The parabolic mean operator and related polynomial sequences 2: applications

In this paper we consider some computational applications of the conjugate Appell-Simpson polynomial sequences, recently introduced. Particularly, the Umbral-Simpson interpolation is studied both in the continuous and discrete case. Two new classes of interpolatory quadrature rules, which include the classic 1/3 Simpson formula and the midpoint formula, are obtained. A new operator of Jakimovski and Leviatan type is derived for the approximation of real functions on the semi-infinite axes. Several numerical examples are given. The results of this paper appear to be new.