A novel class of fractionally orthogonal quasi-polynomials and new fractional quadrature formulas

A novel class of quasi-polynomials orthogonal with respect to the fractional integration operator has been developed in this paper. The related Gaussian quadrature formulas for numerical evaluation of fractional order integrals have also been proposed. By allowing the commensurate order of quasi-polynomials to vary independently of the integration order, a family of fractional quadrature formulas has been developed for each fractional integration order, including novel quadrature formulas for numerical approximation of classical, integer order integrals. A distinct feature of the proposed quadratures is high computational efficiency and flexibility, as will be demonstrated in the paper. As auxiliary results, the paper also presents methods for Lagrangian and Hermitean quasi-polynomial interpolation and Hermitean fractional quadratures. The development is illustrated by numerical examples.