Generalized Laguerre Polynomials Research Articles

The noncommutative Dirac oscillator with a permanent electric dipole moment in the presence of an electromagnetic field

In this work, we investigate the noncommutative Dirac oscillator (NCDO) with a permanent electric dipole moment (EDM) in the presence of an electromagnetic field. We consider a radial magnetic field generated by anti-Helmholtz coils and the uniform electric field of the Stark effect. Next, we determine the exact solutions of the system, given by the Dirac spinor and by the relativistic energy spectrum. We note that such a spinor is written in terms of the generalized Laguerre polynomials, and such a spectrum is a linear function on the potential energy U and depends on the quantum numbers n and m, spin parameter s, and of four angular frequencies: ω, ω¯ , ω θ , and ω η , where ω is the frequency of the DO, ω¯ is a type of ‘cyclotron frequency’, and ω θ and ω η are the NC frequencies of position and momentum. We graphically analyze the behavior of the spectrum as a function of these frequencies for three different values of n, with and without the influence of U. Besides, as an interesting result, another interpretation can be given for the origin of the nonminimal coupling of the DO, which would be through a neutral fermion with EDM in a radial magnetic field. Finally, we also analyze the nonrelativistic limit of our results, and comparing our problem with other works, we verified that our results generalize some particular cases in the literature.

An efficient spectral collocation method based on the generalized Laguerre polynomials to multi-term time fractional diffusion-wave equations

In this study, a spectral collocation method is proposed to solve a multi-term time fractional diffusion-wave equation. The solution is expanded by a series of generalized Laguerre polynomials, and then, by imposing the collocation nodes, the equation is reduced to a linear system of algebraic equations. The coefficients of the expansion can be determined by solving the resulting system. The convergence of the method is proved, and some numerical examples are presented to demonstrate the accuracy and efficiency of the scheme. Finally, conclusions are given.

Stability of selfsimilar solutions to the fragmentation equation with polynomial daughter fragments distribution

We study fragmentation equations with power-law fragmentation rates and polynomial daughter fragments distribution function $ p(s) $. The corresponding selfsimillar solutions are analysed and their exponentially decaying asymptotic behaviour and $ C^{\infty } $ regularity deduced. Stability of selfsimilar solutions (under smooth exponentially decaying perturbations), with sharp exponential decay rates in time are proved, as well as $ C^{\infty } $ regularity of solutions for $ t>0 $. The results are based on explicit expansion in terms of generalized Laguerre polynomials and the analysis of such expansions. For perturbations with power-law decay at infinity stability is also proved. Finally, we consider real analytic $ p(s) $.

Three-dimensional eigenvalues of harmonic oscillator and Coulomb-type potentials from one-dimensional generalized Morse potential: perturbative analysis based on generalized Laguerre polynomials

Three-dimensional eigenvalues of harmonic oscillator and Coulomb-type potentials from one-dimensional generalized Morse potential: perturbative analysis based on generalized Laguerre polynomials

Rational solutions of the fifth Painlevé equation. Generalized Laguerre polynomials

AbstractIn this paper, rational solutions of the fifth Painlevé equation are discussed. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalized Laguerre polynomials, which are the main subject of this paper, and the other in terms of the generalized Umemura polynomials. Both the generalized Laguerre polynomials and the generalized Umemura polynomials can be expressed as Wronskians of Laguerre polynomials specified in terms of specific families of partitions. The properties of the generalized Laguerre polynomials are determined and various differential‐difference and discrete equations found. The rational solutions of the fifth Painlevé equation, the associated σ‐equation, and the symmetric fifth Painlevé system are expressed in terms of generalized Laguerre polynomials. Nonuniqueness of the solutions in special cases is established and some applications are considered. In the second part of the paper, the structure of the roots of the polynomials are investigated for all values of the parameters. Interesting transitions between root structures through coalescences at the origin are discovered, with the allowed behaviors controlled by hook data associated with the partition. The discriminants of the generalized Laguerre polynomials are found and also shown to be expressible in terms of partition data. Explicit expressions for the coefficients of a general Wronskian Laguerre polynomial defined in terms of a single partition are given.

Certain Integral Transforms and Their Applications in Propagation of Laguerre-Gaussian Schell-model Beams

The principal aim of the present work is to investigate a new class of integral transforms involving the product of Bessel function of the first kind of arbitrary order with generalized Laguerre polynomials and logarithmic functions which deduce some new results in terms of the digamma function and Kamp´e de F´eriet functions. A novel expression is found for the Kamp´e de F´eriet function F1:2;1 2:1;0 in terms of hypergeometric functions 1F1, 2F2 and 3F2. Finally, The results obtained are applied in the problem of propagation of Laguerre-Bessel-Gaussian Schell-model beams as an application.

