On Sums, Derivatives, and Flips of Riordan Arrays

TL;DR

This paper explores new operations on Riordan arrays, characterizing their effects on array properties and generating combinatorial identities, thereby expanding understanding of their algebraic structure.

Contribution

It introduces the `Der' and `Flip' operations on Riordan arrays and characterizes their effects on Appell and Lagrange subgroups, providing new tools for combinatorial analysis.

Findings

Characterized when sums of Riordan arrays produce Riordan arrays.

Defined and analyzed `Der' and `Flip' operations on Riordan arrays.

Generated numerous combinatorial identities using these operations.

Abstract

We study three operations on Riordan arrays. First, we investigate when the sum of Riordan arrays yields another Riordan array. We characterize the $A$- and $Z$-sequences of these sums of Riordan arrays, and also identify an analog for $A$-sequences when the sum of Riordan arrays does not yield a Riordan array. In addition, we define the new operations `Der' and `Flip' on Riordan arrays. We fully characterize the Riordan arrays resulting from these operations applied to the Appell and Lagrange subgroups of the Riordan group. Finally, we study the application of these operations to various known Riordan arrays, generating many combinatorial identities in the process.