For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application
$$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
For $n \in \mathbb{N}$ and $x > 0$, express $\left(\delta^n(f)\right)(x)$ using the binomial coefficients $\binom{n}{j}$ and the $f(x+j)$ (where the index $j$ belongs to $\llbracket 0, n \rrbracket$).