On the $r$-shifted central triangles of a Riordan array

TL;DR

This paper investigates a family of matrices derived from a Riordan array by shifting indices, demonstrating they can be factored into products of Riordan arrays and establishing transition relations using Lagrange inversion.

Contribution

It introduces a new family of Riordan arrays obtained by index shifts and provides factorization and transition relations for these matrices.

Findings

Each shifted matrix can be factored into a product of Riordan arrays.

Transition relations connect elements within the family.

Lagrange inversion is key to the proofs.

Abstract

Let $A$ be a proper Riordan array with general element $a_{n,k}$. We study the one parameter family of matrices whose general elements are given by $a_{2n+r, n+k+r}$. We show that each such matrix can be factored into a product of a Riordan array and the original Riordan array $A$, thus exhibiting each element of the family as a Riordan array. We find transition relations between the elements of the family, and examples are given. Lagrange inversion is used as a main tool in the proof of these results.