Zeros and exponential profiles of polynomials II: Examples
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arXiv
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| Format: | Preprint |
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2025
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| author | Jalowy, Jonas Kabluchko, Zakhar Marynych, Alexander |
| author_facet | Jalowy, Jonas Kabluchko, Zakhar Marynych, Alexander |
| contents | In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions.
We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_11248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros and exponential profiles of polynomials II: Examples Jalowy, Jonas Kabluchko, Zakhar Marynych, Alexander Classical Analysis and ODEs Probability Primary: 26C10, Secondary: 60B10, 33C45, 46L54, 30C10, 30C15, 60B20, 11B73 In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices. |
| title | Zeros and exponential profiles of polynomials II: Examples |
| topic | Classical Analysis and ODEs Probability Primary: 26C10, Secondary: 60B10, 33C45, 46L54, 30C10, 30C15, 60B20, 11B73 |
| url | https://arxiv.org/abs/2509.11248 |