Bernoulli polynomial

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

$b_0(x)$$=$$1,$

$b_r^{\prime }(x)$$=$$r{b}_{r-1}(x),r\ge 1,$

${\displaystyle {\int }_0^1}{b}_r(x)𝑑x$$=$$0,r\ge 1$

These assumptions imply the identity

$$\sum _{r=0}^{\mathrm{\infty }}{b}_r(x)\frac{y^r}{r!}=\frac{y{e}^{xy}}{e^y-1}$$

allowing us to calculate the $b_r$. We have

$b_0(x)$$=$$1$

$b_1(x)$$=$$x-{\displaystyle \frac{1}{2}}$

$b_2(x)$$=$$x^2-x+{\displaystyle \frac{1}{6}}$

$b_3(x)$$=$$x^3-{\displaystyle \frac{3}{2}}{x}^2+{\displaystyle \frac{1}{2}}x$

$b_4(x)$$=$$x^4-2x^3+x^2-{\displaystyle \frac{1}{30}}$

$\mathrm{⋮}$