Distribution of roots of Eulerian polynomials

Q: What type of study is this?

This is a Quantitative Study study.

Paul Melotti

doi:10.48550/arxiv.2507.15908

July 21, 2025Open Access

Distribution of roots of Eulerian polynomials

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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.

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Implication

Empirical measures of roots converge to a log-Cauchy distribution, suggesting stability in asymptotic moments.

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/68d46fdc31b076d99fa6a6c3 https://doi.org/10.48550/arxiv.2507.15908

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Distribution of roots of Eulerian polynomials

Key Points

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Implication

Abstract

Discussion

Cite This Study

Distribution of roots of Eulerian polynomials

Key Points

Ask about this paper

Implication

Abstract

Discussion

Cite This Study

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