Asymptotic zero distribution of the polynomials $\widetildeΞ_n$

Abstract

We consider the polynomials $Ξ_n$ 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 $0< x_{n,1} \le \cdots \le x_{n,n-1} < 1$ and forming the empirical measures \[ μ_n := \frac1{n-1}\sum_{k=1}^{n-1}δ_{x_{n,k}}, \] we prove that $(μ_n)_{n\ge2}$ 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~$μ$.

Theorems & Definitions (12)

• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Lemma 1.3
• proof
• Lemma 2.1
• proof
• Theorem 2.2: Limiting density
• proof