Computational aspects of fractional Romanovski–Bessel functions

Nowadays, various groups of classical orthogonal polynomials are part of several spectral algorithms for basic mathematical machinery. It is well known that the polynomial-based spectral methods can provide high accuracy for partial differential equations with smooth solutions, but may not have any advantage when the solutions exhibit weakly singular behavior. However, to establish accurate spectral schemes for problems with non-smooth solutions, one has to enrich the polynomial-based approximation space by introducing special functions that capture the singular behavior of the underlying problem. The main purpose of this paper is to introduce a new finite class of orthogonal functions based on the Romanovski–Bessel polynomials, and to investigate their basic general properties such as the fractional Romanovski–Bessel–Gauss-type quadrature formulae together with fundamental results of approximation for certain weighted projection operators for certain weighted projection operators described in suitable weighted Sobolev spaces. The relationship between such new orthogonal finite set of functions and the other families of infinite fractional orthogonal functions such as fractional Laguerre functions and generalized Laguerre fractional modified functions are derived.