On the halves of a Riordan array and their antecedents

Abstract

Every Riordan array has what we call a horizontal half and a vertical half. These halves of a Riordan array have been studied separately before. Here, we place them in a common context, showing that one may be obtained from the other. Using them, we provide a canonical factorization of elements of the associated or Lagrange subgroup of the Riordan group. The vertical half matrix is shown to be an element of the hitting-time group. We also ask and answer the question: given a Riordan array, when is it the half (either horizontal of vertical) of a Riordan array?