arXiv subjects

K. Castillo

Publications and source records attributed to K. Castillo.

At least 19 recordsLinked to original sources

arXiv2026

A proof of Askey's convexity conjecture

For $-1<\alpha\leq1/2$, let $J_\alpha$ be the Bessel function of the first kind and let $j_{\alpha,2}$ be its second positive zero. Define $\beta(\alpha)<\alpha+1$ by $$ \int_0^{j_{\alpha,2}}u^{-\beta(\alpha)}J_\alpha(u)\,du=0. $$ We prove that $\beta''(\alpha)>0$ for $-1<\alpha\leq1/2$, including the left second derivative at $\alpha=1/2$. The continuous extension $\beta(-1)=0$ is strictly convex on $[-1,1/2]$, strengthening Askey's 1993 convexity conjecture. The analytic argument uses a positive expansion of a primitive and a vanishing weighted sum. Increasing ratios of consecutive weights give a negative covariance term, while one comparison controls slope and curvature. The remaining algebraic step proves eight rational inequalities by finite exact polynomial calculations, reproduced in an appendix.

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arXiv2026

Symmetric products of Laguerre zeros

We prove and substantially extend a conjecture of Gazeau, Josse-Michaux, and Monceau concerning the extreme positive zeros of Hermite polynomials, which arose from a finite-dimensional quantisation of the phase plane. For the generalised Laguerre polynomial $L_m^{(\alpha)}$ of degree $m$ and parameter $\alpha>-1$, the product of its $j$th smallest and $j$th largest zeros is strictly increasing with $m$ for each fixed positive integer $j$, once $m\ge2j-1$. The classical Hermite-Laguerre identities then prove the original conjecture for the degree-$N$ Hermite polynomial $H_N$, in both parities and for every $N\ge4$. As a further consequence, a one-parameter extension involving a deformation of the corresponding Jacobi matrix is settled.

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arXiv2026

Affine Scaling of Jacobi Zeros: Sharp Orderings Beyond Gautschi's Conjectures

We settle two conjectures of Gautschi on the degree dependence of the zeros of the Jacobi polynomials $P_n^{(\alpha,\beta)}$, $\alpha,\beta>-1$, and obtain results substantially stronger than those conjectured. The conjectures stem from a line of questions originating in spherical cubature and hyperinterpolation. A Liouville transformation and Sturm comparison yield affine comparison principles with exact thresholds for the pointwise monotonicity of the rescaled potential. We prove that an increasing affine ordering with a degree-independent shift exists if and only if $|\beta|\leq1/2$. For the spectral scale $n+(\alpha+\beta+1)/2$, we determine the exact parameter regions for the two opposite orderings and show that no uniform spectral ordering is possible outside them. We also characterise all equality cases and derive finite-degree bounds in terms of Bessel zeros. The resulting classifications are exact and cannot be enlarged: outside the stated parameter regions the corresponding uniform zero orderings necessarily fail.

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arXiv2026

Degree-uniform regions for Gautschi's conjecture on subrange Jacobi polynomials

Let $\pi_n$ be the monic polynomial of degree $n$ orthogonal on $[-c,c]$, $0<c\leq 1$, with respect to the Jacobi weight $(1-x)^\alpha(1+x)^\beta$, where $-1<\alpha<\beta$. Gautschi conjectured that $$ \left[ \frac{\pi_n(-c)}{\pi_n(c)} \right]^2 \left(\frac{1-c}{1+c}\right)^{\beta-\alpha} <1. $$ By his variation formula, this inequality is sufficient for every positive zero of $\pi_n$ to move to the right as $c$ increases. For $0<c<1$, we prove the conjecture, uniformly in the degree, throughout $0<\alpha<\beta$. Combined with the region $\alpha\leq 0\leq\beta$, recorded by Gautschi on the basis of an unpublished communication from Milovanovi'c, and with Milovanovi'c's published criterion, this settles the full admissible range $\beta\geq 0$. In the remaining negative wedge, writing $\alpha=-r-\lambda$ and $\beta=-r+\lambda$, we prove the conjecture whenever $$ c^2\leq\frac{3}{3+r}. $$ Consequently, it holds for every admissible pair of parameters when $0<c\leq\sqrt{3}/2$. For arbitrary admissible parameters, we also establish the degree-one case and eventual validity as $n\to\infty$. The proof combines an exact boundary identity, a first-crossing argument based on Pearson and root-motion identities, and, in the negative wedge, a Ward identity with positive association for an MTP$_2$ orthogonal polynomial ensemble. The case $c=1$ is immediate.

