arXiv subjects

Anier Soria-Lorente

Publications and source records attributed to Anier Soria-Lorente.

11 recordsLinked to original sources

arXiv2026

Mehler--Heine asymptotics for finite differences of mass-modified Charlier and Meixner polynomials

Point-mass perturbations and finite-difference operations alter the finite-degree structure of discrete orthogonal polynomials, but their combined effect at the Mehler--Heine scale is not immediate. We study the monic Charlier and Meixner families under a pure Uvarov modification by a fixed mass $Aδ_{0}$ at the endpoint of the support and determine the asymptotic behaviour of their forward and backward differences of arbitrary fixed order. Using a rank-one connection formula, we show that the perturbation coefficient decays factorially in the Charlier case and exponentially, with an algebraic prefactor, in the Meixner case. In both families this decay is faster than every algebraic power of $n^{-1}$ and therefore suppresses the growth produced by any fixed number of finite differences. Consequently, for every fixed $A\geq0$ and $k\in\Nzero$, the mass-modified and classical families have the same locally uniform Mehler--Heine limits in $\C$. Forward differences preserve the reciprocal-Gamma profile up to the factor $(-1)^k$, whereas backward differences translate the limiting argument by $k$, shifting the limiting zero lattice from $\Nzero$ to $k+\Nzero$. We also derive a first-order shift equation for the common forward limit and characterize its entire solution space. Complex-plane portraits and real-axis computations illustrate the predicted limiting profiles, the two zero lattices, and the asymptotic disappearance of the fixed endpoint mass. These results identify a scale-separation mechanism governing the fixed-order asymptotic stability of the Charlier and Meixner families under rank-one endpoint perturbations.

math.CA↗

arXiv2026

ADPSO-ERLS: A Hybrid Discrete PSO with Enhanced Local Search for the Traveling Salesman Problem

The Traveling Salesman Problem is a canonical setting for studying how a population-based method should allocate a fixed search budget between exploration and progressively stronger local intensification. We propose ADPSO-ERLS, a discrete swarm algorithm that treats this allocation as an explicit, tunable design variable. It couples memory-guided swap mutation, heterogeneous initialization, selective candidate-restricted 2-opt during evolution, and an incumbent-only final refinement combining candidate-restricted and optional full 2-opt with double-bridge perturbations. The method is PSO-inspired, using personal and global memories yet dispensing with velocity, inertia, and acceleration coefficients. All six algorithms are implemented in Rust, run on identical hardware, and stopped at a strict, recorded limit of 100,000 candidate-solution assessments, so that programming language, hardware, and evaluation budget are held common across methods; wall-clock time is reported separately because equal assessment counts need not correspond to equal arithmetic work. Over 50 runs on five symmetric TSPLIB instances under the integer \texttt{EUC\_2D} convention, ADPSO-ERLS attains the lowest best and mean cost on every instance, with best-tour Gap of $1.93$--$4.17\%$ and relative error of $3.04$--$5.50\%$. It ranks first under the Friedman test, and all twenty-five multiplicity-controlled Wilcoxon comparisons favor it with large, near-complete distributional separation. A paired ablation with common seeds links initialization, in-run local search, and final refinement to quality gains, while candidate restriction chiefly cuts runtime, by up to a factor of roughly $38$. Further experiments up to $16{,}862$ cities keep best-tour Gaps below $6.7\%$, solving the largest case in under eleven minutes.

math.OC↗

arXiv2024

On zero behavior of higher-order Sobolev-type discrete q-Hermite I orthogonal polynomials

In this work, we investigate the sequence of monic q-Hermite I-Sobolev type orthogonal polynomials of higher-order, denoted as $\{\mathbb{H}_{n}(x;q)\}_{n\geq 0}$, which are orthogonal with respect to the following non-standard inner product involving q-differences: \begin{equation*} \langle p,q\rangle_{λ}=\int_{-1}^{1}f\left( x\right) g\left(x\right) (qx,-qx;q)_{\infty }d_{q}(x)+λ\,(\mathscr{D}_{q}^{j}f)(α)(\mathscr{D}_{q}^{j}g)(α), \end{equation*} where $α\in \mathbb{R}\backslash (-1,1)$, $λ$ belongs to the set of positive real numbers, $\mathscr{D}_{q}^{j}$ denotes the $j$-th $q $-discrete analogue of the derivative operator, and $(qx,-qx;q)_{\infty}d_{q}(x)$ denotes the orthogonality weight with its points of increase in a geometric progression. We proceed to obtain the hypergeometric representation of $\mathbb{H}_{n}(x;q)$ and explicit expressions for the corresponding ladder operators. From the latter, we obtain a novel kind of three-term recurrence formula with rational coefficients associated with these polynomial family. Moreover, for certain real values of $α$, we present some results concerning the location of the zeros of $\mathbb{H}_n(x;q)$ and we perform a comprehensive analysis of their asymptotic behavior as the parameter $λ$ varies from zero to infinity.

math.CA↗

arXiv2023

A dual watermaking scheme based on Sobolev type orthogonal moments for document authentication

A dual watermarking scheme based on Sobolev type orthogonal moments, Charlier and Meixner, is proposed based on different discrete measures. The existing relation through the connection formulas allows to provide with structure and recurrence relations, together with two difference equations satisfied by such families. Weighted polynomials derived from them are being applied in an embedding and extraction watermarking algorithm, comparing the results obtained in imperceptibly and robustness tests with other families of polynomials.

cs.MM↗

arXiv2021

On second order q-difference equations for high-order Sobolev-type q-Hermite orthogonal polynomials

