24 Bernoulli and Euler Polynomials Properties 24.10 Arithmetic Properties 24.12 Zeros

§24.11 Asymptotic Approximations

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Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials, asymptotic approximations
Notes:: For (24.11.1)–(24.11.4) see Leeming (1977). For (24.11.5) and (24.11.6) see Dilcher (1987a).
Referenced by:: Erratum (V1.0.11) for References
Permalink:: http://dlmf.nist.gov/24.11
Replacement (effective with 1.0.11):: In the sentence at the end of this section the original reference to López and Temme (1999b) was replaced with a reference to López and Temme (1999c). (The reference to López and Temme (1999c) was moved to §24.16(i).)
See also:: Annotations for Ch.24

As $n\to\infty$

24.11.1	$\displaystyle(-1)^{n+1}B_{2n}$	$\displaystyle\sim\frac{2(2n)!}{(2\pi)^{2n}},$
ⓘ Symbols: $B_{\NVar{n}}$ : 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: TeX, pMML, png See also: Annotations for §24.11 and Ch.24
24.11.2	$\displaystyle(-1)^{n+1}B_{2n}$	$\displaystyle\sim 4\sqrt{\pi n}\left(\frac{n}{\pi e}\right)^{2n},$
ⓘ Symbols: $B_{\NVar{n}}$ : 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: TeX, pMML, png See also: Annotations for §24.11 and Ch.24
24.11.3	$\displaystyle(-1)^{n}E_{2n}$	$\displaystyle\sim\frac{2^{2n+2}(2n)!}{\pi^{2n+1}},$
ⓘ Symbols: $E_{\NVar{n}}$ : 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: TeX, pMML, png See also: Annotations for §24.11 and Ch.24
24.11.4	$\displaystyle(-1)^{n}E_{2n}$	$\displaystyle\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.$
ⓘ Symbols: $E_{\NVar{n}}$ : 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: TeX, pMML, png See also: Annotations for §24.11 and Ch.24

Also,

24.11.5	$\displaystyle(-1)^{\left\lfloor n/2\right\rfloor-1}\frac{(2\pi)^{n}}{2(n!)}B_{% n}\left(x\right)$	$\displaystyle\to\begin{cases}\cos\left(2\pi x\right),&n\text{ even},\\ \sin\left(2\pi x\right),&n\text{ odd},\end{cases}$
ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\sin\NVar{z}$ : sine function, $n$ : integer and $x$ : real or complex Referenced by: §24.11 Permalink: http://dlmf.nist.gov/24.11.E5 Encodings: TeX, pMML, png See also: Annotations for §24.11 and Ch.24
24.11.6	$\displaystyle(-1)^{\left\lfloor(n+1)/2\right\rfloor}\frac{\pi^{n+1}}{4(n!)}E_{% n}\left(x\right)$	$\displaystyle\to\begin{cases}\sin\left(\pi x\right),&n\text{ even},\\ \cos\left(\pi x\right),&n\text{ odd},\end{cases}$
ⓘ Symbols: $E_{\NVar{n}}\left(\NVar{x}\right)$ : Euler polynomials, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $!$ : factorial (as in $n!$ ), $\left\lfloor\NVar{x}\right\rfloor$ : floor of $x$ , $\sin\NVar{z}$ : sine function, $n$ : integer and $x$ : real or complex Referenced by: §24.11 Permalink: http://dlmf.nist.gov/24.11.E6 Encodings: TeX, pMML, png See also: Annotations for §24.11 and Ch.24

uniformly for $x$ on compact subsets of $\mathbb{C}$ .

For further results see Temme (1995b) and López and Temme (1999c).