The efficacy of numerical methods like integral estimates via Gaussian quadrature formulas depends on the localization of the zeros of the associated family of orthogonal polynomials. In this regard, following the renewed interest in quadrature formulas on the unit circle, and $R_{II}$-type polynomials, which include the complementary Romanovski-Routh polynomials, in this work we present a collection of properties of their zeros. Our results include extreme bounds, convexity, and density, alongside the connection of such polynomials to classical orthogonal polynomials via asymptotic formulas.

翻译：数值方法（如通过高斯求积公式进行积分估计）的有效性取决于相关正交多项式族零点的定位。为此，随着对单位圆上求积公式以及包含互补的Romanovski-Routh多项式的$R_{II}$型多项式兴趣的重新兴起，本文展示了其零点的一系列性质。我们的结果包括极值界限、凸性和密度，同时通过渐近公式揭示了此类多项式与经典正交多项式的联系。