The q-Hermite I-Sobolev type polynomials of higher order are consider for their study. Their hypergeometric representation is provided together with further useful properties such as several structure relations which give rise to a three-term recurrence relation of their elements. Two different q-difference equations satisfied by the q-Hermite I-Sobolev type polynomials of higher order are also established.

math.CA↗

arXiv2021

The annihilation operator for certain family of q-Hermite Sobolev-type orthogonal polynomials

We present a new family $\left\{ S_{n}(x;q)\right\} _{n\geq 0}$ of monic polynomials in $x$, orthogonal with respect to a Sobolev-type inner product related to the $q$-Hermite I orthogonal polynomials, involving a first-order $q$-derivative on a mass-point $α\in \mathbb{R}$ located out of the corresponding orthogonality interval $[-1,1]$, for some fixed real number $q \in (0, 1)$. We present connection formulas, and the annihilation operator for this non-standard orthogonal polynomial family.

math.CA↗

arXiv2020

On difference equations of Kravchuk-Sobolev type polynomials of higher order

In this contribution we consider sequences of monic polynomials orthogonal with respect to Sobolev-type inner product \[ \left\langle f,g\right\rangle _{λ,μ}\!=\!\sum_{x=0}^Nf(x)g(x)\frac{Γ(N+1) p^x(1-p)^{N-x} }{Γ(N-x+1) Γ(x+1) }+λΔ^j f(0)Δ^j g(0)+μΔ^j f(N)Δ^j g(N), \] where $0<p <1$, $λ,μ\in \mathbb R_{+}$, $n\leq N\in \mathbb Z_{+}$, $j\in \mathbb Z_{+}$ and $Δ$ denotes the forward difference operators. We derive an explicit representation for these polynomials. In addition, the ladder operators associated with these polynomials are obtained. As a consequence, the linear difference equations of second order are also given.

math.CA↗

arXiv2020

On second order q-difference equations satisfied by Al-Salam-Carlitz I-Sobolev type polynomials of higher order

This contribution deals with the sequence $\{\mathbb{U}_{n}^{(a)}(x;q,j)\}_{n\geq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary number of $q$-derivatives on the two boundaries of the corresponding orthogonality interval. We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order $q$-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by $\mathbb{U}_{n}^{(a)}(x;q,j)$, which paves the way to establish an appealing generalization of the so-called $J$-fractions to the framework of Sobolev-type orthogonality.

math.CA↗

arXiv2019

Hiding data inside images using orthogonal moments

In this contribution we propose a novel steganographic method based on several orthogonal polynomials and their combinations. The steganographic algorithm embeds a secrete message at the first eight coefficients of high frequency image. Moreover, this embedding method uses the Beta chaotic map to determine the order of the blocks where the secret bits will be inserted. In addition, from a 128-bit private key and the steps of a cryptography algorithm according to the Advanced Encryption Standard (AES) to generate the key expansion, the proposed method generates a key expansion of 2560 bits, with the purpose to permute the first eight coefficients of high frequency before the insertion. The insertion takes eventually place at the first eight high frequency coefficients in the transformed orthogonal moments domain. Before the insertion of the message the image undergoes a series of transformations. After the insertion the inverse transformations are applied to the original transformations in reverse order. The experimental work on the validation of the algorithm consists of the calculation of the Peak Signal-to-Noise Ratio (PSNR), the Universal Image Quality Index (UIQI), the Image Fidelity (IF), and the Relative Entropy (RE), comparing the same characteristics for the cover and stego image. The proposed algorithm improves the level of imperceptibility and security analyzed through the PSNR and RE values, respectively.

cs.MM↗

arXiv2018

New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials

In this contribution we consider the sequence $\{Q_{n}^λ\}_{n\geq 0} $ of monic polynomials orthogonal with respect to the following inner product involving differences \begin{equation*} \langle p,q\rangle _λ=\int_{0}^{\infty}p\left(x\right) q\left(x\right) dψ^{(a)}(x)+λ\,Δp(c)Δq(c), \end{equation*} where $λ\in \mathbb{R}_{+}$, $Δ$ denotes the forward difference operator defined by $Δf\left(x\right) =f\left(x+1\right) -f\left(x\right) $, $ψ^{(a)}$ with $a>0$ is the well known Poisson distribution of probability theory% \begin{equation*} dψ^{(a)}(x)=\frac{e^{-a}a^{x}}{x!}\quad \text{at}x=0,1,2,\ldots, \end{equation*}% and $c\in \mathbb{R}$ is such that $ψ^{(a)}$ has no points of increase in the interval $(c,c+1)$. We derive its corresponding hypergeometric representation. The ladder operators and two different versions of the linear difference equation of second order corresponding to these polynomials are given. Recurrence formulas of five and three terms, the latter with rational coefficients, are presented. Moreover, for real values of $c$ such that $c+1<0$, we obtain some results on the distribution of its zeros as decreasing functions of $λ$, when this parameter goes from zero to infinity.

math.CA↗

arXiv2018

A single parameter Hermite-Padé series representations for Apéry's constant

Inspired by the results of Rhin and Viola (2001), the purpose of this work is to elaborate on a series representation for $ζ\left( 3\right)$ which only depends on one single integer parameter. This is accomplished by deducing a Hermite-Padé approximation problem using ideas of Sorokin (1998). As a consequence we get a new recurrence relation for the approximation of $ζ(3)$ as well as a corresponding new continued fraction expansion for $ζ(3)$, which do no reproduce Apéry's phenomenon, i.e., though the approaches are different, they lead to the same sequence of diophantine approximations to $ζ\left( 3\right) $. Finally, the convergence rates of several series representations of $ζ(3)$ are compared.

math.NT↗