Asymptotic for the rightmost zeros of Bell and Eulerian polynomials

Write ζₘ(n), 1≤ m≤ n-1, for the negative zeros of the n-th Bell polynomial, ordered in decreasing size. In this paper, we prove the following asymptotic: for a positive integer m we have limn→ ∞ζₘ(n)-m(/m+1)ⁿ⁻¹=1. The approach used to find this asymptotic applies to many other significant families of polynomials. In particular, analogous asymptotics are also proved for the negative rightmost zeros of Eulerian polynomials, r-Bell polynomials, linear combinations of K consecutive Bell polynomials and many others.