A study on fractional tumor-immune interaction model related to lung cancer via generalized Laguerre polynomials

BackgroundCancer, a complex and deadly health concern today, is characterized by forming potentially malignant tumors or cancer cells. The dynamic interaction between these cells and their environment is crucial to the disease. Mathematical models can enhance our understanding of these interactions, helping us predict disease progression and treatment strategies.MethodsIn this study, we develop a fractional tumor-immune interaction model specifically for lung cancer (FTIIM-LC). We present some definitions and significant results related to the Caputo operator. We employ the generalized Laguerre polynomials (GLPs) method to find the optimal solution for the FTIIM-LC model. We then conduct a numerical simulation and compare the results of our method with other techniques and real-world data.ResultsWe propose a FTIIM-LC model in this paper. The approximate solution for the proposed model is derived using a series of expansions in a new set of polynomials, the GLPs. To streamline the process, we integrate Lagrange multipliers, GLPs, and operational matrices of fractional and ordinary derivatives. We conduct a numerical simulation to study the effects of varying fractional orders and achieve the expected theoretical results.ConclusionThe findings of this study demonstrate that the optimization methods used can effectively predict and analyze complex phenomena. This innovative approach can also be applied to other nonlinear differential equations, such as the fractional Klein–Gordon equation, fractional diffusion-wave equation, breast cancer model, and fractional optimal control problems.

A collocation method using generalized Laguerre polynomials for solving nonlinear optimal control problems governed by integro-differential equations

The Laguerre polynomials represent a new set of basic functions adopted to solve a class of optimal control problems (OCP) specified by integro-differential equation namely IOCP. The corresponding operational matrices of derivatives are computed to extend the solution of the problem in terms of Laguerre polynomials. Using the results of the considered basis functions with a joint application of collocation method, the IOCP is reduced to solve a system of nonlinear algebraic system. Come acrossing an error bound and convergence analysis for the approximate optimal value of the performance index have been proved by the proposed method. Finally, some examples are given to bring about the validity and applicability of this technique.

An optimization method for studying fractional-order tuberculosis disease model via generalized Laguerre polynomials.

Tuberculosis (TB) is a deadly contagious disease that affects vital organs of the body, especially the lungs. Although the disease is preventable, there are still concerns about its continued spread. Without effective prevention or appropriate treatment, TB infection can be fatal to humans. This paper presents a fractional-order TB disease (FTBD) model to analyze TB dynamics and a new optimization method to solve it. The method is based on the basis functions of generalized Laguerre polynomials (GLPs) and some new operational matrices of derivatives in the Caputo sense. Finding the optimal solution to the FTBD model is reduced to solving a system of nonlinear algebraic equations with the aid of GLPs using the Lagrange multipliers method. A numerical simulation is also carried out to determine the impact of the presented method on the susceptible, exposed, infected without treatment, infected with treatment, and recovered cases in the population.

Quaternion fractional-order weighted generalized Laguerre–Fourier moments and moment invariants for color image analysis

Orthogonal moments based on Laguerre polynomials, especially the quaternion representation of fractional-order orthogonal moments have attracted interest in the imaging field. However, most of the existing quaternion fractional-order color orthogonal moments (QFr-COMs) based on Laguerre polynomials are constructed on Cartesian coordinate space and do not have direct rotation geometric invariance. Inspired by the idea of polynomial transformation, we present a weighted radial normalized fractional-order generalized Laguerre polynomials (WRNFr-GLPs) in polar coordinate space. On the basis, a new set of quaternion fractional-order weighted generalized Laguerre–Fourier moments (QFr-WGLFMs) is proposed and a fast and reliable computing method for geometric invariance of QFr-WGLFMs also is introduced in this paper. Theoretical analyses and experimental results showed that the proposed QFr-WGLFMs and moment invariants offer enhanced image reconstruction and pattern recognition compared with the existing quaternion fractional-order-based orthogonal moments and deep learning-based​ methods, especially under the conditions of image processing with noise and smooth filtering.

Potentials from the Polynomial Solutions of the Confluent Heun Equation

Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function Hc(p,β,γ,δ,σ;z). One of these recovers the generalized Laguerre polynomials LN(α), and another one the rationally extended X1 type Laguerre polynomials L^N(α). The two remaining solutions represent previously unknown polynomials that do not form an orthogonal set and exhibit features characteristic of semi-classical orthogonal polynomials. A standard method of generating exactly solvable potentials in the one-dimensional Schrödinger equation is applied to the CHE, and all known potentials with solutions expressed in terms of the generalized Laguerre polynomials within, or outside the Natanzon confluent potential class, are recovered. It is also found that the potentials generated from the two new polynomial systems necessarily depend on the N quantum number. General considerations on the application of the Heun type differential differential equations within the present framework are also discussed.

