  In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$
orthogonal with respect to the weight function $w(x) = x^{\alpha} e^{-N cx}$
supported on the infinite nodes $L_N = \{ x_{k,N} = \frac{k^2}{N^2}, k \in
\mathbb{N} \}$. We focus on the "band-saturated region" situation when the
parameter $c > \frac{\pi^2}{4}$. As $n \to \infty$, uniform expansions for
$P_{n, n}(z)$ are achieved for $z$ in different regions in the complex plane.
Typically, the Airy-function expansions and Gamma-function expansions are
derived for $z$ near the endpoints of the band and the origin, respectively.
The asymptotics for the normalizing coefficient $h_{n, N}$, recurrence
coefficients $\mathscr{B}_{n, N}$ and $\mathscr{A}_{n, N}^2$, are also
obtained. Our method is based on the Deift-Zhou steepest descent method for
Riemann-Hilbert problems.
