  We approach Riordan arrays and their generalizations via umbral symbolic
methods. This new approach allows us to derive fundamental aspects of the
theory of Riordan arrays as immediate consequences of the umbral version of the
classical Abel's identity for polynomials. In particular, we obtain a novel
non-recursive formula for Riordan arrays and derive, from this new formula,
some known recurrences and a new recurrence relation for Riordan arrays.
