Asymptotic for the rightmost zeros of Bell and Eulerian polynomials

Abstract

Write ${\zeta }_m(n)$, $1\le m\le n-1$, 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 $$\underset{n\to \infty }{\lim }\frac{{\zeta }_m(n)}{-m{\left(\frac{m}{m+1}\right)}^{n-1}}=1.$$ 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.