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In this paper, by using Riordan arrays and a particular model of lattice paths, we are able to generalize in several ways an identity proposed by Lou Shapiro by giving both an algebraic and a combinatorial proof. The identities studied in this paper allow us to move from an arithmetic progression, and other C-finite sequences, to a geometric progression in terms of Riordan array transformations and vice versa, via the Riordan array inverse. [ABSTRACT FROM AUTHOR]

Abstract: In the realm of the Riordan group, we consider the characterization of Riordan arrays by means of the - and -sequences. It corresponds to a horizontal construction of a Riordan array, whereas the traditional approach is through column generating functions. We show how the - and -sequences of the product of two Riordan arrays are derived from those of the two factors; similar results are obtained for the inverse. We also show how the sequence characterization is applied to construct easily a Riordan array. Finally, we give the characterizations relative to some subgroups of the Riordan group, in particular, of the hitting-time subgroup. [Copyright &y& Elsevier]

Abstract: In this paper we present the theory of implicit Riordan arrays, that is, Riordan arrays which require the application of the Lagrange Inversion Formula to be dealt with. We show several examples in which our approach gives explicit results, both in finding closed expressions for sums and, especially, in solving classes of combinatorial sum inversions. [Copyright &y& Elsevier]

Abstract: In this paper we present the theory of implicit Riordan arrays, that is, Riordan arrays which require the application of the Lagrange Inversion Formula to be dealt with. We show several examples in which our approach gives explicit results in solving classes of combinatorial sum inversions. [Copyright &y& Elsevier]

Abstract: We study many properties of Cauchy numbers in terms of generating functions and Riordan arrays and find several new identities relating these numbers with Stirling, Bernoulli and harmonic numbers. We also reconsider the Laplace summation formula showing some applications involving the Cauchy numbers. [Copyright &y& Elsevier]

Abstract: Recently, the concept of the complementary array of a Riordan array (or recursive matrix) has been introduced. Here we generalize the concept and distinguish between dual and complementary arrays. We show a number of properties of these arrays, how they are computed and their relation with inversion. Finally, we use them to find explicit formulas for the elements of many recursive matrices. [Copyright &y& Elsevier]

Abstract: Historically, there exist two versions of the Riordan array concept. The older one (better known as recursive matrix) consists of bi-infinite matrices ( implies ), deals with formal Laurent series and has been mainly used to study algebraic properties of such matrices. The more recent version consists of infinite, lower triangular arrays ( implies ), deals with formal power series and has been used to study combinatorial problems. Here we show that every Riordan array induces two characteristic combinatorial sums in three parameters . These parameters can be specialized and generate an indefinite number of other combinatorial identities which are valid in the bi-infinite realm of recursive matrices. [Copyright &y& Elsevier]