Asymptotic for the rightmost zeros of Bell and Eulerian polynomials

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Write ζm(n), 1≤m≤n−1, for the negative zeros of the nth Bell polynomial, arranged in decreasing order. In this paper, we prove the following asymptotic: for every positive integer m we have limn→∞ζm(n)−mmm+1n−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.