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11: 24.11 Asymptotic Approximations

§24.11 Asymptotic Approximations

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24.11.1 $${(-1)}^{n+1}{B}_{2n}\sim \frac{2(2n)!}{{(2\pi )}^{2n}},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.2 $${(-1)}^{n+1}{B}_{2n}\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.3 $${(-1)}^n{E}_{2n}\sim \frac{2^{2n+2}(2n)!}{{\pi }^{2n+1}},$$

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Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

Permalink:

http://dlmf.nist.gov/24.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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24.11.4 $${(-1)}^n{E}_{2n}\sim 8\sqrt{\frac{n}{\pi }}{\left(\frac{4n}{\pi \mathrm{e}}\right)}^{2n}.$$

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Symbols:

$E_n$: Euler numbers, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Referenced by:

§24.11

Permalink:

http://dlmf.nist.gov/24.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.11 and Ch.24

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12: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

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§26.7(i) Definitions

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§26.7(ii) Generating Function

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§26.7(iii) Recurrence Relation

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§26.7(iv) Asymptotic Approximation

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13: 3.11 Approximation Techniques

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3.11.28 $$S=\sum _{j=1}^J{\left(f(x_j)-p_n(x_j)\right)}^2.$$

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Symbols:

$p_n(x)$: approximation and $J>n+1$: number of points

Permalink:

http://dlmf.nist.gov/3.11.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(v), §3.11 and Ch.3

►Here $x_j$, $j=1,2,\mathrm{\dots },J$, is a given set of distinct real points and $J\ge n+1$. … ►

3.11.32 $$\sum _{j=1}^{J}w(x_j){\left(f(x_j)-{\mathrm{\Phi }}_n(x_j)\right)}^2,$$

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Symbols:

$J>n+1$: number of points, ${\mathrm{\Phi }}_n(x)$: approximant and $w(x)$: weight function

Referenced by:

§3.11(v)

Permalink:

http://dlmf.nist.gov/3.11.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §3.11(v), §3.11 and Ch.3

… ►In consequence of this structure the number of operations can be reduced to $nm=n{\log }_{2}n$ operations. … ►For many applications a spline function is a more adaptable approximating tool than the Lagrange interpolation polynomial involving a comparable number of parameters; see §3.3(i), where a single polynomial is used for interpolating $f(x)$ on the complete interval $[a,b]$. …

14: 24.9 Inequalities

§24.9 Inequalities

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24.9.3 $$4^{-n}|E_{2n}|>{(-1)}^n{E}_{2n}\left(x\right)>0,$$

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Symbols:

$E_n$: Euler numbers, $E_n\left(x\right)$: Euler polynomials, $n$: integer and $x$: real or complex

A&S Ref:

23.1.13

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.6 $$5\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n}>{(-1)}^{n+1}{B}_{2n}>4\sqrt{\pi n}{\left(\frac{n}{\pi \mathrm{e}}\right)}^{2n},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm and $n$: integer

Referenced by:

§24.9, §24.9

Permalink:

http://dlmf.nist.gov/24.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.8 $$\frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{\beta -2n}}\ge {(-1)}^{n+1}{B}_{2n}\ge \frac{2(2n)!}{{(2\pi )}^{2n}}\frac{1}{1-2^{-2n}}$$

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Symbols:

$B_n$: Bernoulli numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $n$: integer and $\beta$: quantity

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

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24.9.10 $$\frac{4^{n+1}(2n)!}{{\pi }^{2n+1}}>{(-1)}^n{E}_{2n}>\frac{4^{n+1}(2n)!}{{\pi }^{2n+1}}\frac{1}{1+3^{-1-2n}}.$$

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Symbols:

$E_n$: Euler numbers, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$) and $n$: integer

A&S Ref:

23.1.15

Referenced by:

§24.9

Permalink:

http://dlmf.nist.gov/24.9.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.9 and Ch.24

15: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

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§26.8(i) Definitions

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§26.8(v) Identities

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§26.8(vi) Relations to Bernoulli Numbers

… ►For asymptotic approximations for $s(n+1,k+1)$ and $S(n,k)$ that apply uniformly for $1\le k\le n$ as $n\to \mathrm{\infty }$ see Temme (1993) and Temme (2015, Chapter 34). …

