Size

… ► $p_k\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. See Table 26.9.1. … ►equivalently, partitions into at most $k$ parts either have exactly $k$ parts, in which case we can subtract one from each part, or they have strictly fewer than $k$ parts. … ►As $n\to \mathrm{\infty }$ with $k$ fixed, …

30: 5.8 Infinite Products

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5.8.1 $$\mathrm{\Gamma }\left(z\right)=\underset{k\to \mathrm{\infty }}{lim}\frac{k!k^z}{z(z+1)\mathrm{\cdots }(z+k)},$$ $z\ne 0,-1,-2,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.2

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

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5.8.2 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=z{\mathrm{e}}^{\gamma z}\prod _{k=1}^{\mathrm{\infty }}\left(1+\frac{z}{k}\right){\mathrm{e}}^{-z/k},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.3

Permalink:

http://dlmf.nist.gov/5.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

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5.8.4 $$\sum _{k=1}^m{a}_k=\sum _{k=1}^m{b}_k,$$

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Defines:

$a_k$: coefficient (locally) and $b_k$: coefficient (locally)

Symbols:

$m$: nonnegative integer and $k$: nonnegative integer

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

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5.8.5 $$\prod _{k=0}^{\mathrm{\infty }}\frac{(a_1+k)(a_2+k)\mathrm{\cdots }(a_m+k)}{(b_1+k)(b_2+k)\mathrm{\cdots }(b_m+k)}=\frac{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_m\right)}{\mathrm{\Gamma }\left(a_1\right)\mathrm{\Gamma }\left(a_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(a_m\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_k$: coefficient and $b_k$: coefficient

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

►provided that none of the $b_k$ is zero or a negative integer.