Asymptotic zero distribution of the polynomials $\widetildeΞ_n$

Abstract: We consider the polynomials $Ξ<em>n$ 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 $\widetildeΞ_n$ in the interval $(0,1)$ as $0&lt; x</em>{n,1} \le \cdots \le x_{n,n-1} &lt; 1$ and forming the empirical measures [ μn := \frac1{n-1}\sum{k=1}^{n-1}δ{x{n,k}}, ] we prove that $(μ<em>n)</em>{n\ge2}$ converges weakly to a deterministic probability measure $μ$ supported on $(0,1)$. We give an explicit formula for the limiting density and the limiting distribution function of~$μ$. The proof is based on a representation of $Ξ_n$ 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~$μ$.