Asymptotic for the rightmost zeros of Bell and Eulerian polynomials

2024/04/04 by Antonio J. Durán, Durán, Antonio J.
Mathematics · #11B73 #11B83 #26C10 #FOS: Mathematics #Functional Equations Stability Results #Mathematical functions and polynomials #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2404.03249

openalex publication_date 2024/04/04 · openalex created_date 2024/04/06 · openalex updated_date 2026/07/30

Abstract

Write ζ_m(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 lim_{n→ ∞}\fracζ_m(n)-m((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.

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