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September 22, 20250 citationsOpen Access

Distribution of roots of Eulerian polynomials

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PMPaul Melotti

Key Points

  • The roots of Eulerian polynomials converge to a specific log-Cauchy distribution, enhancing understanding of their distribution.
  • Empirical measures show that asymptotic moments remain constant, indicating a strong relationship with Nörlund's numbers.
  • The study illustrates the convergence behavior of polynomial roots and their significance in mathematical analysis.
  • These findings may have implications for further research on polynomial roots and their associated distributions.

Abstract

We show that the empirical measures of roots of Eulerian polynomials converge to a certain log-Cauchy distribution. To do so, we show that the moments of the roots of a related family of polynomials not only converge, but are in fact ultimately constant. These asymptotic moments are expressed in terms of Nörlund's numbers.

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Cite This Study

Paul Melotti (2025) studied this question.

synapsesocial.com/papers/68d46fdc31b076d99fa6a6c3https://doi.org/10.48550/arxiv.2507.15908
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