  In this paper, we develop the Riemann-Hilbert method to study the asymptotics
of discrete orthogonal polynomials on infinite nodes with an accumulation
point. To illustrate our method, we consider the Tricomi-Carlitz polynomials
$f_n^{(\alpha)}(z)$ where $\alpha$ is a positive parameter. Uniform
Plancherel-Rotach type asymptotic formulas are obtained in the entire complex
plane including a neighborhood of the origin, and our results agree with the
ones obtained earlier in [{\it SIAM J.\;Math.\;Anal} {\bf 25} (1994)] and
[{{\it Proc.\;Amer.\;Math.\;Soc.\,}{\bf138} (2010)}].
