Asymptotic zero distribution of the polynomials $\widetilde{\Xi}_n$

TL;DR

This paper investigates the asymptotic distribution of zeros of rescaled polynomials related to Eulerian polynomials of type B, proving convergence to a deterministic measure with explicit density and distribution.

Contribution

It establishes the weak convergence of zero distributions of the rescaled polynomials to a specific deterministic measure, with explicit formulas for the limiting density and distribution.

Findings

Zeros of rescaled polynomials converge to a deterministic measure.

Explicit formula for the limiting density of zeros.

Numerical experiments confirm theoretical convergence.

Abstract

We consider the polynomials $\Xi_n$ introduced in~\cite{TallaWaffo2025arxiv2511.02843} and studied in further details in\cite{TallaWaffo2026arxiv2602.16761}, which are expressed in terms of Eulerian polynomials of type~B, and study the zero distribution of the rescaled family \[ \widetilde{\Xi}_n(x) := \Xi_n(\sqrt{x}), \qquad n\ge 2. \] Writing the zeros of $\widetilde{\Xi}_n$ in the interval $(0,1)$ as $0< x_{n,1} \le \cdots \le x_{n,n-1} < 1$ and forming the empirical measures \[ \mu_n := \frac1{n-1}\sum_{k=1}^{n-1}\delta_{x_{n,k}}, \] we prove that $(\mu_n)_{n\ge2}$ converges weakly to a deterministic probability measure $\mu$ supported on $(0,1)$. We give an explicit formula for the limiting density and the limiting distribution function of~$\mu$. The proof is based on a representation of $\Xi_n$ in terms of type~B Eulerian polynomials, a ratio asymptotic for these polynomials derived from a classical series identity, and the Stieltjes transform method. We also provide numerical experiments illustrating the convergence of the empirical zero distributions to~$\mu$.