Asymptotic expansions for random walks conditioned to stay positive

We consider a one-dimensional random walk Sn with i.i.d. increments with zero mean and finite variance. We derive asymptotic expansions for the tail distribution P(τx>n) of the first passage times τx:=inf{n≥1:x+Sn≤0}, x≥0. We also derive asymptotic expansion for local probabilities P(Sn=x,τ0>n). Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.