Orthogonal Polynomials Research Articles

Application of New Orthogonal Shifted Vieta-Lucas Polynomials in Solving Integro-Differential Equations: Methods and Results

Application of New Orthogonal Shifted Vieta-Lucas Polynomials in Solving Integro-Differential Equations: Methods and Results

Orthogonal polynomials associated with the scattering states of the Kratzer potential

Orthogonal polynomials associated with the scattering states of the Kratzer potential

Fluctuating Genetic Influences at Three Different Stages of Development of Dental Arches: A Complex System

Background/Objectives: The development of dental arches is a complex adaptive system with interactions between genetic and environmental factors. At different developmental stages, the relative contribution of these factors varies. The aims of this project were to identify the longitudinal changes of dental arches in the primary, mixed and permanent dentition stages, using curve fitting methods on serial dental casts, and to investigate the contribution of the genotype to dental arch development. Methods: Longitudinal dental records from 125 monozygotic same-sex twin pairs, 89 dizygotic same-sex twin pairs, and 49 opposite-sex dizygotic twin pairs were used. Standardized model photographs were collected, and key landmarks were digitized. Fourth-order orthogonal polynomials were applied to the Cartesian data. Descriptive statistics were calculated, and structural equation models were developed to analyze the individual polynomial coefficients. The final models employed a genetic simplex framework, enabling the evaluation of how genetic and environmental influences changed over time. These changes were examined both quantitatively (e.g., variations in heritability) and qualitatively (e.g., the influence of different genes at various stages). Results: In the primary dentition, arches were typically parabolic, while in the permanent dentition, they tended to be more square-shaped. Asymmetry made a minor contribution to variation across all stages of development. Genetic analysis revealed that a core group of genes influenced arch shape over time, though their impact varied. Additionally, some genes were specific to certain developmental stages, with their relative contributions differing significantly. Notably, there was evidence of sexual heterogeneity in arch shape, particularly in the permanent dentition. Heritability was consistently high, both at individual developmental stages and throughout the overall developmental process. Conclusions: The degree of genetic influence at each developmental stage was substantial but it fluctuated between the primary, mixed, and permanent dentition stages.

Development of globally hyperbolic one-dimensional moment closures based on orthogonal polynomials

Development of globally hyperbolic one-dimensional moment closures based on orthogonal polynomials

Quantum vs classical birth and death processes; exactly solvable examples

A coinless quantisation procedure of continuous and discrete time Birth and Death (BD) processes is presented. The quantum Hamiltonian H is derived by similarity transforming the matrix L describing the BD equation in terms of the square root of the stationary (reversible) distribution. The quantum and classical systems share the entire eigenvalues and the eigenvectors are related one to one. When the birth rate B(x) and the death rate D(x) are chosen to be the coefficients of the difference equation governing the orthogonal polynomials of Askey scheme, the quantum system is exactly solvable. The eigenvectors are the orthogonal polynomials themselves and the eigenvalues are given analytically. Many examples are periodic since their eigenvalues are all integers, or all integers for integer parameters. The situation is very similar to the exactly solvable one dimensional quantum mechanical systems. These exactly solvable BD processes contain many adjustable free parameters which could be helpful for various simulation purposes.

Three-Dimensional Object Recognition Using Orthogonal Polynomials: An Embedded Kernel Approach

Computer vision seeks to mimic the human visual system and plays an essential role in artificial intelligence. It is based on different signal reprocessing techniques; therefore, developing efficient techniques becomes essential to achieving fast and reliable processing. Various signal preprocessing operations have been used for computer vision, including smoothing techniques, signal analyzing, resizing, sharpening, and enhancement, to reduce reluctant falsifications, segmentation, and image feature improvement. For example, to reduce the noise in a disturbed signal, smoothing kernels can be effectively used. This is achievedby convolving the distributed signal with smoothing kernels. In addition, orthogonal moments (OMs) are a crucial technique in signal preprocessing, serving as key descriptors for signal analysis and recognition. OMs are obtained by the projection of orthogonal polynomials (OPs) onto the signal domain. However, when dealing with 3D signals, the traditional approach of convolving kernels with the signal and computing OMs beforehand significantly increases the computational cost of computer vision algorithms. To address this issue, this paper develops a novel mathematical model to embed the kernel directly into the OPs functions, seamlessly integrating these two processes into a more efficient and accurate approach. The proposed model allows the computation of OMs for smoothed versions of 3D signals directly, thereby reducing computational overhead. Extensive experiments conducted on 3D objects demonstrate that the proposed method outperforms traditional approaches across various metrics. The average recognition accuracy improves to 83.85% when the polynomial order is increased to 10. Experimental results show that the proposed method exhibits higher accuracy and lower computational costs compared to the benchmark methods in various conditions for a wide range of parameter values.

