The asymptotics of the Touchard polynomials: a uniform approximation

2016/06/11 by R. B. Paris, R B Paris, Paris, R B
Computer Science · Engineering · Mathematics · #30E15 #33C45 #34E05 #41A30 #41A60 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #math.CA #msc:30E15 #msc:33C45 #msc:34E05 #msc:41A30 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1606.03576

7 pages, 1 figure

openalex publication_date 2016/06/11 · arxiv created 2016/06/28 · arxiv updated 2016/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 \citeP1. 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.

Citations

• The asymptotics of the Touchard polynomials

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