Constructing Pseudo-involutions in the Riordan Group

TL;DR

This paper explores the construction of pseudo-involutions within the Riordan group, linking them to Fibonacci and Lucas numbers, and provides a MATLAB algorithm for their construction.

Contribution

It introduces new methods for constructing pseudo-involutions in the Riordan group based on specific generating functions and their properties.

Findings

Connections between Riordan arrays and Fibonacci/Lucas numbers

A theorem for constructing pseudo-involutions from certain generating functions

A MATLAB algorithm for constructing these pseudo-involutions

Abstract

Riordan arrays, denoted by pairs of generating functions (g(z), f(z)), are infinite lower-triangular matrices that are used as combinatorial tools. In this paper, we present Riordan and stochastic Riordan arrays that have connections to the Fibonacci and modified Lucas numbers. Then, we present some pseudo-involutions in the Riordan group that are based on constructions starting with a certain generating function g(z). We also present a theorem that shows how to construct pseudo-involutions in the Riordan group starting with a certain generating function f(z) whose additive inverse has compositional order 2. The theorem is then used to construct more pseudo-involutions in the Riordan group where some arrays have connections to the Fibonacci and modified Lucas numbers. A MATLAB algorithm for constructing the pseudo-involutions is also given.