The riemannhilbert approach to global asymptotics of discrete orthogonal polynomials with infinite nodes

In this paper, we develop the RiemannHilbert approach to study the global asymptotics of discrete orthogonal polynomials with infinite nodes. We illustrate our method by concentrating on the Charlier polynomials C _n ^(a)(z)$. We first construct a RiemannHilbert problem Y associated with these polynomials and then establish some technical results to transform Y into a continuous RiemannHilbert problem so that the steepest descent method of Deift and Zhou ([8]) can be applied. Finally, we produce three Airy-type asymptotic expansions for C_n ^(a)(z) in three different but overlapping regions whose union is the entire complex z-plane. When z is real, our results agree with the ones given in the literature. Although our approach is similar to that used by Baik, Kriecherbauer, McLaughlin and Miller ([3]), there are crucial differences in the details. For instance, our expansions hold in much bigger regions. Our results are completely new, and one of them answers a question raised in Bo and Wong ([4]). Asymptotic formulas are also derived for large and small zeros of the Charlier polynomials. © 2010 World Scientific Publishing Company.