= Laplace transform of a Laguerre polynomial
{c}
{title2=$\mathcal L[L_n](p)=(p-1)^n/p^{n+1}$}
For the ordinary <Laguerre polynomial> $L_n(t)=\sum_{j=0}^n\binom nj(-t)^j/j!$, apply the <Laplace transform> to each term, using $\mathcal L[t^j](p)=j!/p^{j+1}$ for $\operatorname{Re}p>0$. The <binomial theorem> gives
$$
\mathcal L[L_n](p)=\frac1p\sum_{j=0}^n\binom nj(-p^{-1})^j=\frac1p(1-p^{-1})^n.
$$
This proves the formula for every nonnegative <integer> $n$. In particular $L_2(t)=1-2t+t^2/2$ has transform $(p-1)^2/p^3$.
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