Expansions for random walks conditioned to stay positive

We consider a one-dimensional random walk $S_n$ with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution $\mathbf{P}({\tau }_x>n)$ of the first passage times ${\tau }_x:=\inf \{n\ge 1:x+S_n\le 0\}$ for $x\ge 0.$ We also derive asymptotic expansion for local probabilities $\mathbf{P}(S_n=x,{\tau }_0>n)$. Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.