The asymptotics of the generalised Hermite-Bell polynomials |
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Authors: | RB Paris |
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Institution: | Division of Complex Systems, University of Abertay Dundee, Dundee DD1 1HG, UK |
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Abstract: | The Hermite-Bell polynomials are defined by 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 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 is derived as n→∞. Numerical results are presented to illustrate the accuracy of the various expansions. |
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Keywords: | 33C45 34E05 34E20 |
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