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

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.