On asymptotic expansions in the domain of large deviations for binomial and Poisson distributions

A random variable $\xi$ having binomial distribution with parameters $n$ and $p\ (0 < p < 1)$ is considered. We find an asymptotic estimate (as $n\to\infty$ and $p$ is a constant) for the probability $\mathsf{P}\{\xi\ge k\}$ assuming that $k\to\infty$ $\,(k\in\mathbb{N})$ in such a way that $p<\alpha_0\le \alpha=k/n\le\alpha_1<1$ $\alpha_0$ and $\alpha_1$ are constants). We also consider a random variable $\eta$ having Poisson distribution with parameter $\lambda > 0$. We find asymptotic estimates for the probability $\mathbb{P}\{\eta\ge k\}$, as $\lambda\to +\infty$, assuming that $k\to\infty$ in such a way that $k\in\mathbb{N}$; $1<\gamma_0\le\gamma=k/\gamma\le\gamma_1$ ($\gamma_0$, $\gamma_1$ are constants). By the saddle-point method, expansions of these probabilities into asymptotic series with respect to the variables $n^{-1}$ and $\lambda^{-1}$ are found. Coefficientsof the series satisfy in the complex domain some recurrence relations with certain initial conditions.