The asymptotics of the generalised Hermite–Bell polynomials

Richard B. Paris

Research output: Contribution to journal › Article

  • 4 Citations

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
StatePublished - 15 Oct 2009

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Hermite polynomials
Polynomials
Bell polynomials
Airy functions
Steepest descent
Uniform approximation
Asymptotic approximation
Saddlepoint
Asymptotic expansion
Cover
Numerical results
Generalise
Integer
Zero

Cite this

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

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

Research output: Contribution to journal › Article

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title = "The asymptotics of the generalised Hermite–Bell polynomials",
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.",
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volume = "232",
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journal = "Journal of Computational and Applied Mathematics",
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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. DOI: 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

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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.

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DO - 10.1016/j.cam.2009.05.031

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JO - Journal of Computational and Applied Mathematics

T2 - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

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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. Available from, DOI: 10.1016/j.cam.2009.05.031

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