A convergent series representation for the density of the supremum of a stable process

We study the density of the supremum of a strictly stable L'evy process. We prove that for almost all values of the index $\alpha$ – except for a dense set of Lebesgue measure zero – the asymptotic series which were obtained in A. Kuznetsov (2010) “On extrema of stable processes” are in fact absolutely convergent series representations for the density of the supremum.