Strong asymptotics of the generating polynomials of the stirling numbers of the second kind

Fur the horizontal generating functions P-n(z)=Sigma (n)(k=1) S(n, k)z(k) o f the Stirling numbers of the second kind. strong asymptotics are establish ed. as n --> x. By using the saddle point method for Q(n)(z) = P-n(nz) ther e are two main results: an oscillating asymptotic for z is an element of(-e . 0) and a uniform asymptotic on every compact subset of C\[ -e, 0]. Finall y. an Airy asymptotic. in thc neighborhood of -e is deduced. (C) 2001 Acade mic Press.