Pseudo-involutions in the Riordan group and Chebyshev polynomials

TL;DR

This paper explores pseudo-involutions in the Riordan group, develops methods to find their B-functions involving Chebyshev polynomials, and applies these techniques to derive new and existing results efficiently.

Contribution

It introduces a general method for finding B-functions of Riordan pseudo-involutions involving Chebyshev polynomials, extending previous work.

Findings

Derived new pseudo-involutions in the Riordan group.

Established a connection between B-functions and Chebyshev polynomials.

Provided efficient derivations of known results using the new method.

Abstract

Generalizing the results in our previous paper, we consider pseudo-involutions in the Riordan group where the generating function $g$ for the first column of a Riordan array satisfies a functional equation of certain types involving a polynomial. For those types of equations, we find the pseudo-involutory companion of $g$. We also develop a general method for finding B-functions of Riordan pseudo-involutions in the cases we consider, and show that these B-functions involve Chebyshev polynomials. We apply our method for several families of Riordan arrays, obtaining new results and deriving known results more efficiently.