The asymptotics of the generalised Hermite-Bell polynomials
type: Item
instance of: scholarly article
title: The asymptotics of the generalised Hermite-Bell polynomials
zbMATH Open document ID: 1176.33014
author: Richard B. Paris
published in: Journal of Computational and Applied Mathematics
publication date: 2009-09-29T00:00:00Z
review text: The Hermite-Bell polynomials are defined by \(H^r_n(x)=(-1)^n\exp(x^r)(d/dx)^n\exp(-x^r)\) for \(n=0, 1, 2, \ldots\) and integer \(r\geq 2\) and generalise the classical Hermite polynomials corresponding to \(r = 2\). These polynomials satisfy a differential equation of order \(r\) and are generated by \[ \exp(x^r-(x-z)^r)=\sum_{n=0}^\infty H_n^r(x)\,{z^n \over n!}\;. \] The authors obtain an asymptotic expansion for \(H^r_n(x)\) as \(n\rightarrow\infty\) 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 \(H^r_n(x)\) is derived as \(n\rightarrow\infty\). Numerical results are presented to illustrate the accuracy of the various expansions.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 34E05
zbMATH DE Number: 5608820
zbMATH Keywords: Hermite-Bell polynomials
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: uniform approximation
zbMATH Keywords: extreme zeros
zbMATH Keywords: Airy functions
reviewed by: W. Koepf
MaRDI profile type: Publication
full work available at URL: j.cam.2009.05.031
OpenAlex ID: W2083599961
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DOI: 10.1016/J.CAM.2009.05.031
description: scientific article; zbMATH DE number 5608820