Optimal Approximation of Fractional Order Brain Tumor Model Using Generalized Laguerre Polynomials

A brain tumor occurs when abnormal cells form within the brain. Glioblastoma (GB) is an aggressive and fast-growing type of brain tumor that invades brain tissue or spinal cord. GB evolves from astrocytic glial cells in the central nervous system. GB can occur at almost any age, but the occurrence increases with advancing age in older adults. Its symptoms may include nausea, vomiting, headaches, or even seizures. GB, formerly known as glioblastoma multiforme, currently has no cure with a high rate of resistance to therapy in clinical treatment. However, treatments can slow tumor progression or alleviate the signs and symptoms. In this paper, a fractional order brain tumor model was considered. The optimal solution of the model was obtained using an optimization method based on operational matrices. The solution to the problem under study was expanded in terms of generalized Laguerre polynomials (GLPs). The study problem was shifted to a system of nonlinear algebraic equations by the use of Lagrange multipliers combined with operational matrices based on GLPs. The analysis of convergence was discussed. In the end, some numerical examples were presented to justify theoretical statements along with the patterns of biological behavior.

TRIPLE SERIES EQUATIONS INVOLVING GENERALIZED LAGUERRE POLYNOMIALS

In this paper, the solutions of two sets of triple series equations involving generalized Laguerre polynomials have been obtained by reducing them to a Fredholm integral equation of second kind. In each case, the problem is reduced to the solution of a Fredholm integral equation of the second kind. We consider certain triple series equations involving generalized Laguerre polynomials which are generalization of those considerd by Lawndes-Srivastava .Connected to this work solutions have been considered by Sneddon, Lowndes and Srivastava ,Dwivedi and Trivedi, Singh et al., Narain, Srivastava - Panda etc

A Comparative Study of Thermal Performance of Different Nanofluids: An Analytic Approach

The purpose of this study was to determine an exact solution for the fluid flow and heat transfer of laminar steady magnetohydrodynamics (MHD) nanofluid flow over a stretching/shrinking surface. Appropriate similarity transformations were used to transform the governing partial differential equations into coupled nonlinear ordinary differential equations. The current study showed good correspondence with previously published work. The solution was deduced from the solution of the flow field and temperature field. Furthermore, the dimensionless skin friction coefficient and Nusselt number were derived. The solution of the temperature field was deduced in terms of the generalized Laguerre polynomial. The value of the generalized Laguerre polynomial was calculated using the “LaguerreL” command in MuPAD. The impact of different physical parameters of the symmetry on the thermal performance, including the nanoparticle volume fraction parameter, magnetic parameter, mass suction/injection parameter and stretching/shrinking parameter, is discussed in detail for different nanoparticles. Furthermore, the effect of nanoparticle type on the fluid velocity component, temperature distribution, skin friction coefficient and Nusselt number was studied in detail. Four different nanoparticles were considered in this study. This work reveals that the nanoparticles within the base fluid have the potential to increase the heat transfer ability of many liquids. The results indicate that silver and titanium oxide nanoparticles had the largest and lowest skin friction coefficients, respectively, in the shrinking surface case, exhibiting opposite behavior in the stretching surface case among all the nanoparticles considered. The results also indicate that silver and titanium oxide nanoparticles had the largest and lowest Nusselt numbers, respectively, for both the stretching and the shrinking surface cases. It is suggested silver nanoparticles are not used for optimum heat transfer.

Expansion of some well-known functions and classical polynomials in series of the generalized Laguerre polynomials.1

The generalized Laguerre polynomials form a complete set (orthogonal and normalized) in the space ** with respect to a certain weighting function because they are the Eigen functions of a second-order differential operator. Here we shall show how to expand some well-known classical polynomials such as the Legendre and the Hermite polynomials in series of generalized Laguerre polynomials. Since the generalized Laguerre orthogonal polynomials are set of polynomials that are mutually orthogonal to each other with respect to a measure of weighting function that is just the integrand of the gamma function, thus it grants us the guarantee of the ability to expand the first kind of Bessel functions and the gamma function in terms of the generalized Laguerre polynomials. The series expansion in terms of the generalized Laguerre polynomials can be achieved by following various approaches. For instance, the series expansion of the first kind Bessel functions is gained by the generalized hypergeometric function approach, whereas the series expansion of gamma function was obtained directly by the usual way that is by calling the orthogonality and orthonormality properties of the generalized Laguerre polynomials. Another powerful technique to gain the series expansion of the classical polynomials is the generating function approach as it has been followed here to obtain a series expansion of the Legendre and the Hermite polynomials in terms of the generalized Laguerre polynomials. The series expansion in series of the generalized Laguerre polynomials has a variety of applications in mathematics, physics, and engineering

