On Romanovski-Jacobi polynomials and their related approximation results

The aim of this article is to present the essential properties of a finite class of orthogonal polynomials related to the probability density function of theF-distribution over the positive real line. We introduce some basic properties of the Romanovski-Jacobi polynomials, the Romanovski-Jacobi-Gauss type quadrature formulae and the associated interpolation, discrete transforms, spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kinds of finite orthogonal polynomials and other classes of infinite orthogonal polynomials. Moreover, we derive spectral Galerkin schemes based on a Romanovski-Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line. Two numerical examples demonstrate the robustness and accuracy of the schemes.