Uniform asymptotic expansions

In this lecture, I first review an extension of the method of steepest descents given by Chester, Friedman and Ursell (1957). This extension allows one to derive asymptotic expansions which hold uniformly in regions where two saddle points may coalesce at a single point. As an illustration, I give an outline of a derivation of a uniform asymptotic expansion of the Hermite polynomial H-n (root2n+1t). Next, I present a modification of the method of Chester, Friedman and Ursell, which can handle situations where two saddle points may coalesce at two distinct locations. Such situations occur in the cases of Meixner, Meixner-Pollaczek, and Krawtchouk polynomials.