Characteristic Curves and the exponentiation in the Riordan Lie group: A connection through examples

TL;DR

This paper explores the use of the characteristic method to compute matrix exponentials within the Riordan Lie group, linking classical PDE techniques with algebraic structures and analyzing dynamical systems with symmetries.

Contribution

It introduces a novel application of the characteristic method to Riordan group matrices and connects this with Lie algebra structures and dynamical systems analysis.

Findings

Matrix exponential computation via characteristic method in Riordan group

Identification of symmetries and involutions in linear dynamical systems

Dynamical properties assigned to Pascal Triangle

Abstract

We point out how to use the classical characteristic method, that is used to solve quasilinear PDE's, to obtain the matrix exponential of some lower triangle infinite matrices. We use the Lie Frechet structure of the Riordan group described in [4]. After that we describe some linear dynamical systems in $\mathbb{K}[[x]]$ with a concrete involution being a symmetry or a time-reversal symmetry for them. We take this opportunity to assign some dynamical properties to the Pascal Triangle.