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Pseudo-involutions in the Riordan group and Chebyshev polynomials

Alexander Burstein, Louis W. Shapiro

TL;DR

This work develops a unified framework for pseudo-involutions in the Riordan group by analyzing the first-column generating function $g$ through functional equations involving a γ-function. The authors derive the pseudo-involutory companion $f$ and the corresponding B-function $B_f$, revealing that Chebyshev polynomials underpin the B-function in key cases. They treat Laurent-polynomial γ and rational $g$ cases, obtaining explicit B-functions for several classical combinatorial families (e.g., Catalan, Schröder, Motzkin) and providing a general polynomial machinery for these constructions. The results extend and streamline prior theory, offering efficient derivations and new constructions across multiple Riordan-subgroup contexts while highlighting deep connections to Chebyshev-structured recurrences.

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.

Pseudo-involutions in the Riordan group and Chebyshev polynomials

TL;DR

This work develops a unified framework for pseudo-involutions in the Riordan group by analyzing the first-column generating function through functional equations involving a γ-function. The authors derive the pseudo-involutory companion and the corresponding B-function , revealing that Chebyshev polynomials underpin the B-function in key cases. They treat Laurent-polynomial γ and rational cases, obtaining explicit B-functions for several classical combinatorial families (e.g., Catalan, Schröder, Motzkin) and providing a general polynomial machinery for these constructions. The results extend and streamline prior theory, offering efficient derivations and new constructions across multiple Riordan-subgroup contexts while highlighting deep connections to Chebyshev-structured recurrences.

Abstract

Generalizing the results in our previous paper, we consider pseudo-involutions in the Riordan group where the generating function 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 . 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.
Paper Structure (8 sections, 12 theorems, 137 equations)

This paper contains 8 sections, 12 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Basics of Riordan arrays
  3. Organization of the paper
  4. Functions associated with a Riordan array
  5. The $\gamma$ function of a Riordan array
  6. $B$-function from $\gamma$-function
  7. $B$-function given a Laurent polynomial $\gamma$
  8. B-function given a rational function $g$

Key Result

Theorem 2.2

For any pseudo-involutory function $f\ne-z$, we have Moreover, $f=h\circ\widehat{h}$ for $h\in\mathcal{F}_1$ if an only if $h=\widehat{(\sqrt{zf})}\circ \phi$, where $\phi$ is an arbitrary odd function, i.e. $\phi(-z)=-\phi(z)$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • ...and 28 more