# Euler polynomials

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##### 1: 24.1 Special Notation
###### Euler Numbers and Polynomials
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 24.3.2: 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)). Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).
##### 4: 24.4 Basic Properties
24.4.2 $E_{n}\left(x+1\right)+E_{n}\left(x\right)=2x^{n}.$
###### §24.4(ii) Symmetry
24.4.4 $E_{n}\left(1-x\right)=(-1)^{n}E_{n}\left(x\right).$
24.4.6 $(-1)^{n+1}E_{n}\left(-x\right)=E_{n}\left(x\right)-2x^{n}.$
Next, …
##### 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(ii) EulerPolynomials
24.13.8 $\int_{0}^{1}E_{n}\left(t\right)\,\mathrm{d}t=-2\frac{E_{n+1}\left(0\right)}{n+% 1}=\frac{4(2^{n+2}-1)}{(n+1)(n+2)}B_{n+2},$
##### 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 $\ell$ : …
$E^{(\ell)}_{n}=E^{(\ell)}_{n}\left(0\right).$
##### 8: 24.2 Definitions and Generating Functions
$\widetilde{E}_{n}\left(x\right)=E_{n}\left(x\right)$ , $0\leq x<1$,
$\widetilde{E}_{n}\left(x+1\right)=-\widetilde{E}_{n}\left(x\right)$ , $x\in\mathbb{R}$.
##### 9: 24.17 Mathematical Applications
###### §24.17 Mathematical Applications
24.17.2 $R_{m}(n)=\frac{1}{2(m-1)!}\int_{a}^{n}f^{(m)}(x)\widetilde{E}_{m-1}\left(h-x% \right)\,\mathrm{d}x.$
24.17.3 $S_{n}(x)=\frac{\widetilde{E}_{n}\left(x+\tfrac{1}{2}n+\tfrac{1}{2}\right)}{% \widetilde{E}_{n}\left(\tfrac{1}{2}n+\tfrac{1}{2}\right)},$ $n=0,1,\dots$,
##### 10: 24.14 Sums
###### §24.14 Sums
24.14.3 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)E_{n-k}\left(x\right)=2(E_{n+1}% \left(x+h\right)-(x+h-1)E_{n}\left(x+h\right)),$
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 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)B_{n-k}\left(x\right)=2^{n}B_{n}% \left(\tfrac{1}{2}(x+h)\right),$