Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and d-orthogonal polynomial sequences
Abstract
We study when a polynomial sequence {pn(x)} satisfying a linear homogeneous recurrence with polynomial coefficients admits an ordinary Riordan array as its coefficient matrix. For second-order recurrences pn+2(x)=a(x)pn+1(x)+b(x)pn(x), we give a complete characterization. We extend this to a necessary condition and a full factorization criterion for recurrences of arbitrary order ℓ ≥ 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 PA−1=ASA−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.
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BibTeX
@misc{he2026riordan,
title = {{Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and $d$-orthogonal polynomial sequences}},
author = {Tian-Xiao He},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28834},
eprint = {2608.28834},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28834},
}
RIS
TY - UNPB
TI - Riordan array representation of recursive polynomial sequences, orthogonal polynomial sequences, and d-orthogonal polynomial sequences
AU - Tian-Xiao He
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28834
UR - https://arxiv.org/abs/2608.28834
AB - 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 ℓ ≥ 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=ASA^−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.
ER -