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ABSTRACT. We give several new characterizations of Riordan Arrays, the most important of which is: if fd nÒk g nÒk2N is a lower triangular array whose generic element d nÒk linearly depends on the elements in a well-defined though large area of the array, then fd nÒk g nÒk2N is Riordan. We also provide some applications of these characterizations to the lattice path theory.

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In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the A-and Z-sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the A-and Z-sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. Finally, we give the characterizations relative to the some subgroups of the Riordan group, in particular of the hitting-time subgroup.

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