On asymptotic expansions in local limit theorems for equiprobable schemes of allocating particles to distinguishable cells

We consider equiprobable schemes of allocating $n$ indistinguishable and distinguishable particles to $N$ distinguishable cells. Under the condition that $n,N\to\infty$ so that $N-k\to\infty$ and
$$ 0<\alpha_0\le\alpha=(n-kr)/(N-k)\le\alpha_1<\infty, $$
where $\alpha_0$, $\alpha_1$ are constants, we arrive at asymptotic expansions in local theorems on large deviations which approximate the probabilities $\mathsf P\{\theta_r(n,N)=k\}$ and $\mathsf P\{\mu_r(n,N)=k\}$, where $\theta_r(n,N)$ and $\mu_r(n,N)$ are the random variables equal to the number of cells with exactly $r$ particles each in the schemes under consideration, $r$ is fixed.