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

The potential function and ladder variables of a recurrent random walk on $\mathbb{Z}$ with infinite variance