  Using the steepest descent method of Deift-Zhou, we derive locally uniform
asymptotic formulas for the Meixner polynomials. These include an asymptotic
formula in a neighborhood of the origin, a result which as far as we are aware
has not yet been obtained previously. This particular formula involves a
special function, which is the uniformly bounded solution to a scalar
Riemann-Hilbert problem, and which is asymptotically (as the polynomial degree
$n$ tends to infinity) equal to the constant $"1"$ except at the origin.
Numerical computation by using our formulas, and comparison with earlier
results, are also given.
