Mathematica Eterna
Open Access

Abstract

The asymptotics of the Touchard polynomials

R. B. Paris

We examine the asymptotic expansion of the Touchard polynomials Tn(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 Tn(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.