24 Bernoulli and Euler Polynomials Properties 24.10 Arithmetic Properties 24.12 Zeros

§24.11 Asymptotic Approximations

Notes:: For (24.11.1)–(24.11.4) see Leeming (1977). For (24.11.5) and (24.11.6) see Dilcher (1987a).
Keywords:: Bernoulli numbers, Bernoulli polynomials, Euler numbers, Euler polynomials
Permalink:: http://dlmf.nist.gov/24.11

As $n\to\infty$

24.11.1 $(-1)^{{n+1}}\mathop{B_{{2n}}\/}\nolimits\sim\frac{2(2n)!}{(2\pi)^{{2n}}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : : factorial, $\sim$ : asymptotic equality and : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E1
Encodings:: TeX, pMML, png

24.11.2 $(-1)^{{n+1}}\mathop{B_{{2n}}\/}\nolimits\sim 4\sqrt{\pi n}\left(\frac{n}{\pi e% }\right)^{{2n}},$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, : base of exponential function, $\sim$ : asymptotic equality and : integer
Permalink:: http://dlmf.nist.gov/24.11.E2
Encodings:: TeX, pMML, png

24.11.3 $(-1)^{n}\mathop{E_{{2n}}\/}\nolimits\sim\frac{2^{{2n+2}}(2n)!}{\pi^{{2n+1}}},$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : : factorial, $\sim$ : asymptotic equality and : integer
Permalink:: http://dlmf.nist.gov/24.11.E3
Encodings:: TeX, pMML, png

24.11.4 $(-1)^{n}\mathop{E_{{2n}}\/}\nolimits\sim 8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{% \pi e}\right)^{{2n}}.$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, : base of exponential function, $\sim$ : asymptotic equality and : integer
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E4
Encodings:: TeX, pMML, png

Also,

24.11.5 $(-1)^{{\left\lfloor n/2\right\rfloor-1}}\frac{(2\pi)^{n}}{2(n!)}\mathop{B_{{n}% }\/}\nolimits\!\left(x\right)\to\begin{cases}\mathop{\cos\/}\nolimits\!\left(2% \pi x\right),&n\text{ even},\\ \mathop{\sin\/}\nolimits\!\left(2\pi x\right),&n\text{ odd},\end{cases}$

Symbols:: $\mathop{B_{{n}}\/}\nolimits\!\left(x\right)$ : Bernoulli polynomials, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , $\mathop{\sin\/}\nolimits z$ : sine function, : integer and : real or complex
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E5
Encodings:: TeX, pMML, png

24.11.6 $(-1)^{{\left\lfloor(n+1)/2\right\rfloor}}\frac{\pi^{{n+1}}}{4(n!)}\mathop{E_{{% n}}\/}\nolimits\!\left(x\right)\to\begin{cases}\mathop{\sin\/}\nolimits\!\left% (\pi x\right),&n\text{ even},\\ \mathop{\cos\/}\nolimits\!\left(\pi x\right),&n\text{ odd},\end{cases}$

Symbols:: $\mathop{E_{{n}}\/}\nolimits\!\left(x\right)$ : Euler polynomials, $\mathop{\cos\/}\nolimits z$ : cosine function, : : factorial, $\left\lfloor x\right\rfloor$ : floor of , $\mathop{\sin\/}\nolimits z$ : sine function, : integer and : real or complex
Referenced by:: §24.11
Permalink:: http://dlmf.nist.gov/24.11.E6
Encodings:: TeX, pMML, png

uniformly for on compact subsets of $\Complex$ .

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

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