Jordan Canonical Forms of Riordan Arrays

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

Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups

In a first part, we are concerned with the relationships between polynomials in the two generators of the algebra of Heisenberg—Weyl, its Bargmann—Fock representation with differential operators and the associated one-parameter group.Upon this basis, the paper is then devoted to the groups of Riordan matrices associated to the related transformations of matrices (i.e., substitutions with prefunctions). Thereby, various properties are studied arising in Riordan arrays, in the Riordan group and, more specifically, in the "striped" Riordan subgroups; further, a striped quasigroup and a semigroup are also examined. A few applications to combinatorial structures are also briefly addressed in the Appendix.