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G. Gordillo-Núñez

Publications and source records attributed to G. Gordillo-Núñez.

8 recordsLinked to original sources
arXiv2026

Old structures, a vanishing formula

We identify a normalization defect in a determinantal construction of fourth-order Hahn difference equations and compare its scalar equation with Foupouagnigni's 1998 elimination and its published 2001 counterpart. On an explicit positive orthogonal family, all five prescribed coefficients vanish: the formula gives \(0=0\) and imposes no condition. Along an explicit regular perturbation, division by \(\varepsilon^2\) before specialization recovers a nonzero equation on that family, the same scalar equation as the correctly executed old shift-and-eliminate procedure. On the common nondegenerate domain specified below, both constructions give the same scalar operator up to a nonzero rational factor. The first-order degree bounds follow directly from valid identities of 1998; preserving cancellations gives a bound that can be arbitrarily sharper on explicit admissible inputs, together with exact top coefficients and cancellation criteria. Explicit coefficient formulas do not by themselves yield a new scalar equation on this domain. Our comparison does not exclude corrective or expository contributions; additional computational gain or exceptional coverage must be demonstrated.

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arXiv2026

The manufacture of examples II: the "para-classical" orthogonal polynomials

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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arXiv2026

The manufacture of examples: the $(-1)$-classical orthogonal polynomials

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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arXiv2026

The multiple Markov theorem on Angelesco sets

Addressing a problem described by Ismail as a "worthwhile research project", this note extends Markov's theorem to multiple orthogonal polynomials on Angelesco sets. The interaction $\mathcal{Z}$-matrix is shown to be a nonsingular $\mathcal{M}$-matrix; hence the theorem holds for positive Borel measures satisfying the natural support condition, including finite supports, without restriction on the number of intervals. Beyond separated supports, a dual variation identity gives necessary and sufficient tail conditions. An explicit AT example disproves the componentwise extension, while for Nikishin systems we prove positive results for certain deformations of the first generator and give a counterexample for an internal generator.

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arXiv2026

Yet Another Characterisation of Classical Orthogonal Polynomials?

The NIST Handbook of Mathematical Functions (2010) and the NIST Digital Library of Mathematical Functions (2025) classify classical orthogonal polynomials through Bochner's 1929 algebraic-differential characterisation and its discretisation. Yet this classification rests on a narrow reading of Bochner's work and on a restricted notion of orthogonality that becomes inadequate once polynomials are characterised by their algebraic properties. As a result, algebraically equivalent families are treated as distinct, parameter domains are restricted, and families already implicit in Bochner's scheme are excluded. In the mid-1980s, Maroni challenged this view by extending the notion of classical orthogonal polynomials through duality theory on locally convex spaces, thereby reaching the algebraic limits latent in Bochner's framework. Yet when the notion was later enlarged to include further families, Maroni's criteria and rationale were largely set aside. To clarify this history, we revisit a less familiar line of development and use it to obtain a classification of classical orthogonal polynomials on linear lattices within Maroni's functional-analytic setting, beyond the positive-definite case. This classification recovers all known families as special cases, preserves orthogonality and the defining algebraic properties, places supposedly new families in their proper structural context, and shows that algebraically identical polynomials are often treated as distinct. Moreover, through a limit process in the weak topology of the continuous dual, we recover families implicit in Bochner's work and unify the continuous and discrete cases within a dual-topological framework. Thus, neither Bochner's classical characterisation nor its discrete analogue is modified to produce ad hoc families; both are recovered at the level of their intrinsic algebraic structure.

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arXiv2026

Wendroff's theorem beyond consecutive degrees and related inverse spectral problems

A classical theorem of Wendroff shows that one may reconstructs a sequence of orthogonal polynomials on the real line from two non-constant polynomials of consecutive degrees whose zeros strictly interlace on the real line. In this note we extend this result to arbitrary non-constant polynomials. The reconstruction may be formulated via a Vandermonde-type linear system and recast as an underdetermined inverse spectral problem, in which the spectra of a finite Jacobi matrix and of one of its leading principal submatrices are prescribed. In addition, the analogous result on the unit circle is established by reconstructing a sequence of paraorthogonal polynomials from two arbitrary non-constant polynomials whose zeros strictly interlace on the unit circle. In this setting, the Jacobi matrix is replaced by a finite unitary pentadiagonal matrix, and the spectral data consist of the spectrum of the full matrix together with that of a rank-one perturbation of a leading principal submatrix. Strict interlacing of zeros is shown to be a necessary and sufficient condition for solvability, and explicit constructions of the associated polynomial families and matrices are provided. Finally, an algorithm and several illustrative examples are presented.

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arXiv2025

On recurrence coefficients of classical orthogonal polynomials

In Lett. Math. Phys. 114, 54 (2024) and 115, 70 (2025), the author introduces what is presented as a novel method for determining whether a sequence of orthogonal polynomials is "classical", based solely on its initial recurrence coefficients. This note demonstrates that all the results contained in those works are already encompassed by two general theorems previously established in J. Math. Anal. Appl. 515 (2022), Article 126390. A symbolic algorithm, implemented in Mathematica, is also provided to enable automated verification of the classical character of orthogonal polynomial sequences on quadratic lattices. As an application, it is shown that the so-called para-Krawtchouk polynomials on bi-lattices, discussed in Lett. Math. Phys. 115, 70 (2025), constitute a particular instance of a classical orthogonal family on a linear lattice. Consequently, their algebraic properties follow as a specific case of one of the main theorems established in J. Math. Anal. Appl. 515 (2022), Article 126390.

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