A symbolic treatment of Riordan arrays

Abstract

Abstract. We approach Riordan arrays and their generalizations via umbral symbolic methods. This new approach allows us to derive fundamental aspects of the theory of Riordan arrays as immediate consequences of the umbral version of the classical Abel's identity for polynomials. In particular, we obtain a novel non-recursive formula for Riordan arrays and derive, from this new formula, some known recurrences and a new recurrence relation for Riordan arrays.

On One-Parameter Catalan Arrays

“…The literature on Riordan arrays is vast and constantly growing (see, for instance, [1,4,5,7,9,10,12,13,14,15,16,18,20,21,22]). In this section we recall the fundamental theorem of Riordan arrays, and discuss some general facts regarding the generalized Sheffer polynomial sequence associated with any Riordan array.…”

“…The classical Pascal array P of binomial numbers n k is a Riordan array, since we can write P = (p(z), zp(z)), where p(z) = 1 1−z . Given any real number r, the Riordan array P (r) = p(rz), zp(rz) = 1 1−rz , z 1−rz is known as the generalized Pascal array of parameter r. Its entries are given by P (r…”

“…Let us denote them by C and B, respectively. By way of 1 Corresponding author. 2 Work performed within the activities of Centro de Análise Funcional, Estruturas Lineares e Aplicações (Faculdade de Ciências, Universidade de Lisboa) and supported by the fellowship SFRH/BPD/48223/2008 provided by the Portuguese Foundation for Science and Technology (FCT).…”