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 [25] and the production matrix defined in [7]. By modifying the marked succession rule introduced in [18], 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 [17], 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. AMS Subject Classification: 05A15, 05A05, 11B39, 11B73, 15B36, 15A06, 05A19, 11B83.