A Symbolic Treatment of Riordan Arrays

TL;DR

This paper introduces an umbral symbolic approach to Riordan arrays, deriving fundamental properties and new formulas, including a non-recursive expression, enhancing understanding of their structure and relations.

Contribution

It presents a novel umbral symbolic method for Riordan arrays, leading to new formulas and recurrences that simplify their analysis.

Findings

Derived a non-recursive formula for Riordan arrays

Established new recurrence relations for Riordan arrays

Connected Riordan array properties to umbral Abel's identity

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.