We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting
$$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$
Using the result $A_k'(X) = A_{k-1}(X-a)$, deduce, for $j$ and $k$ elements of $\llbracket 0, n \rrbracket$, the value of $A_k^{(j)}(ja)$. Distinguish according to whether $j < k$, $j = k$ or $j > k$.