# §24.2 Definitions and Generating Functions

## §24.2(i) Bernoulli Numbers and Polynomials

 24.2.1 $\displaystyle\frac{t}{e^{t}-1}$ $\displaystyle=\sum_{n=0}^{\infty}B_{n}\frac{t^{n}}{n!},$ $|t|<2\pi$. 24.2.2 $\displaystyle B_{2n+1}$ $\displaystyle=0$, $\displaystyle(-1)^{n+1}B_{2n}$ $\displaystyle>0$, $n=1,2,\dots$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers and $n$: integer Referenced by: §2.10(i), (25.11.7) Permalink: http://dlmf.nist.gov/24.2.E2 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §24.2(i), §24.2 and Ch.24 24.2.3 $\displaystyle\frac{te^{xt}}{e^{t}-1}$ $\displaystyle=\sum_{n=0}^{\infty}B_{n}\left(x\right)\frac{t^{n}}{n!},$ $|t|<2\pi$. 24.2.4 $\displaystyle B_{n}$ $\displaystyle=B_{n}\left(0\right),$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$: Bernoulli polynomials and $n$: integer A&S Ref: 23.1.2 Referenced by: (25.11.7) Permalink: http://dlmf.nist.gov/24.2.E4 Encodings: TeX, pMML, png See also: Annotations for §24.2(i), §24.2 and Ch.24 24.2.5 $\displaystyle B_{n}\left(x\right)$ $\displaystyle=\sum_{k=0}^{n}{n\choose k}B_{k}x^{n-k}.$

 24.2.6 $\displaystyle\frac{2e^{t}}{e^{2t}+1}$ $\displaystyle=\sum_{n=0}^{\infty}E_{n}\frac{t^{n}}{n!},$ $|t|<\tfrac{1}{2}\pi$, 24.2.7 $\displaystyle E_{2n+1}$ $\displaystyle=0$, $\displaystyle(-1)^{n}E_{2n}$ $\displaystyle>0$. ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers and $n$: integer Referenced by: §24.9 Permalink: http://dlmf.nist.gov/24.2.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §24.2(ii), §24.2 and Ch.24 24.2.8 $\displaystyle\frac{2e^{xt}}{e^{t}+1}$ $\displaystyle=\sum_{n=0}^{\infty}E_{n}\left(x\right)\frac{t^{n}}{n!},$ $|t|<\pi$, 24.2.9 $\displaystyle E_{n}$ $\displaystyle=2^{n}E_{n}\left(\tfrac{1}{2}\right)=\text{integer},$ ⓘ Symbols: $E_{\NVar{n}}$: Euler numbers, $E_{\NVar{n}}\left(\NVar{x}\right)$: Euler polynomials and $n$: integer A&S Ref: 23.1.2 Permalink: http://dlmf.nist.gov/24.2.E9 Encodings: TeX, pMML, png See also: Annotations for §24.2(ii), §24.2 and Ch.24 24.2.10 $\displaystyle E_{n}\left(x\right)$ $\displaystyle=\sum_{k=0}^{n}{n\choose k}\frac{E_{k}}{2^{k}}(x-\tfrac{1}{2})^{n% -k}.$
 24.2.11 $\displaystyle\widetilde{B}_{n}\left(x\right)$ $\displaystyle=B_{n}\left(x\right)$, $\displaystyle\widetilde{E}_{n}\left(x\right)$ $\displaystyle=E_{n}\left(x\right)$, $0\leq x<1$,
 24.2.12 $\displaystyle\widetilde{B}_{n}\left(x+1\right)$ $\displaystyle=\widetilde{B}_{n}\left(x\right),$ $\displaystyle\widetilde{E}_{n}\left(x+1\right)$ $\displaystyle=-\widetilde{E}_{n}\left(x\right)$, $x\in\mathbb{R}$.