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...

Several important combinatorial arrays, after inserting some minus signs, turn out to be involutions when considered as lower triangular matrices. Among these are the Pascal, RNA, and directed animal matrices. These examples and many others are in the Bell subgroup of the Riordan group. We characterize all such pseudo-involutions by means of a single sequence called the ∆-sequence. Finally we c...

We give recurrence relations for any family of generalized Appell polynomials unifying so some known recurrences of many classical sequences of polynomials. Our main tool to get our goal is the Riordan group. We use the product of Riordan matrices to interpret some relationships between different families of polynomials. Moreover using the Hadamard product of series we get a general recurrence ...

My talk will be divided into two parts, one concerning the diagonals top-left to bottom-right of Riordan arrays, and the other a short history of these matrices. It is important to realize that the results obtained can be extended to recursive matrices, which seems the most natural setting where these properties should be studied. Riordan arrays have been studied from many points of view, corre...

Here presented are the definitions of (c)-Riordan arrays and (c)-Bell polynomials which are extensions of the classical Riordan arrays and Bell polynomials. The characterization of (c)-Riordan arrays by means of the Aand Z-sequences is given, which corresponds to a horizontal construction of a (c)Riordan array rather than its definition approach through column generating functions. There exists...

Abstract We consider the possible Jordan canonical forms of Riordan arrays. prove that there are, in fact, only two such forms. Moreover, transition matrix is group case when given array has one some three specific

The Krawtchouk polynomials play an important role in various areas of mathematics. Notable applications occur in coding theory [11], association schemes [4], and in the theory of group representations [21, 22]. In this note, we explore links between the Krawtchouk polynomials and Riordan arrays, of both ordinary and exponential type, and we study integer sequences defined by evaluating the Kraw...

Riordan group concepts are combined with the basic properties of convolution families of polynomials and Sheffer sequences, to establish a duality law, canonical forms ρ(n,m) = ( n m ) cFn−m(m), c 6= 0, and extensions ρ(x, x − k) = (−1) xcFk(x), where the Fk(x) are polynomials in x, holding for each ρ(n,m) in a Riordan array. Examples ρ(n,m) = ( n m ) Sk(x) are given, in which the Sk(x) are “or...