# Bernoulli 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. …
##### 3: 24.16 Generalizations
For $\ell=0,1,2,\dotsc$, Bernoulli and Euler polynomials of order $\ell$ are defined respectively by …
$B^{(\ell)}_{n}=B^{(\ell)}_{n}\left(0\right),$
For extensions of $B^{(\ell)}_{n}\left(x\right)$ to complex values of $x$, $n$, and $\ell$, and also for uniform asymptotic expansions for large $x$ and large $n$, see Temme (1995b) and López and Temme (1999b, 2010b). … $B^{(x)}_{n}$ is a polynomial in $x$ of degree $n$. …
##### 4: 24.4 Basic Properties
24.4.1 $B_{n}\left(x+1\right)-B_{n}\left(x\right)=nx^{n-1},$
24.4.3 $B_{n}\left(1-x\right)=(-1)^{n}B_{n}\left(x\right),$
24.4.5 $(-1)^{n}B_{n}\left(-x\right)=B_{n}\left(x\right)+nx^{n-1},$
24.4.21 $B_{n}\left(x\right)=2^{n-1}\left(B_{n}\left(\tfrac{1}{2}x\right)+B_{n}\left(% \tfrac{1}{2}x+\tfrac{1}{2}\right)\right),$
##### 5: 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)). …
##### 6: 25.1 Special Notation
 $k,m,n$ nonnegative integers. … Bernoulli number and polynomial (§24.2(i)). periodic Bernoulli function $B_{n}\left(x-\left\lfloor x\right\rfloor\right)$. …
##### 7: 24.13 Integrals
###### §24.13(i) BernoulliPolynomials
24.13.6 $\int_{0}^{1}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t=\frac{(-1)^{n-% 1}m!n!}{(m+n)!}B_{m+n}.$
For integrals of the form $\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t$ and $\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)B_{k}\left(t\right)\,\mathrm% {d}t$ see Agoh and Dilcher (2011). …
##### 9: 24.2 Definitions and Generating Functions
$\widetilde{B}_{n}\left(x\right)=B_{n}\left(x\right)$ ,
$\widetilde{B}_{n}\left(x+1\right)=\widetilde{B}_{n}\left(x\right),$
##### 10: 24.17 Mathematical Applications
###### §24.17 Mathematical Applications
24.17.5 $M_{n}(x)=\begin{cases}\widetilde{B}_{n}\left(x\right)-B_{n},&n\text{ even},\\ \widetilde{B}_{n}\left(x+\frac{1}{2}\right),&n\text{ odd}.\end{cases}$