The asymptotics of the generalised Hermite-Bell polynomials
The Hermite-Bell polynomials are defined by \(H^r_n(x)=(-1)^n\exp(x^r)(d/dx)^n\exp(-x^r)\) for \(n=0, 1, 2, \ldots\) and integer \(r\geq 2\) and generalise the classical Hermite polynomials corresponding to \(r = 2\). These polynomials satisfy a differential equation of order \(r\) and are generated by \[ \exp(x^r-(x-z)^r)=\sum_{n=0}^\infty H_n^r(x)\,{z^n \over n!}\;. \] The authors obtain an asymptotic expansion for \(H^r_n(x)\) as \(n\rightarrow\infty\) 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 \(n\rightarrow\infty\). Numerical results are presented to illustrate the accuracy of the various expansions.
scientific article; zbMATH DE number 5608820
scientific article