Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and $d$-orthogonal polynomial sequences

Abstract

We study when a polynomial sequence $\{p_n(x)\}$ satisfying a linear homogeneous recurrence with polynomial coefficients admits an ordinary Riordan array as its coefficient matrix. For second-order recurrences $p_{n+2}(x)=a(x)p_{n+1}(x)+b(x)p_n(x)$, we give a complete characterization. We extend this to a necessary condition and a full factorization criterion for recurrences of arbitrary order $\ell \ge 3$. We then develop a production-matrix framework for arbitrary orthogonal polynomial sequences (OPS), not restricted to the Riordan-array-type case: for any lower-triangular invertible coefficient matrix $A$ of an OPS, the production matrix $P_{A^{-1}}=AS{A}^{-1}$ is always the tridiagonal Jacobi matrix encoding the three-term recurrence, and the first column of $A^{-1}$ always gives the moment sequence -- even when, as for the Legendre polynomials, the recurrence coefficients are non-constant and no Riordan array representation of the coefficient matrix exists. We illustrate this with Legendre and Chebyshev polynomials as, respectively, non-Riordan-type and Riordan-type examples, and give an explicit closed-form coefficient matrix and its inverse for the generalized Gegenbauer--Humbert OPS. Finally, we extend the framework to $d$-orthogonal polynomial sequences, showing that the coefficient matrix of a $d$-orthogonal sequence is always invertible, that its associated production matrix is $(d+2)$-banded (lower Hessenberg) and encodes the corresponding $(d+2)$-term recurrence, and that the first $d$ columns of the inverse matrix recover the moment sequences of the defining vector functional -- identifying $d$-orthogonality with $(d+1)$-Hessenberg generalized Riordan arrays and extending the classical $d=1$ tridiagonal correspondence.