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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]