# Bernoulli and Euler polynomials

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##### 1: 24.1 Special Notation
###### Bernoulli Numbers and Polynomials
The origin of the notation $B_{n}$, $B_{n}\left(x\right)$, is not clear. …
###### Euler Numbers and Polynomials
Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_{n}$, $E_{n}\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
##### 2: 24.3 Graphs Figure 24.3.1: Bernoulli polynomials B n ⁡ ( x ) , n = 2 , 3 , … , 6 . Magnify Figure 24.3.2: Euler polynomials E n ⁡ ( x ) , n = 2 , 3 , … , 6 . Magnify
##### 3: 24.18 Physical Applications
###### §24.18 Physical Applications
Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)). …
##### 4: 24.4 Basic Properties
###### §24.4(iii) Sums of Powers
Let $P(x)$ denote any polynomial in $x$, and after expanding set $(B(x))^{n}=B_{n}\left(x\right)$ and $(E(x))^{n}=E_{n}\left(x\right)$. …
24.4.39 $E_{n}\left(x+h\right)=(E(x)+h)^{n}.$
##### 6: 24.13 Integrals
24.13.3 $\int_{x}^{x+(1/2)}B_{n}\left(t\right)\,\mathrm{d}t=\frac{E_{n}\left(2x\right)}% {2^{n+1}},$
24.13.8 $\int_{0}^{1}E_{n}\left(t\right)\,\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+% 1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},$
24.13.9 $\int_{0}^{1/2}E_{2n}\left(t\right)\,\mathrm{d}t=-\frac{E_{2n+1}\left(0\right)}% {2n+1}=\frac{2(2^{2n+2}-1)B_{2n+2}}{(2n+1)(2n+2)},$
24.13.11 $\int_{0}^{1}E_{n}\left(t\right)E_{m}\left(t\right)\,\mathrm{d}t=(-1)^{n}4\frac% {(2^{m+n+2}-1)m!n!}{(m+n+2)!}B_{m+n+2}.$
##### 7: 24.16 Generalizations
###### §24.16 Generalizations
For $\ell=0,1,2,\dotsc$, Bernoulli and Euler polynomials of order $\ell$ are defined respectively by …When $x=0$ they reduce to the Bernoulli and Euler numbers of order $\ell$ : …
##### 9: 24.17 Mathematical Applications
###### §24.17(iii) Number Theory
Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and $L$-series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); $p$-adic analysis (Koblitz (1984, Chapter 2)).
##### 10: 24.14 Sums
###### §24.14 Sums
24.14.4 $\sum_{k=0}^{n}{n\choose k}E_{k}E_{n-k}=-2^{n+1}E_{n+1}\left(0\right)=-2^{n+2}(% 1-2^{n+2})\frac{B_{n+2}}{n+2}.$
24.14.5 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)B_{n-k}\left(x\right)=2^{n}B_{n}% \left(\tfrac{1}{2}(x+h)\right),$