For all $n \in \mathbb{N}$ and all $z \in \mathbb{C}$, let
$$Q_{n}(z) = n! \int_{0}^{1} \frac{\mathrm{e}^{z\gamma_{n}(t)}}{(\mathrm{e}^{\gamma_{n}(t)} - 1)\gamma_{n}(t)^{n-1}} \,\mathrm{d}t$$
where $\gamma_{n}(t) = (2n+1)\pi\,\mathrm{e}^{2\mathrm{i}\pi t}$. Show that
$$\forall n \in \mathbb{N}^{*},\, \forall z \in \mathbb{C}, \quad Q_{n}(z+1) - Q_{n}(z) = n z^{n-1}.$$