Asymptotic Estimates of Stirling Numbers and related asymptotic problems

\def\stn{Stirling numbers } The purpose of the talk is to present new expansions for the \stn which hold uniformly with respect to $m$. The method is based on a modification of the saddle point method. Short tables are given to show the results for $n=10$; further computer experiments confirm the uniform character of our estimates.\\ We describe the underlying type of integrals on which the asymptotic method for the \stn is based. One integral is the Laplace transform $$F_\lambda(z)={1\over \Gamma(\lambda)}\int_0^\infty t^{\lambda-1}e^{-zt}f(t)\,dt.$$\\ We describe the uniform expansion of this integral as $z\to\infty$; $\lambda$ may range through the interval $[0,\infty)$. A uniform asymptotic expansion of a similar contour integral is the basis of the method for the Stirling numbers.