A Three-parameter Family Of Involutions In The Riordan Group Defined By Orthogonal Polynomials

TL;DR

This paper introduces a method to construct involutions in the Riordan group using orthogonal polynomials and pseudo-involutions, expanding the understanding of the group's structure and its connections to polynomial families.

Contribution

It provides a new framework for defining involutions in the Riordan group based on orthogonal polynomials and pseudo-involutions, with explicit formulas and broader applicability.

Findings

Constructed involutions from Riordan group elements and pseudo-involutions.

Linked orthogonal polynomial families to involutions in the Riordan group.

Provided explicit forms of the involutions derived from polynomial families.

Abstract

We show how to define, for every Riordan group element $(g(x), f(x))$, an involution in the Riordan group. More generally, we show that for every pseudo-involution $P$ in the Riordan group, we can define a new involution beginning with an arbitrary element $(g(x), f(x))$ in the Riordan group. We then use this result to show that certain two-parameter families of orthogonal polynomials defined by a Riordan array can lead to involutions in the Riordan group, and we give an explicit form of these involutions.