Polynomial sequences of integral type and recursive matrices

We show how the theory of recursive matrices—bi-infinite matrices in which each row can be recursively computed from the previous one—can be used to formulate a version of the umbral calculus that is also suited for the study of polynomials p(x) taking integer values when the variable x is an integer. In this way, most results of the classical umbral calculus—such as expansion theorems and closed formulas—can be seen as immediate consequences of the two main properties of recursive matrices, namely, the Product Rule and the Double Recursion Theorem.