Method of undetermined coefficients for circuits and filters using Legendre functions

This article presents a new way to implement matching networks and filters using the method of undetermined coefficients. A method is proposed for approximating the transmission coefficient of the synthesized filter, taking into account the required amplitude-frequency characteristics. To synthesize the filter, an approximating function (AF) was used using orthogonal Legendre polynomials, which is a mathematical description using a system of equations. Filter properties whose implementation is based on modified Legendre approximating functions usually depend on the interval on which they are defined and have the property that they are orthogonal on this interval. An example of seventh order filter synthesis using modified Legendre approximating functions is given. The filter circuit is implemented, the elements of the filter circuit are calculated based on the selected approximating modified function. The criteria used were minimization of the unevenness of the group delay time (GDT) and minimization of the complex approximation error for given values of the AF parameters. As a result, the number of filter elements, the group delay value and the complex approximation error are significantly reduced.

Krein–Sobolev Orthogonal Polynomials II

In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials {Kn}n=0∞—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space H1[−1,1] with respect to the (positive-definite) inner product (f,g)1,c:=−f(1)−f(−1)g¯(1)−g¯(−1)2+∫−11(f′(x)g¯′(x)+cf(x)g¯(x))dx, where c is a fixed, positive constant. These polynomials generalize the Althammer (or Legendre–Sobolev) polynomials first studied by Althammer and Schäfke. The Krein–Sobolev polynomials were found as a result of a left-definite spectral study of the self-adjoint Krein Laplacian operator Kc(c>0) in L2(−1,1). Other than K0 and K1, these polynomials are not eigenfunctions of Kc. As shown by Littlejohn and Quintero, the sequence {Kn}n=0∞ forms a complete orthogonal set in the first left-definite space (H1[−1,1],(·,·)1,c) associated with (Kc,L2(−1,1)). Furthermore, they show that, for n≥1,Kn(x) has n distinct zeros in (−1,1). In this note, we find an explicit formula for Krein–Sobolev polynomials {Kn}n=0∞.

Dynamical boundary value problem for a viscoelastic half-space with cut

The dynamical boundary value problem for viscoelastic half-space with cut in the form of a strip is considered. The problem is reduced to the singular integral equation of first kind. Using the method of orthogonal polynomials, the integral equation is reduced to an infinite system of linear algebraic equations. The quasi-completely regularity of the obtained system is proved and the reduction method for approximate solution is developed.

Analytical Bounds for Mixture Models in Cauchy–Stieltjes Kernel Families

Mixture models are widely used in mathematical statistics and theoretical probability. However, the mixture probability distribution is rarely explicit in its formula. One must then decide whether to keep the parent probability distribution or to obtain an approximation of the mixture probability distribution. In such cases, it is essential to estimate or evaluate the distance between a mixture probability distribution and its parent probability distribution. On the other hand, orthogonal polynomials offer a versatile mathematical tool for approximating, fitting, and analyzing mixture models, facilitating more accurate and efficient modeling in statistics and data science. This article considers mixture models in Cauchy–Stieltjes Kernel (CSK) families. Using a suitable basis of polynomials, we obtain an expression for the distance in the L2-norm between the mixed probability distribution and its parent probability distribution which belongs to a given CSK family. For the distance between the corresponding distribution functions, bounds are derived in L1-norm. The results are illustrated by some examples from quadratic CSK families.

Orthogonal Polynomial Approximation and Extended Dynamic Mode Decomposition in Chaos

Orthogonal Polynomial Approximation and Extended Dynamic Mode Decomposition in Chaos

Matching Polynomials of Symmetric, Semisymmetric, Double Group Graphs, Polyacenes, Wheels, Fans, and Symmetric Solids in Third and Higher Dimensions

The primary objective of this study is the computation of the matching polynomials of a number of symmetric, semisymmetric, double group graphs, and solids in third and higher dimensions. Such computations of matching polynomials are extremely challenging problems due to the computational and combinatorial complexity of the problem. We also consider a series of recursive graphs possessing symmetries such as D2h-polyacenes, wheels, and fans. The double group graphs of the Möbius types, which find applications in chemically interesting topologies and stereochemistry, are considered for the matching polynomials. Hence, the present study features a number of vertex- or edge-transitive regular graphs, Archimedean solids, truncated polyhedra, prisms, and 4D and 5D polyhedra. Such polyhedral and Möbius graphs present stereochemically and topologically interesting applications, including in chirality, isomerization reactions, and dynamic stereochemistry. The matching polynomials of these systems are shown to contain interesting combinatorics, including Stirling numbers of both kinds, Lucas polynomials, toroidal tree-rooted map sequences, and Hermite, Laguerre, Chebychev, and other orthogonal polynomials.

