Series representations and asymptotic expansions for the density of the supremum of a stable process

We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $\alpha$ is not rational. In the case when parameters $\alpha$ and $\rho =\text{p}(X_1>0)$ satisfy $\rho +k=l\mathrm{/}\alpha$ for some integers $k,l\ge 1$ we prove that these asymptotic expansions are in fact convergent series representations of the density of supremum.