For an operator $A: V \rightarrow V$ in the finite-dimensional vector space $V$, we define the exponential function $\exp (A)$ as the following operator in $V$.
$$ \exp (A)=\sum_{k=0}^{\infty} \frac{A^{k}}{k !} $$
We consider the following two operators $D$ and $S$ in the vector space $\mathcal{P}_{n}$ of polynomials of degree $\leq n$ :
$$ D f(t)=\frac{d f}{d t}(t) \quad \text { and } \quad S f(t)=f(t+1),$$
for each $f \in \mathcal{P}_{n}$. Now I want to calculate $\exp (D)$ but I have trouble doing so. Any hints would be appreciated.
The Taylor expansion at the point $a$ is $$ f(x)=\sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!} (x-a)^k. $$ If we calculate it for $f(x+1)$ at the point $x$, we get. $$ f(x+1)=\sum_{k=0}^{\infty} \frac{f^{(k)}(x)}{k!} (x+1-x)^k=\sum_{k=0}^{\infty} \frac{D^{k}f(x)}{k!} 1^k=\exp(D)f(x), $$ so that $\exp (D)=S. \quad \blacksquare$