16: 26.14 Permutations: Order Notation

… ►As an example, $35247816$ is an element of ${𝔖}_8.$ The inversion number is the number of pairs of elements for which the larger element precedes the smaller: … ► ►The Eulerian number, denoted , is the number of permutations in ${𝔖}_n$ with exactly $k$ descents. …It is also equal to the number of permutations in ${𝔖}_n$ with exactly $k+1$ weak excedances. … ►

§26.14(iii) Identities

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17: 24.4 Basic Properties

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24.4.7 $$\sum _{k=1}^m{k}^n=\frac{B_{n+1}\left(m+1\right)-B_{n+1}}{n+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $k$: integer, $m$: integer and $n$: integer

A&S Ref:

23.1.4

Referenced by:

§24.17(iii)

Permalink:

http://dlmf.nist.gov/24.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.4(iii), §24.4 and Ch.24

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§24.4(iv) Finite Expansions

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24.4.26 $$E_n\left(0\right)=-E_n\left(1\right)=-\frac{2}{n+1}(2^{n+1}-1)B_{n+1},$$ $n>0$.

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials and $n$: integer

A&S Ref:

23.1.20

Referenced by:

§24.9, Erratum (V1.0.5) for Equation (24.4.26)

Permalink:

http://dlmf.nist.gov/24.4.E26

Encodings:

pMML, png, TeX

Errata (effective with 1.0.5):

This equation is true only for $n>0$. Previously, $n=0$ was also allowed.

Reported 2012-05-14 by Vladimir Yurovsky

See also:

Annotations for §24.4(vi), §24.4 and Ch.24

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§24.4(ix) Relations to Other Functions

►For the relation of Bernoulli numbers to the Riemann zeta function see §25.6, and to the Eulerian numbers see (26.14.11).

18: DLMF Project News

error generating summary

19: 24.5 Recurrence Relations

§24.5 Recurrence Relations

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§24.5(ii) Other Identities

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24.5.6 $$\sum _{k=2}^n\left(\genfrac{}{}{0pt}{}{n}{k-2}\right)\frac{B_k}{k}=\frac{1}{(n+1)(n+2)}-B_{n+1},$$ $n=2,3,\mathrm{\dots }$,

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Symbols:

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

Referenced by:

§24.5(ii)

Permalink:

http://dlmf.nist.gov/24.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(ii), §24.5 and Ch.24

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24.5.7 $$\sum _{k=0}^n\left(\genfrac{}{}{0pt}{}{n}{k}\right)\frac{B_k}{n+2-k}=\frac{B_{n+1}}{n+1},$$ $n=1,2,\mathrm{\dots }$,

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Symbols:

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

Referenced by:

§24.5(ii)

Permalink:

http://dlmf.nist.gov/24.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §24.5(ii), §24.5 and Ch.24

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§24.5(iii) Inversion Formulas

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20: 24.13 Integrals

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24.13.4 $${\int }_0^{1/2}{B}_n\left(t\right)dt=\frac{1-2^{n+1}}{2^n}\frac{B_{n+1}}{n+1},$$

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

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24.13.5 $${\int }_{1/4}^{3/4}{B}_n\left(t\right)dt=\frac{E_n}{2^{2n+1}}.$$

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $E_n$: Euler numbers, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Referenced by:

§24.13(i)

Permalink:

http://dlmf.nist.gov/24.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

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24.13.6 $${\int }_0^1{B}_n\left(t\right)B_m\left(t\right)dt=\frac{{(-1)}^{n-1}m!n!}{(m+n)!}{B}_{m+n}.$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $m$: integer, $n$: integer and $t$: real or complex

A&S Ref:

23.1.12

Referenced by:

§24.13(i)

Permalink:

http://dlmf.nist.gov/24.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(i), §24.13 and Ch.24

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24.13.8 $${\int }_0^1{E}_n\left(t\right)dt=-2\frac{E_{n+1}\left(0\right)}{n+1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}{B}_{n+2},$$

ⓘ

Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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24.13.9 $${\int }_0^{1/2}{E}_{2n}\left(t\right)dt=-\frac{E_{2n+1}\left(0\right)}{2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},$$

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Symbols:

$B_n$: Bernoulli numbers, $E_n\left(x\right)$: Euler polynomials, $dx$: differential of $x$ , $\int$: integral, $n$: integer and $t$: real or complex

Permalink:

http://dlmf.nist.gov/24.13.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.13(ii), §24.13 and Ch.24

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