On the partial sums of Riordan arrays

TL;DR

This paper explores partial sums of Riordan arrays, characterizing resulting matrices, revealing new Riordan arrays and Hessenberg matrices, and connecting these concepts to Fibonacci numbers and phyllotaxis.

Contribution

It introduces two notions of partial sums for Riordan arrays, characterizes the resulting matrices, and links the findings to Fibonacci numbers and biological patterns.

Findings

New Riordan array derived from partial sums of rows

Rectangular array with inverse as a lower Hessenberg matrix

Connection established between these matrices and Fibonacci numbers

Abstract

We define two notions of partial sums of a Riordan array, corresponding respectively to the partial sums of the rows and the partial sums of the columns of the Riordan array in question. We characterize the matrices that arise from these operations. On the one hand, we obtain a new Riordan array, while on the other hand, we obtain a rectangular array which has an inverse that is a lower Hessenberg matrix. We examine the structure of these Hessenberg matrices. We end with a generalization linked to the Fibonacci numbers and phyllotaxis.