The manufacture of examples: the (−1)-classical orthogonal polynomials
Abstract
The big (−1)-Jacobi, Bannai-Ito, and (−1)-Meixner-Pollaczek polynomials have appeared in the mathematical physics literature through differential- or difference-reflection eigenvalue problems. We show that these three families are governed by a single alternating mechanism, formulated at the level of orthogonality functionals, transpose operators, and structural identities. Starting from an ordinary orthogonal polynomial sequence in the quadratic variable, the resulting algorithm constructs the direct alternating family and transfers its moments, squared norms, and recurrence coefficients. One regular Geronimus inversion then produces the big (−1)-Jacobi and Bannai-Ito families, whereas the (−1)-Meixner-Pollaczek family is obtained directly.
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BibTeX
@misc{castillo2026manufacture,
title = {{The manufacture of examples: the $(-1)$-classical orthogonal polynomials}},
author = {K. Castillo and G. Gordillo-Núñez},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.29792},
eprint = {2605.29792},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.29792},
note = {CC0},
}
RIS
TY - UNPB
TI - The manufacture of examples: the (−1)-classical orthogonal polynomials
AU - K. Castillo
AU - G. Gordillo-Núñez
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2605.29792
UR - https://arxiv.org/abs/2605.29792
AB - The big (−1)-Jacobi, Bannai-Ito, and (−1)-Meixner-Pollaczek polynomials have appeared in the mathematical physics literature through differential- or difference-reflection eigenvalue problems. We show that these three families are governed by a single alternating mechanism, formulated at the level of orthogonality functionals, transpose operators, and structural identities. Starting from an ordinary orthogonal polynomial sequence in the quadratic variable, the resulting algorithm constructs the direct alternating family and transfers its moments, squared norms, and recurrence coefficients. One regular Geronimus inversion then produces the big (−1)-Jacobi and Bannai-Ito families, whereas the (−1)-Meixner-Pollaczek family is obtained directly.
ER -