PreprintCC BY 4.0arXiv
Orthogonal polynomials on two or more intervals: the Sprugnoli context
Abstract
We show how the group of Sprugnoli arrays on three elements can be used to define families of polynomials that are orthogonal on two intervals, and we indicate how the group can be extended in the hierarchy of Sprugnoli groups to yield families of orthogonal polynomials with more complex supports. In particular, we examine the defining three term recurrences and the Jacobi matrices of these families of orthogonal polynomials, while detailing the Sprugnoli matrix structure of the coefficient matrices of the polynomial families. We also consider integral representations of the principal moments on the defining intervals. Sprugnoli arrays are shown to relate to periodic recurrences where the parameters remain cyclically constant.
Related
- Long intervals of consecutive composite values of polynomials
- Matrix orthogonal polynomials related to diffference operators for L-orthogonal polynomials
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BibTeX
@misc{barry2026orthogonal,
title = {{Orthogonal polynomials on two or more intervals: the Sprugnoli context}},
author = {Paul Barry},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05673},
eprint = {2611.05673},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05673},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Orthogonal polynomials on two or more intervals: the Sprugnoli context
AU - Paul Barry
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05673
UR - https://arxiv.org/abs/2611.05673
AB - We show how the group of Sprugnoli arrays on three elements can be used to define families of polynomials that are orthogonal on two intervals, and we indicate how the group can be extended in the hierarchy of Sprugnoli groups to yield families of orthogonal polynomials with more complex supports. In particular, we examine the defining three term recurrences and the Jacobi matrices of these families of orthogonal polynomials, while detailing the Sprugnoli matrix structure of the coefficient matrices of the polynomial families. We also consider integral representations of the principal moments on the defining intervals. Sprugnoli arrays are shown to relate to periodic recurrences where the parameters remain cyclically constant.
ER -