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Generalized Riordan arrays. (English) Zbl 1158.05008
Summary: We generalize the concept of Riordan array. A generalized Riordan array with respect to \(c_n\) is an infinite, lower triangular array determined by the pair \((g(t),f(t))\) and has the generic element \(d_{n,k}=[t^n/c_n]g(t)(f(t))^k/c_k\), where \(c_n\) is a fixed sequence of non-zero constants with \(c_{0}=1\).
We demonstrate that the generalized Riordan arrays have similar properties to those of the classical Riordan arrays. Based on the definition, the iteration matrices related to the Bell polynomials are special cases of the generalized Riordan arrays and the set of iteration matrices is a subgroup of the Riordan group. We also study the relationships between the generalized Riordan arrays and the Sheffer sequences and show that the Riordan group and the group of Sheffer sequences are isomorphic. From the Sheffer sequences, many special Riordan arrays are obtained. Additionally, we investigate the recurrence relations satisfied by the elements of the Riordan arrays. Based on one of the recurrences, some matrix factorizations satisfied by the Riordan arrays are presented. Finally, we give two applications of the Riordan arrays, including the inverse relations problem and the connection constants problem.

MSC:
05A15 Exact enumeration problems, generating functions
05A40 Umbral calculus
05A19 Combinatorial identities, bijective combinatorics
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