# Bernoulli polynomial

The Bernoulli polynomials are the sequence $\{b_{r}(x)\}_{r=0}^{\infty}$ of polynomials defined on $[0,1]$ by the conditions:

 $\displaystyle b_{0}(x)$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle b^{\prime}_{r}(x)$ $\displaystyle=$ $\displaystyle rb_{r-1}(x),r\geq 1,$ $\displaystyle\int_{0}^{1}b_{r}(x)dx$ $\displaystyle=$ $\displaystyle 0,r\geq 1$

These assumptions imply the identity

 $\sum_{r=0}^{\infty}b_{r}(x)\frac{y^{r}}{r!}=\frac{ye^{xy}}{e^{y}-1}$

allowing us to calculate the $b_{r}$. We have

 $\displaystyle b_{0}(x)$ $\displaystyle=$ $\displaystyle 1$ $\displaystyle b_{1}(x)$ $\displaystyle=$ $\displaystyle x-\frac{1}{2}$ $\displaystyle b_{2}(x)$ $\displaystyle=$ $\displaystyle x^{2}-x+\frac{1}{6}$ $\displaystyle b_{3}(x)$ $\displaystyle=$ $\displaystyle x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x$ $\displaystyle b_{4}(x)$ $\displaystyle=$ $\displaystyle x^{4}-2x^{3}+x^{2}-\frac{1}{30}$ $\displaystyle\vdots$
Title Bernoulli polynomial BernoulliPolynomial 2013-03-22 11:45:51 2013-03-22 11:45:51 KimJ (5) KimJ (5) 12 KimJ (5) Definition msc 11B68 msc 65-01 BernoulliNumber