Riordan arrays, the $A$-matrix, and Somos $4$ sequences

TL;DR

This paper explores the structure of Riordan arrays through their A-matrices and rho sequences, proposing a conjecture on A-matrix forms that generate Somos 4 sequences, and analyzing their relation to orthogonal polynomials.

Contribution

It introduces a characterization of Riordan arrays via A-matrices and rho sequences, and conjectures a form of A-matrix that produces Somos 4 sequences, advancing understanding of their algebraic structure.

Findings

Identified A-matrices that generate Riordan quasi-involutions

Proposed a conjecture on the form of A-matrices for Somos 4 sequences

Analyzed moment matrices of perturbed orthogonal polynomials

Abstract

We characterize certain Riordan arrays by their $A$-matrices and $\rho$ sequences. We conjecture the form of a generic $A$-matrix which leads to Somos $4$ sequences. We find an $A$-matrix that produces a Riordan quasi-involution, and we study the $A$-matrices and $\rho$ sequences of the moment matrices of certain perturbed orthogonal polynomials.