  Let $X$ be a L\'evy process with absolutely continuous L\'evy measure $\nu$.
Small time polynomial expansions of order $n$ in $t$ are obtained for the tails
$P(X_{t}\geq{}y)$ of the process, assuming smoothness conditions on the L\'evy
density away from the origin. By imposing additional regularity conditions on
the transition density $p_{t}$ of $X_{t}$, an explicit expression for the
remainder of the approximation is also given. As a byproduct, polynomial
expansions of order $n$ in $t$ are derived for the transition densities of the
process. The conditions imposed on $p_{t}$ require that its derivatives remain
uniformly bounded away from the origin, as $t\to{}0$; such conditions are shown
to be satisfied for symmetric stable L\'evy processes as well as for other
related L\'evy processes of relevance in mathematical finance. The expansions
seem to correct asymptotics previously reported in the literature.
