Right edge rates of the zeros of $\widetilde{\Xi}_n$ and $\widetilde{\Lambda}_n$

Abstract

We consider the two families of even polynomials $\Xi_n$ and $\Lambda_n$ studied in~\cite{TallaWaffo2026arxiv2602.16761}, together with the rescaled polynomials $\widetilde{\Xi}_n(x):=\Xi_n(\sqrt{x})$ and $\widetilde{\Lambda}_n(x):=\Lambda_n(\sqrt{x})$, $n\ge2$. Their zeros are real, simple, and contained in $(0,1)$. Writing them as $0<x^{(\Xi)}_{1,n}<\cdots<x^{(\Xi)}_{n-1,n}<1$ and $0<x^{(\Lambda)}_{1,n}<\cdots<x^{(\Lambda)}_{n-1,n}<1$, we study the asymptotic behaviour of the largest zeros $x^{(\Xi)}_{n-1,n}$ and $x^{(\Lambda)}_{n-1,n}$. We prove that the two families have different exponential rates at the right endpoint: \[ \frac{1}{n-1}\log\bigl(1-x^{(\Lambda)}_{n-1,n}\bigr)\to-\log4, \qquad \frac{1}{n-1}\log\bigl(1-x^{(\Xi)}_{n-1,n}\bigr)\to-\log9. \] Thus, although the two families share the same global limiting zero distribution, their extreme right zeros approach $1$ on different exponential scales. The proof is based on the representation of $\Xi_n$ and $\Lambda_n$ in terms of Eulerian polynomials of type~B and type~A, respectively, and on an elementary estimate for the smallest negative zero in terms of the first non-constant coefficient.