### 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 |
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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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## Cite this

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