The manufacture of examples II: the "para-classical" orthogonal polynomials
Abstract
Under the stated regularity assumptions, the full orthogonality functionals of the five finite para-families considered here are classical for the respective affine, quadratic, q-quadratic, q-linear and alternating calculi. For each of the para-Krawtchouk, para-Racah, q-para-Racah and q-para-Krawtchouk families, two finite solutions of one global structural equation have N+1 nodes altogether and common moments through degree N−1. Their relative mass changes only the central recurrence coefficients. We prove regularity for every normalised combination with both component masses non-zero and recover the full published functionals and recurrences. The alternating construction identifies para-Bannai-Ito within the same structural framework. For the four non-alternating families, equal positive combinations give persymmetric Jacobi matrices; the known spectral gap criterion then decides perfect state transfer. Explicit parameter correspondences locate the classical components and resonant limits of the full systems. A positive seven-point example shows how further named families can be manufactured without a new governing principle. Counterexamples to mathematical assertions in three of four recent preprints expose the consequences of insufficient hypotheses for finite representations and physical models. In an era of AI-assisted symbolic production, we call for a standard that rewards structural explanation and verification rather than the proliferation of examples.
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BibTeX
@misc{castillo2026manufacture,
title = {{The manufacture of examples II: the "para-classical" orthogonal polynomials}},
author = {K. Castillo and G. Gordillo-Núñez},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16704},
eprint = {2610.16704},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16704},
note = {CC0},
}
RIS
TY - UNPB
TI - The manufacture of examples II: the "para-classical" orthogonal polynomials
AU - K. Castillo
AU - G. Gordillo-Núñez
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16704
UR - https://arxiv.org/abs/2610.16704
AB - Under the stated regularity assumptions, the full orthogonality functionals of the five finite para-families considered here are classical for the respective affine, quadratic, q-quadratic, q-linear and alternating calculi. For each of the para-Krawtchouk, para-Racah, q-para-Racah and q-para-Krawtchouk families, two finite solutions of one global structural equation have N+1 nodes altogether and common moments through degree N−1. Their relative mass changes only the central recurrence coefficients. We prove regularity for every normalised combination with both component masses non-zero and recover the full published functionals and recurrences. The alternating construction identifies para-Bannai-Ito within the same structural framework. For the four non-alternating families, equal positive combinations give persymmetric Jacobi matrices; the known spectral gap criterion then decides perfect state transfer. Explicit parameter correspondences locate the classical components and resonant limits of the full systems. A positive seven-point example shows how further named families can be manufactured without a new governing principle. Counterexamples to mathematical assertions in three of four recent preprints expose the consequences of insufficient hypotheses for finite representations and physical models. In an era of AI-assisted symbolic production, we call for a standard that rewards structural explanation and verification rather than the proliferation of examples.
ER -