Strong Asymptotics of the Generating Polynomials of the Stirling Numbers of the Second Kind Original Research Article

For the horizontal generating functions Pn(z)=∑nk=1 S(n, k) zk of the Stirling numbers of the second kind, strong asymptotics are established, as n→∞. By using the saddle point method for Qn(z)=Pn(nz) there are two main results: an oscillating asymptotic for z∈(−e, 0) and a uniform asymptotic on every compact subset of C\[−e, 0]. Finally, an Airy asymptotic in the neighborhood of −e is deduced.