Reconstruction of neutron multiplicity distributions from low-order statistical information

A point-kinetic methodology for rapid and accurate computation of neutron multiplicity distributions in fixed time intervals is demonstrated. The approach uses an optimal combination of low-order count probabilities and low-order statistical moments, both of which are inexpensively computed from a backward Master equation, to reconstruct distributions valid for all count numbers. For high count numbers a generalized Laguerre polynomial representation is shown to accurately reconstruct the multiplicity distribution from five or fewer statistical moments when the zero-count probability is negligible. For intermediate count numbers and when the zero-count probability is significant, maximum information entropy reconstruction constrained by low-order moments and low-order count probabilities reproduces the overall count distribution very accurately. Numerical results for count number distributions and reconstruction accuracy as a function of number of constraints, relative to a system state-updating stochastic simulation MC algorithm (SSA), are presented to demonstrate the efficacy of this approach.

Predictor-Feedback Prescribed-Time Stabilization of LTI Systems With Input Delay

This article first deals with the problem of prescribed-time stability of linear systems without delay. The analysis and design involve the <i>Bell polynomials</i>, <i>the generalized Laguerre polynomials</i>, <i>the Lah numbers</i>, and a suitable <i>polynomial-based Vandermonde matrix</i>. The results can be used to design a new controller&#x2014;with time-varying gains&#x2014;ensuring prescribed-time stabilization of controllable linear time-invariant (LTI) systems. The approach leads to similar results compared to Holloway <i>et al.</i> 2019, but offers an alternative and compact control design (especially for the choice of the time-varying gains). Based on the preliminary results for the delay-free case, we then study controllable LTI systems with single input and subject to a constant input delay. We design a predictor feedback with time-varying gains. To achieve this, we model the input delay as a transport partial differential equation (PDE) and build on the cascade PDE&#x2013;ordinary differential equation setting (inspired by Krstic 2009) so as the design of the prescribed-time predictor feedback is carried out based on the backstepping approach, which makes use of <i>time-varying kernels</i>. We guarantee the bounded invertibility of the backstepping transformation, and we prove that the closed-loop solution converges to the equilibrium in a prescribed time. A simulation example illustrates the results.

Generalized Gamma-Laguerre Polynomial Chaos to Model Random Bending of Wearable Antennas

A novel generalized Gamma-Laguerre polynomial chaos expansion is proposed to account for the effect of random variations in lower bounded design parameters on antenna performance. After fitting a shifted generalized Gamma distribution to datasets of such random variables, a predistorted polynomial chaos expansion is generated based on a set of orthogonal generalized Laguerre polynomials. The new statistical methodology is applied to assess the random change in resonance frequency when bending a wearable antenna around different parts of the human body, such as a leg, an arm, and a head. For different datasets, an excellent statistical fit is found to both the estimated probability density function of the bending radius and the resulting statistical distribution of the resonance frequency, while requiring up to 480 times fewer sample evaluations.

Families of Laguerre polynomials with alternating group as Galois group

For an arbitrary real number α and a positive integer n, the Generalized Laguerre Polynomials (GLP) is a family of polynomials defined byLn(α)(x)=(−1)n∑j=0n(n+α)(n−1+α)⋯(j+1+α)(n−j)!j!(−x)j. Following the work of Banerjee, Filaseta, Finch and Leidy [2] which described the set A0∪A∞ of integer pairs (n,α) for which the discriminant of Ln(α)(x) is a nonzero square, where A0 is finite and A∞ is explicitly given infinite set, it was conjectured by Banerjee in [1] that for α≠−1, the only pair (n,α)∈A∞ for which the associated Galois group of Ln(α)(x) is not An is (4,23). In this paper, we verify this conjecture for (α,n)∈A∞ with α∈{−2n,−2n−2,−2n−4}.In fact, we prove more general results concerning the irreducibility and Galois groups of the Generalized Laguerre polynomials Ln(α)(x) for n∈N and integers α such that α∈[−2n−4,−2n]. The case α=−2n−1 corresponds to the Bessel polynomials which have been studied earlier by Filaseta and Trifonov [7] and Grosswald [9]. Our ideas give a simpler proof of their results.

Limits of Jensen polynomials for partitions and other sequences

It was discovered recently by Griffin, Ono, Rolen and Zagier that the Jensen polynomials associated to many sequences have Hermite polynomial limits. We develop this theory in detail, based on the log-polynomial property which is a refinement of log-concavity and log-convexity. Applications to various partition sequences are given. An application to the sequence of factorials leads naturally to evaluating limits of generalized Laguerre polynomials.