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arXiv2026

A proof of the Szeg\H{o} conjecture on Jacobi extrema

For Jacobi polynomials with parameters greater than $-1/2$, and with the relative extrema enumerated from the endpoint $x=1$, the normalised modulus at the $k$th extremum of degree $n+1$ is proved to be strictly smaller than that at the $k$th extremum of degree $n$, for $1\leq k\leq n$. This proves the Szeg\H{o} conjecture, recorded in the 1975 fourth edition of his classic monograph Orthogonal Polynomials, and strengthens it by removing the ordering assumption on the parameters. The proof transforms the Jacobi equation to angular form and compares Pr\"ufer amplitudes at equal phase. Combined with a reduction of de Oliveira Filho and a separate quadratic-transformation argument for the boundary case, the result also settles a question concerning the Lov\'asz theta number of spherical distance graphs in every dimension at least four.

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arXiv2026

A solution to Meneguette's polynomial problem

In 1979, Jeltsch [Numer. Math. 32 (1979), 167-181] conjectured that every zero-stable Brown method is stiffly stable. For Brown $(K,L)$ methods, Meneguette subsequently reduced the relevant $A_0$-stability question to a zero-location property for a family of perturbed characteristic polynomials. A closely related perturbation question already appeared in his 1987 Oxford doctoral thesis, and he later posed a broader purely polynomial version of the problem in 1994 [SIAM Review 36 (1994), 656-657]. We provide a complete solution to Meneguette's polynomial problem.

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arXiv2026

On a measure-theoretic reading of $\beta$-Gr\"uss-type inequalities

On its absolute-integrability domain, the positive $\beta$-integral is integration with respect to a finite positive purely atomic measure. After normalisation, its Chebyshev functional is a covariance, and its $L^p$-spaces are canonically isometric to direct sums of weighted sequence spaces. On the natural product- and square-integrability domains, the previously formulated $\beta$-Gr\"uss inequalities reduce to Korkine's identity, H\"older's inequality, Cauchy-Schwarz, and elementary variance bounds. On the induced countably atomic probability space, the optimal fixed-grid coefficient is $\kappa_\beta=\sup_A P_\beta(A)(1-P_\beta(A))\leq 1/4$, the countably atomic counterpart of the classical finite weighted coefficient; it may be strictly smaller than $1/4$. The same reduction corrects a coefficient previously claimed to be best possible and identifies a missing sign hypothesis in a related convexity estimate. For the Riemann--Stieltjes $\beta$-integral, within the class of finite induced signed measures, the $\beta$-Lipschitz condition is equivalent to $|\nu_u|\leq L\mu_\beta$. This reduces the principal centred signed estimate to total variation and yields its exact fixed-grid coefficient $2\kappa_\beta$. Finally, truncation of the two atomic orbits gives positive quadrature rules with explicit tail masses. A fixed-point correction yields computable H\"older error bounds, while the uncorrected geometrically graded rule accommodates integrable singularities at the fixed point.

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arXiv2026

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

The $(-1)$-Jacobi, Bannai-Ito, and $(-1)$-Meixner-Pollaczek polynomials are studied in [Trans. Amer. Math. Soc. 364 (2012), 5491-5507], [Adv. Math. 229 (2012), 2123-2158], and [Stud. Appl. Math. 153 (2024), e12728], respectively, through polynomial eigenfunctions of first-order Dunkl operators. The purpose of the present note is to show that these families are not isolated phenomena, but particular instances of a single alternating mechanism which is most naturally formulated at the level of orthogonality functionals, transpose operators, and structural identities. This functional-analytic point of view also leads to an explicit algorithm which, starting from an ordinary orthogonal polynomial sequence in the quadratic variable, systematises the construction of such $(-1)$-classical families.

