The asymptotics of the Touchard polynomials: a uniform approximation

TL;DR

This paper derives uniform asymptotic expansions for Touchard polynomials for large degree and complex arguments, especially near saddle point coalescence, using Airy functions for improved approximation accuracy.

Contribution

It provides a new uniform two-term asymptotic approximation involving Airy functions for Touchard polynomials near saddle point coalescence.

Findings

Derived uniform asymptotic expansion involving Airy functions.

Numerical results confirm high accuracy of the approximations.

Analyzed behavior near the saddle point coalescence at n/x=1/e.

Abstract

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 $z$, where $|z|$ may be finite or allowed to be large like $O(n)$, has been recently considered in \cite{P1}. When $z=-x$ is negative, it is found that there is a coalesence of two contributory saddle points when $n/x=1/e$. Here we determine the expansion when $n$ and $x$ satisfy this condition and also a uniform two-term approximation involving the Airy function in the neighbourhood of this value. Numerical results are given to illustrate the accuracy of the asymptotic approximations obtained.