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The asymptotics of the generalised Hermite–Bell polynomials

✍ Scribed by R.B. Paris

Publisher
Elsevier Science
Year
2009
Tongue
English
Weight
992 KB
Volume
232
Category
Article
ISSN
0377-0427

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✦ Synopsis

The Hermite-Bell polynomials are defined by H r n (x) = (-) n exp(x r )(d/dx) 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 r n (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 r n (x) is derived as n → ∞.

Numerical results are presented to illustrate the accuracy of the various expansions.

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