Asymptotic, analysis of the distributions in problems with two boundaries. I

Let $\{\xi_k\}_{k=1}^{\infty}$ be a sequence of i. i. d. integer valued random variables, $\mathbf M\xi_i=0$, $S_n=\xi_1+\dots+\xi_n\ (S_0=0)$, and the function $\mathbf M(\lambda^{\xi_1}\colon\xi_1>0)$ is rational. For $a>0$, $b>0$ we introduce the random variable
$$ N=\min\{k\colon S_k\notin[-a,b)\}. $$

The complete asymptotic (as $n\to\infty$) expansions of the probabilities
$$ \mathbf P\{S_n=k,\ N>n\},\ k\in[-a,b),\quad \mathbf P\{S_N=k,\ N=n\},\ k\notin[-a,b), $$
are obtained for $a=a(n)=o(n)$, $b=b(n)=o(n)$, $a\to\infty$, $b\to\infty$, $a+b\ge C\sqrt n$.