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 𝐏(τ_x>n) of the first passage times τ_x:=inf{n≥1:x+S_n≤0} for x≥0. We also derive asymptotic expansion for local probabilities 𝐏(S_n=x,τ_0>n). Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.