  In this paper we consider an irreducible random walk on the integer lattice
$\mathbb{Z}$ that is in the domain of normal attraction of a strictly stable
process with index $\alpha\in (1, 2)$ and obtain the asymptotic form of the
distribution of the hitting time of the origin and that of the transition
probability for the walk killed when it hits a finite set. The asymptotic forms
obtained are valid uniformly in the natural domain of the space and time
variables.
