For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, we define $$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$ where $\omega(t) = e^{2i\pi t}$, and for all $p \in \mathbb{Z}$, $$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$ Show that, for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, $$B_{n}(z) = n! \sum_{k=0}^{n} \frac{z^{k}}{k!} I_{k-n}.$$ Deduce that $B_{n}$ is a monic polynomial of degree $n$.
For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, we define
$$B_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\omega(t)}}{(\mathrm{e}^{\omega(t)} - 1)\omega(t)^{n-1}} \,\mathrm{d}t$$
where $\omega(t) = e^{2i\pi t}$, and for all $p \in \mathbb{Z}$,
$$I_{p} = \int_{0}^{1} \frac{\omega(t)^{p+1}}{\mathrm{e}^{\omega(t)} - 1} \,\mathrm{d}t.$$
Show that, for all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$,
$$B_{n}(z) = n! \sum_{k=0}^{n} \frac{z^{k}}{k!} I_{k-n}.$$
Deduce that $B_{n}$ is a monic polynomial of degree $n$.