Complementary Romanovski-Routh polynomials and their zeros

TL;DR

This paper investigates the zeros of complementary Romanovski-Routh polynomials, providing bounds, density, and asymptotic connections, which are crucial for numerical integration methods like Gaussian quadrature.

Contribution

It offers new properties and insights into the zeros of these polynomials, enhancing understanding of their behavior and applications in numerical analysis.

Findings

Derived extreme bounds for zeros

Analyzed zero density and convexity

Connected polynomials to classical orthogonal polynomials asymptotically

Abstract

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.