For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, the Bernoulli polynomial is defined by
$$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$
where $\omega(t) = e^{2i\pi t}$. Show that, for all $n \in \mathbb{N}^{*}$ and all $z \in \mathbb{C}$,
$$B_{n}(z+1) - B_{n}(z) = n z^{n-1}.$$