Ultrametrics, Banach's fixed point theorem and the Riordan group

We interpret the reciprocation process in K[[x]] as a fixed point problem related to contractive functions for certain adequate ultrametric spaces. This allows us to give a dynamical interpretation of certain arithmetical triangles introduced herein. Later we recognize, as a special case of our construction, the so called Riordan group which is a device used in combinatorics. In this manner we give a new and alternative way to construct the proper Riordan arrays. Our point of view allows us to give a natural metric on the Riordan group turning this group into a topological group. This construction allows us to recognize a countable descending chain of normal subgroups.

• Type: Journal
• Year: 2008
• Where: DAM
• Authors: Ana Luzón, Manuel A. Morón
• Keywords: Certain Arithmetical Triangles, DAM 2008, Fixed Point Problem, Riordan Group