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

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 \[ \widetildeΞ_n(x) := Ξ_n(\sqrt{x}), \qquad n\ge 2. \] Writing the zeros of ${\tilde{\Xi }}_n$ in the interval $(0,1)$ as $0<x_{n,1}\le \cdots \le x_{n,n-1}<1$ and forming the empirical measures \[ μ_n := \frac1{n-1}\sum_{k=1}^{n-1}δ_{x_{n,k}}, \] we prove that $({\mu }_n{)}_{n\ge 2}$ 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~$\mu$.