The asymptotics of the generalised Hermite–Bell polynomials

The Hermite–Bell polynomials are defined by H n r ( x ) = ( − ) n exp ( x r ) ( d / d x ) n exp ( − x r ) 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 H n r ( x ) as n → ∞ using the method of s...

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Bibliographic Details
Published inJournal of computational and applied mathematics Vol. 232; no. 2; pp. 216 - 226
Main Author Paris, R.B.
Format Journal Article
LanguageEnglish
Published Kidlington Elsevier B.V 15.10.2009
Elsevier
Subjects
Approximations and expansions
Asymptotic expansion
Exact sciences and technology
Extreme zeros
Hermite–Bell polynomials
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Sciences and techniques of general use
Special functions
Uniform approximation
Hermite–Bell polynomials
33C45
Asymptotic expansion
34E20
Extreme zeros
Uniform approximation
34E05
Hermite polynomial
Hermite-Bell polynomials
Integer
Bell polynomial
Numerical analysis
Applied mathematics
Descent method
Asymptotic approximation
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Summary:The Hermite–Bell polynomials are defined by H n r ( x ) = ( − ) n exp ( x r ) ( d / d x ) n exp ( − x r ) 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 H n r ( x ) 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 H n r ( x ) is derived as n → ∞ . Numerical results are presented to illustrate the accuracy of the various expansions.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 23
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2009.05.031