Orthogonal Polynomials on Radial Rays in the Complex Plane: Construction, Properties and Applications

Orthogonal polynomials on radial rays in the complex plane were introduced and studied intensively in several papers almost three decades ago. This paper presents an account of such kinds of orthogonality in the complex plane, as well as a number of new results and examples. In addition to several types of standard orthogonality, the concept of orthogonality on arbitrary radial rays is introduced, some or all of which may be infinite. A general method for numerical constructing, the so-called discretized Stieltjes–Gautschi procedure, is described and several interesting examples are presented. The main properties, zero distribution and some applications are also given. Special attention is paid to completely symmetric cases. Recurrence relations for such kinds of orthogonal polynomials and their zero distribution, as well as a connection with the standard polynomials orthogonal on the real line, are derived, including the corresponding linear differential equation of the second order. Finally, some applications in physics and electrostatics are mentioned.

Extensions of Riordan Arrays and Their Applications

The Riordan group of Riordan arrays was first described in 1991, and since then, it has provided useful tools for the study of areas such as combinatorial identities, polynomial sequences (including families of orthogonal polynomials), lattice path enumeration, and linear recurrences. Useful extensions of the idea of a Riordan array have included almost Riordan arrays, double Riordan arrays, and their generalizations. After giving a brief overview of the Riordan group, we define two further extensions of the notion of Riordan arrays, and we give a number of applications for these extensions. The relevance of these applications indicates that these new extensions are worthy of study. The first extension is that of the reverse symmetrization of a Riordan array, for which we give two applications. The first application of this symmetrization is to the study of a family of Riordan arrays whose symmetrizations lead to the famous Robbins numbers as well as to numbers associated with the 20 vertex model of mathematical physics. We provide closed-form expressions for the elements of these arrays, and we also give a canonical Catalan factorization for them. We also describe an alternative family of Riordan arrays whose symmetrizations lead to the same integer sequences. The second application of this symmetrization process is to the area of the enumeration of lattice paths. We remain with the applications to lattice paths for the second extension of Riordan arrays that we introduce, which is the interleaved Riordan array. The methods used include generating functions, linear algebra, weighted compositions, and linear recurrences. In the case of the symmetrization process applied to Riordan arrays, we focus on the principal minor sequences of the resulting square matrices in the context of integrable lattice models.

X1-Jacobi Differential Polynomial Systems and Related Double-Step Shape-Invariant Liouville Potentials Solvable by Exceptional Orthogonal Polynomials

This paper develops a new formalism to treat both infinite and finite exceptional orthogonal polynomial (EOP) sequences as X-orthogonal subsets of X-Jacobi differential polynomial systems (DPSs). The new rational canonical Sturm–Liouville equations (RCSLEs) with quasi-rational solutions (q-RSs) were obtained by applying rational Rudjak–Zakhariev transformations (RRZTs) to the Jacobi equation re-written in the canonical form. The presented analysis was focused on the RRZTs leading to the canonical form of the Heun equation. It was demonstrated that the latter equation preserves its form under the second-order Darboux–Crum transformation. The associated Sturm–Liouville problems (SLPs) were formulated for the so-called ‘prime’ SLEs solved under the Dirichlet boundary conditions (DBCs). It was proven that one of the two X1-Jacobi DPSs composed of Heun polynomials contains both the X1-Jacobi orthogonal polynomial system (OPS) and the finite EOP sequence composed of the pseudo-Wronskian transforms of Romanovski–Jacobi (R-Jacobi) polynomials, while the second analytically solvable Heun equation does not have the discrete energy spectrum. The quantum-mechanical realizations of the developed formalism were obtained by applying the Liouville transformation to each of the SLPs formulated in such a way.

