Some explicit formulas in the theory of branching stochastic processes with discrete time and one-type particles

Let $Z_0=1$, $Z_n$, $n=1,2,\dots,$ be the number of particles belonging to the $n$-th generation of a Halton–Watson branching process, $p_{nr}=\mathbf P\{Z_n=r\}$ and $F_n(z)=\sum_{r\ge0}p_{nr}z^r$. It is supposed that $m=\mathbf EZ_1\ne1$ and $F(z)=F_1(z)$ be regular at the point $z=1$ for $m<1$. In the paper some formulas and an asymptotic expansion for the probabilities $p_{nr}$ are obtained.