Matrix characterizations of Riordan arrays

Here we discuss two matrix characterizations of Riordan arrays, P-matrix characterization and A-matrix characterization. P-matrix is an extension of the Stieltjes matrix defined in [28] and the production matrix defined in [8]. By modifying the marked succession rule introduced in [21], a combinatorial interpretation of the P-matrix is given. The P-matrix characterizations of some subgroups of Riordan group are presented, which are used to find some algebraic structures of the subgroups. We also give the P-matrix characterizations of the inverse of a Riordan array and the product of two Riordan arrays. A-matrix characterization is defined in [20], and it is proved to be a useful tool for a Riordan array, while, on the other side, the A-sequence characterization is very complex sometimes. By using the fundamental theorem of Riordan arrays, a method of construction of A-matrix characterizations from Riordan arrays is given. The converse process is also discussed. Several examples and applications of two matrix characterizations are presented.