Accelerated Modeling of Transients in Electromagnetic Devices Based on Magnetoelectric Substitution Circuits

During switching in electrical systems, transient electromagnetic processes occur. The resulting dangerous current surges are best studied by computer simulation. However, the time required for computer simulation of such processes is significant for complex electromagnetic devices, which is undesirable. The use of spectral methods can significantly speed up the calculation of transient processes and ensure high accuracy. At present, we are not aware of publications showing the use of spectral methods for calculating transient processes in electromagnetic devices containing ferromagnetic cores. The purpose of the work: The objective of this work is to develop a highly effective method for calculating electromagnetic transient processes in a coil with a ferromagnetic magnetic core connected to a voltage source. The method involves the use of nonlinear magnetoelectric substitution circuits for electromagnetic devices and a spectral method for representing solution functions using orthogonal polynomials. Additionally, a schematic model for applying the spectral method is developed. Obtained Results: A method for calculating transients in magnetoelectric circuits based on approximating solution functions with algebraic orthogonal polynomial series is proposed and studied. This helps to transform integro-differential state equations into linear algebraic equations for the representations of the solution functions. The developed schematic model simplifies the use of the calculation method. Representations of true electric and magnetic current functions are interpreted as direct currents in the proposed substitution circuit. Based on these methods, a computer program is created to simulate transient processes in a magnetoelectric circuit. Comparing the application of various polynomials enables the selection of the optimal polynomial type. The proposed method has advantages over other known methods. These advantages include reducing the simulation time for electromagnetic transient processes (in the examples considered, by more than 12 times than calculations using the implicit Euler method) while ensuring the same level of accuracy. The simulation of processes over a long time interval demonstrate error reduction and stabilization. This indicates the potential of the proposed method for simulating processes in more complex electromagnetic devices, (for example, transformers).

Lattice Fermions with Solvable Wide Range Interactions

Abstract Exactly solvable (spinless) lattice fermions with wide range interactions are constructed explicitly based on exactly solvable stationary and reversible Markov chains $\mathcal {K}^R$ reported a few years earlier by Odake and myself. The reversibility of $\mathcal {K}^R$ with the stationary distribution $\pi$ leads to a positive classical Hamiltonian $\mathcal {H}^R$. The exact solvability of $\mathcal {H}^R$ warrants that of a spinless lattice fermion $c_x$, $c_x^\dagger$, $\mathcal {H}^R_f=\sum _{x,y\in \mathcal {X}}c_x^\dagger \mathcal {H}^R(x,y) c_y$ based on the principle advocated recently by myself. The reversible Markov chains $\mathcal {K}^R$ are constructed by convolutions of the orthogonality measures of the discrete orthogonal polynomials of the Askey scheme. Several explicit examples of the fermion systems with wide range interactions are presented.

Orthogonal polynomials: From Heun equations to Painlevé equations

In this paper, we study four kinds of polynomials: orthogonal with the singularly perturbed Gaussian weight wSPG(x), the deformed Freud weight wDF(x), the jumpy Gaussian weight wJG(x), and the Jacobi-type weight wJC(x). The second order linear differential equations satisfied by these orthogonal polynomials and the associated Heun equations are presented. Utilizing the method of isomonodromic deformations given by Dereziński et al. [Symmetry Integr. Geom. Methods Appl. 17, 056 (2021)], we transform these Heun equations into Painlevé equations. It is interesting that the Painlevé equations obtained by the way in this work are same as the results satisfied by the related three term recurrence coefficients or the auxiliaries studied by other authors. In addition, we discuss the asymptotic behaviors of the Hankel determinant generated by the first weight, wSPG(x), under a suitable double scaling for large s and small s, where the Dyson’s constant is recovered.

Finite Markov chains and multiple orthogonal polynomials

Finite Markov chains and multiple orthogonal polynomials

Analysis of sweet corn growth and yield with the use of organic cow manure fertilizer

Sweet corn is a horticultural crop that is widely cultivated and consumed by Indonesian people. Meeting the demand for sweet corn is still reliant on imports; therefore, increasing production by improving soil properties through proper fertilization. Cow manure with a content of N 2.33%, P 0.61%, and K 1.58% is able to add nutrients to the soil. This research aimed to obtain an optimum dose of organic fertilizer for sweet corn growth and yield. The doses of cow manure consisting of 5 levels: control, 10, 20, 30, and 40 tons ha⁻¹ were assigned in the randomized complete block design with 5 replications. Data analysis deployed analysis of variance (Anova) at 5% level, and if there were significant differences, it would be proceeded to Duncan Multiple Range Test 5% and orthogonal polynomial. The results showed that a dose of 20 tons ha⁻¹ was able to increase growth in plant height, stem diameter and number of leaves by 12.92%, 26.29%, and 15.67%, respectively, compared to the control. A dose of 35 tons ha⁻¹ can produce a weight of cob-husk and cob without-husk per plot, respectively, 4.38 kg plot⁻¹ and 2.64 kg plot⁻¹, and a productivity of cob-husk of 5.30 tons ha⁻¹. A dose of 38 tons ha⁻¹ can produce a cob without-husk productivity of 3.29 tons ha⁻¹.