The asymptotics of the Touchard polynomials

TL;DR

This paper investigates the asymptotic behavior of Touchard polynomials for large degrees and complex arguments, using steepest descents and saddle point analysis to derive accurate expansions.

Contribution

It provides a detailed asymptotic analysis of Touchard polynomials for large n and complex z, including the influence of saddle points and numerical validation.

Findings

Asymptotic expansions depend on the number of contributing saddle points.

Method of steepest descents effectively derives expansions for large n and complex z.

Numerical results confirm the accuracy of the asymptotic approximations.

Abstract

We examine the asymptotic expansion of the Touchard polynomials $T_n(z)$ (also known as the exponential polynomials) for large $n$ and complex values of the variable $z$. In our treatment $|z|$ may be finite or allowed to be large like $O(n)$. We employ the method of steepest descents to a suitable integral representation of $T_n(z)$ and find that the number of saddle points that contribute to the expansion depends on the values of $n$ and $z$. Numerical results are given to illustrate the accuracy of the various expansions.