The asymptotics of the generalised Hermite–Bell polynomials

Richard B. Paris

Research output: Contribution to journal › Article

5 Citations (Scopus)

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 languageEnglish
Pages (from-to)216-226
Number of pages11
JournalJournal of Computational and Applied Mathematics
Volume232
Issue number2
DOIs
Publication statusPublished - 15 Oct 2009

Fingerprint

Generalized Polynomials
Polynomials
Airy Functions
Steepest Descent
Uniform Approximation
Hermite Polynomials
Asymptotic Approximation
Saddlepoint
Asymptotic Expansion
Cover
Numerical Results
Generalise
Polynomial
Integer
Zero

Cite this

Paris, Richard B. / The asymptotics of the generalised Hermite–Bell polynomials. In: Journal of Computational and Applied Mathematics. 2009 ; Vol. 232, No. 2. pp. 216-226.
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Paris, RB 2009, 'The asymptotics of the generalised Hermite–Bell polynomials', Journal of Computational and Applied Mathematics, vol. 232, no. 2, pp. 216-226. https://doi.org/10.1016/j.cam.2009.05.031

The asymptotics of the generalised Hermite–Bell polynomials. / Paris, Richard B.

In: Journal of Computational and Applied Mathematics, Vol. 232, No. 2, 15.10.2009, p. 216-226.

Research output: Contribution to journal › Article

TY - JOUR

T1 - The asymptotics of the generalised Hermite–Bell polynomials

AU - Paris, Richard B.

PY - 2009/10/15

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N2 - 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.

AB - 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.

U2 - 10.1016/j.cam.2009.05.031

DO - 10.1016/j.cam.2009.05.031

M3 - Article

VL - 232

SP - 216

EP - 226

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

SN - 0377-0427

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Paris RB. The asymptotics of the generalised Hermite–Bell polynomials. Journal of Computational and Applied Mathematics. 2009 Oct 15;232(2):216-226. https://doi.org/10.1016/j.cam.2009.05.031

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