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 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.