  We consider a recurrent random walk of i.i.d. increments on the
one-dimensional integer lattice and obtain a formula relating the hitting
distribution of a half-line with the potential function, $a(x)$, of the random
walk. Applying it, we derive an asymptotic estimate of $a(x)$ and thereby a
criterion for $a(x)$ to be bounded on a half-line. The application is also made
to estimate some hitting probabilities as well as to derive asymptotic
behaviour for large times of the walk conditioned never to visit the origin.
