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：通过 Gaussian 二次曲线公式进行整体估计等数字方法的效力取决于正方形圆圆形相关组合零点的本地化。 在这方面,在对单位圆的二次曲线公式和美元(R ⁇ II})美元类型的多面形公式重新产生兴趣之后,我们在此工作中展示了一个零属性的集合。 我们的结果包括极端界限、共性和密度,以及这种多面形的极端界限、共性和密度,同时将这种多面形与古典或多面形的单面公式连接在一起。