The Characterization of Riordan Arrays and Sheffer-type Polynomial Sequences

Here we present a characterization of Sheffer-type polynomial sequences based on the isomorphism between the Riordan group and Sheffer group and the sequence characterization of Riordan arrays. We also give several alternative forms of the characterization of the Riordan group, Sheffer group and their subgroups. Formulas for the computation of the generating functions of Riordan arrays and Sheffer-type polynomial sequences from the characteristics are shown. Furthermore, the applications of the characteristics to lattice walks and recursive construction of Sheffer-type polynomial sequences are also given.