Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients
Abstract
We present a new classification of the class of all the hypergeometric orthogonal polynomial sequences. The classification uses properties of the coefficients αn and βn in the three-term recurrence relation satisfied by the orthogonal polynomial sequences. Such coefficients are rational functions of n and are determined by a set of six parameters. The partial fraction decomposition of αn is a linear combination of five linearly independent functions of n with coefficients dj, for 0 ≤ j ≤ 4, that are polynomials in our six parameters. For each value of a parameter r, associated with the eigenvalues, we classify the αn according to the set of coefficients dj that are nonzero. There are certain particular values of r that must be considered separately. Our classification yields a collection of 53 disjoint classes and it is different from the Askey scheme in several aspects. It is similar to the classifications proposed recently by Koornwinder.
Related
- Hypergeometric Mixed-Type Multiple Orthogonal Polynomials
- Infinite and finite series involving central binomial coefficients and closed forms of generalized hypergeometric functions
- Matrix orthogonal polynomials related to diffference operators for L-orthogonal polynomials
- Identities and congruences involving orthogonal polynomials and Apéry-like numbers
- Bounds for Orthogonal Expansion Coefficients via Weighted Primitives
BibTeX
@misc{verdestar2026classification,
title = {{Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients}},
author = {Luis Verde-Star},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.18578},
eprint = {2606.18578},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.18578},
}
RIS
TY - UNPB
TI - Classification of the hypergeometric orthogonal polynomials via their recurrence coefficients
AU - Luis Verde-Star
PY - 2026
DA - 2026/08/29
PB - arXiv
DO - 10.48550/arxiv.2606.18578
UR - https://arxiv.org/abs/2606.18578
AB - We present a new classification of the class of all the hypergeometric orthogonal polynomial sequences. The classification uses properties of the coefficients α_n and β_n in the three-term recurrence relation satisfied by the orthogonal polynomial sequences. Such coefficients are rational functions of n and are determined by a set of six parameters. The partial fraction decomposition of α_n is a linear combination of five linearly independent functions of n with coefficients d_j, for 0 ≤ j ≤ 4, that are polynomials in our six parameters. For each value of a parameter r, associated with the eigenvalues, we classify the α_n according to the set of coefficients d_j that are nonzero. There are certain particular values of r that must be considered separately. Our classification yields a collection of 53 disjoint classes and it is different from the Askey scheme in several aspects. It is similar to the classifications proposed recently by Koornwinder.
ER -