 Factorial monomial basis for polynomial differentiation

On polynomials of degree at most $n$, the derivative-evaluation functionals $e_j(\varphi )={\varphi }^{(j)}(0)$ are dual to $p_j=x^j/j!$, since $e_i(p_j)={\delta }_{ij}$. Differentiation maps $p_j$ to $p_{j-1}$, with $p_0$ mapped to zero. Its matrix is a single nilpotent Jordan block with ones above the diagonal, and its dual has the transpose matrix.