Laplace transform of a Laguerre polynomial ($\mathcal{L}[L_n](p)=(p-1{)}^n/p^{n+1}$)

For the ordinary Laguerre polynomial $L_n(t)={\sum }_{j=0}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)(-t{)}^j/j!$, apply the Laplace transform to each term, using $\mathcal{L}[t^j](p)=j!/p^{j+1}$ for $\mathrm{Re}p>0$. The binomial theorem givesThis proves the formula for every nonnegative integer $n$. In particular $L_2(t)=1-2t+t^2/2$ has transform $(p-1{)}^2/p^3$.