The asymptotics of the generalised Hermite–Bell polynomials

The Hermite–Bell polynomials are defined by Hnr(x)=(−)nexp(xr)(d/dx)nexp(−xr) for n=0,1,2,…n=0,1,2,… and integer r≥2r≥2 and generalise the classical Hermite polynomials corresponding to r=2r=2. We obtain an asymptotic expansion for Hnr(x) as n→∞n→∞ using the method of steepest descents. For a certain value of xx, 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 Hnr(x) is derived as n→∞n→∞. Numerical results are presented to illustrate the accuracy of the various expansions.