Abstract
The Hermite–Bell polynomials are defined by [formula missing] for n=0,1,2,… and integer r≥2 and generalise the classical Hermite polynomials corresponding to r=2. We obtain an asymptotic expansion for [formula missing] as n→∞ using the method of steepest descents. For a certain value of x, two saddle points coalesce and a uniform approximation in terms of Airy functions is given to cover this situation. An asymptotic approximation for the largest positive zeros of [formula missing] is derived as n→∞. Numerical results are presented to illustrate the accuracy of the various expansions.
| Original language | English |
|---|---|
| Pages (from-to) | 216-226 |
| Number of pages | 11 |
| Journal | Journal of Computational and Applied Mathematics |
| Volume | 232 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 15 Oct 2009 |
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Paris, R. B. (2009). The asymptotics of the generalised Hermite–Bell polynomials. Journal of Computational and Applied Mathematics, 232(2), 216-226. https://doi.org/10.1016/j.cam.2009.05.031