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arXiv2026

Half-Step-Invariant Sets, Admissible Maps, and Classical Orthogonal Polynomials

This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual reappraisal of the subject from a perspective in which certain persistent distortions become plainly visible. The theory is developed on subsets of the complex plane that are stable under half-step translations, and both classicality and orthogonality are understood in the continuous dual of a suitable locally convex space of polynomials. The repeated reappearance of ostensibly new families of classical orthogonal polynomials arising from exotic maps, algebraically equivalent families artificially separated, unnecessary parameter restrictions inherited from positive-definite models, apparently distinct phenomena associated with root-of-unity values of $q$ in the $q$-exponential case, naive $q\to -1$ limits in that same setting, finite truncations of otherwise infinite orthogonal polynomial sequences, and geometric recastings of the local half-step relation in terms of plane conic curves all make the need for a broader structural framework increasingly clear. The present paper seeks to articulate such a framework in a way that allows the reader to distinguish the genuinely new from the merely artificial, without resorting to an exhaustive case-by-case examination of prior work. In the ordinary quadratic and non-torsion $q$-exponential regimes, the infinite regularity criteria retain their full force as if-and-only-if statements; finite resonances, by contrast, are governed by the exact moment system and the non-vanishing of its Hankel determinants.

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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

A note on lower bounds for numerical series

After passage to the representing measure, the three principal theorems of Esstafa and Sfaxi, J. Math. Anal. Appl. 556 (2026), 130199, reduce respectively to Jensen's inequality, one quadratic identity, and a one-point measure. We solve the underlying reciprocal-moment problem sharply, determine all extremisers, and obtain a strictly stronger Hurwitz-zeta bound. We also identify the gap between formal inversion and evaluation at 1, and give explicit counterexamples to further analytic claims.

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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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arXiv2025

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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arXiv2024

On the product of the extreme zeros of Laguerre polynomials

The purpose of this note is twofold: firstly, it intends to bring to light an apparently unknown property of the product of the extreme zeros of Laguerre polynomials, which in a very particular case leads to a twenty-year-old conjecture for Hermite polynomials posed by Gazeau, Josse-Michaux, and Moncea while developing numerical methods in quantum mechanics; and secondly to progress towards the solution of this problem as an application of a parametric eigenvalue problem.

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arXiv2024

Extensions of certain Markov inequalities to Toeplitz matrices

We establish finite Toeplitz analogues of Markov's inequalities for Hankel determinants and zeros of orthogonal polynomials. We characterise the real symmetric positive-definite Toeplitz matrices whose determinants dominate those of all such matrices below them in the alternating moment order: non-negativity of the diagonal sums of deleted minors is necessary and sufficient. Equality holds only when the matrices coincide, and a quantitative estimate controls the determinant gap. Paraorthogonal zeros in the right half-plane provide a geometric sufficient condition in both parities. For differentiable finite Hermitian moment data, we derive individual and relative angular-velocity identities. For real symmetric data, these identities yield a finite comparison of all upper zeros under simultaneous changes in the moments: a semicircle condition at one endpoint is inherited by the other, and every angular inequality is strict when the matrices differ. No regularity assumption on a representing measure is required. Coefficient criteria also establish nearest-zero monotonicity. An exact Rogers-Szeg\H{o} counterexample shows that this need not extend to every upper zero.

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arXiv2023

On classical orthogonal polynomials on bi-lattices

In [J. Phys. A: Math. Theor. 45 (2012)], while looking for spin chains that admit perfect state transfer, Vinet and Zhedanov found an apparently new sequence of orthogonal polynomials, that they called para-Krawtchouk polynomials, defined on a bilinear lattice. In this note we present necessary and sufficient conditions for the regularity of solutions of the corresponding functional equation. Moreover, the functional Rodrigues formula and a closed formula for the recurrence coefficients are presented. As a consequence, we characterize all solutions of the functional equation, including as very particular cases the Meixner, Charlier, Krawtchouk, Hahn, and para-Krawtchouk polynomials.

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arXiv2023

Epilegomena to the study of semiclassical orthogonal polynomials

In his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005 (paperback edition 2009)], Ismail conjectured that certain structure relations involving the Askey-Wilson operator characterize proper subsets of the set of all $\mathcal{D}_q$-classical orthogonal polynomials, here to be understood as the Askey-Wilson polynomials and their limit cases. In this paper we give two characterization theorems for $\mathcal{D}_q$-semiclassical (and classical) orthogonal polynomials in consonance with the pioneering works by Maroni [Ann. Mat. Pura. Appl. (1987)] and Bonan, Lubinsky, and Nevai [SIAM J. Math. Anal. 18 (1987)] for the standard derivative, re-establishing in this context the perfect "symmetry" between the standard derivative and the Askey-Wilson operator. As an application, we present a sequence of $\mathcal{D}_q$-semiclassical orthogonal polynomials of class two that disproves Ismail's conjectures. Further results are presented for Hahn's operator.

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