Abstract
Our concern in this paper is with the essential properties of a finite class of orthogonal polynomials with respect to a weight function related to the probability density function of the inverse gamma distribution over the positive real line. We present some basic properties of the Romanovski-Bessel polynomials, the Romanovski-Bessel-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 finite and infinite orthogonal polynomials. Moreover, we propose adaptive spectral tau and collocation methods based on Romanovski-Bessel polynomial basis for solving linear and nonlinear high-order differential equations, respectively.