The asymptotics of the generalised Hermite–Bell polynomials

Richard B. Paris

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    5 Citations (Scopus)


    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 languageEnglish
    Pages (from-to)216-226
    Number of pages11
    JournalJournal of Computational and Applied Mathematics
    Issue number2
    Publication statusPublished - 15 Oct 2009


    • Asymptotic expansion
    • Uniform approximation
    • Extreme zeros
    • Hermite-Bell polynomials


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