Let \(\left(U_{\alpha}ight)_{\alpha>0}\) be the \(\alpha\)-potential operator of a \(\mathrm{BM}^{d}\). Use the resolvent equation to prove the following formulae for \(f \in \mathcal{B}_{b}\) and \(x \in \mathbb{R}^{d}\) :

\[\frac{d^{n}}{d \alpha^{n}} U_{\alpha} f(x)=n !(-1)^{n} U_{\alpha}^{n+1} f(x)\]

and

\[\frac{d^{n}}{d \alpha^{n}}\left(\alpha U_{\alpha}ight) f(x)=n !(-1)^{n+1}\left(\mathrm{id}-\alpha U_{\alpha}ight) U_{\alpha}^{n} f(x)\]