We consider the family of polynomials
$$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that, for $k, l \in \llbracket 0, n \rrbracket$,
$$\delta^k\left(H_l\right)(0) = \begin{cases} 1 & \text{if } k = l \\ 0 & \text{if } k \neq l \end{cases}$$