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.

Structural analysis of finite para-families in arXiv:1110.6475v3, 1211.2461v2, 1302.6142v1, 1507.05919v1, 1509.08965v1, 1511.05215v1, 1612.04038v1, 1708.03368v1, 2005.06961v1, 2008.03266v2, 2103.09631v1, 2104.13960v1, 2108.09386v1, 2209.10725v3, critically examining first versions 2609.38211v1, 2609.30619v1, 2609.30624v1, 2609.30636v1