A uniform asymptotic expansion of the single variable Bell polynomials

In this paper, we investigate the uniform asymptotic behavior of the single variable Bell polynomials on the negative real axis, to which all zeros belong. It is found that there exists an ascending sequence {Zk}1∞⊂(−e,0) such that the polynomials are represented by a finite sum of infinite asymptotic series, each in terms of the Airy function and its derivative, and the number of series under this sum is 1 in the interval (−∞,Z1) and k+1 in [Zk,Zk+1), k⩾1. Furthermore, it is shown that an asymptotic expansion, also in terms of Airy function and its derivative, completed with error bounds, holds uniformly in (−∞,−δ] for positive δ.