Asymptotic for the rightmost zeros of Bell and Eulerian polynomials
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910611717226496 |
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| author | Durán, Antonio J. |
| author_facet | Durán, Antonio J. |
| contents | Write $ζ_m(n)$, $1\le m\le n-1$, 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 $$ \lim_{n\to \infty}\frac{ζ_m(n)}{-m\left(\displaystyle\frac{m}{m+1}\right)^{n-1}}=1. $$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03249 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic for the rightmost zeros of Bell and Eulerian polynomials Durán, Antonio J. Number Theory 11B83, 11B73, 26C10, Write $ζ_m(n)$, $1\le m\le n-1$, 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 $$ \lim_{n\to \infty}\frac{ζ_m(n)}{-m\left(\displaystyle\frac{m}{m+1}\right)^{n-1}}=1. $$ 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. |
| title | Asymptotic for the rightmost zeros of Bell and Eulerian polynomials |
| topic | Number Theory 11B83, 11B73, 26C10, |
| url | https://arxiv.org/abs/2404.03249 |