Laguerre Polynomials Research Articles

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

Estimate for the largest zeros of the D’Arcais polynomials

The zeros of the nth D’Arcais polynomial, also known in combinatorics as the Nekrasov–Okounkov polynomial, dictate the vanishing properties of the nth Fourier coefficients of all complex powers x of the Dedekind η\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta $$\\end{document}-function. In this paper, we prove that these coefficients are non-vanishing for |x|>κ(n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\vert x \\vert > \\kappa \\, (n-1)$$\\end{document} and κ≈9.7225\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa \\approx 9.7225$$\\end{document}. Numerical computations imply that 9.72245 is a lower bound for κ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa $$\\end{document}. This significantly improves previous results by Kostant, Han, and Heim–Neuhauser. The polynomials studied in this paper include Chebyshev polynomials of the second kind, 1-associated Laguerre polynomials, Hermite polynomials, and polynomials associated with overpartitions and plane partitions.

Can inflation predict energy price volatility?

Fluctuations in energy prices impact production costs and inflation. This study examines whether inflation data can predict volatility in energy markets. Both inflation and energy market volatility exhibit complex behaviour over time, including structural shifts due to demand and supply shocks. Accounting for differences in data frequencies, we use an extended GARCH model (MIDAS) with Laguerre polynomials for time-varying parameters. The empirical results demonstrate that including low-frequency inflation data enhances energy model predictability particularly during periods of high volatility and extreme price fluctuations. Considering inflation improves forecasting for energy market models, benefiting portfolio management and helping policymakers manage inflation.

Galerkin spectral method for linear second‐kind Volterra integral equations with weakly singular kernels on large intervals

This paper considers the Galerkin spectral method for solving linear second‐kind Volterra integral equations with weakly singular kernels on large intervals. By using some variable substitutions, we transform the mentioned equation into an equivalent semi‐infinite integral equation with nonsingular kernel, so that the inner products from the Galerkin procedure could be evaluated by means of Gaussian quadrature based on scaled Laguerre polynomials. Furthermore, the error analysis is based on the Gamma function and provided in the weighted ‐norm, which shows the spectral rate of convergence is attained. Moreover, several numerical experiments are presented to validate the theoretical results.

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

Zeros transfer for recursively defined polynomials

The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.

New operators based on Laguerre polynomials

New operators based on 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.

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

Laplace transform solution of the time-dependent annular Couette flow with dynamic wall slip

The annular Couette flow has several industrial applications, particularly for the characterization of the fluid flow and deformation behavior of fluids. The inclusion of the dynamic wall slip into the flow boundary conditions seems to be necessary for an efficient continuum description of motion of nanofluidics as it reflects the importance of fluid–structure interface related phenomena. Dynamic wall slip introduces a dissipative boundary condition and thus increases the difficulties of finding solutions to related problems. In the present work we investigate the behavior of fluid flow between two infinitely long coaxial circular cylinders, when the inner cylinder is axially moving due to sudden constant velocity, while the outer cylinder is held stationary. The boundary condition on the outer cylinder is that of dynamic wall slip, in addition to the usual Navier slip. The medium considered here is a Newtonian viscous fluid. The solution of the governing equations, initial and boundary conditions for this flow is obtained using the Laplace transform technique and inversion by Laguerre polynomials. This method may be useful, when applied in conjunction with perturbation methods, to solve nonlinear Couette flow problems involving temperature changes. Numerical results are presented and discussed.

Cascaded metasurfaces for high-purity vortex generation

We introduce a new paradigm for generating high-purity vortex beams with metasurfaces. By applying optical neural networks to a system of cascaded phase-only metasurfaces, we demonstrate the efficient generation of high-quality Laguerre-Gaussian (LG) vortex modes. Our approach is based on two metasurfaces where one metasurface redistributes the intensity profile of light in accord with Rayleigh-Sommerfeld diffraction rules, and then the second metasurface matches the required phases for the vortex beams. Consequently, we generate high-purity LGp,l optical modes with record-high Laguerre polynomial orders p = 10 and l = 200, and with the purity in p, l and relative conversion efficiency as 96.71%, 85.47%, and 70.48%, respectively. Our engineered cascaded metasurfaces suppress greatly the backward reflection with a ratio exceeding −17 dB. Such higher-order optical vortices with multiple orthogonal states can revolutionize next-generation optical information processing.

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.

Exact Solution of the Position-Dependent Mass Schrödinger Equation with the Completely Positive Oscillator-Shaped Quantum Well Potential

"Two exactly-solvable confined models of the completely positive oscillator-shaped quantum well are proposed. Exact solutions of the position-dependent mass Schrodinger equation corresponding to the proposed quantum well potentials are ¨ presented. It is shown that the discrete energy spectrum expressions of both models depend on certain positive confinement parameters. The spectrum exhibits positive equidistant behavior for the model confined only with one infinitely high wall and nonequidistant behavior for the model confined with the infinitely high wall from both sides. Wavefunctions of the stationary states of the models under construction are expressed through the Laguerre and Jacobi polynomials. In general, the Jacobi polynomials appearing in wavefunctions depend on parameters a and b, but the Laguerre polynomials depend only on the parameter a. Some limits and special cases of the constructed models are discussed."

Comparative Study of the Mass Spectra of Heavy Quarkonium System with an Interacting Potential Model

In this work, the comparison of the mass spectra of heavy quarkonium system with an interacting potential (Class of Yukawa potential) was studied. The Schrodinger equation is analytically solved using Nikiforov- Uvarov (NU) method and series expansion method (SEM). The approximate solutions of the eigen energy equation and corresponding eigenfunction in terms of Laguerre polynomials were obtained using the NU method and the solutions of the eigen energy equation were also obtained with the SEM. The mass spectra for heavy quarkonium system (HQS) for the potential under study were obtained for bottomonium and charmonium HQS. We compared the results obtained between NU and SEM. It was noticed that SEM solutions yield mass spectra very close to experimental data compared to solutions with NU method. The obtained results were also compared with works by some other authors and were found to be improved. This study can be extended by using other exponential-type potential models with other analytical approach and a different approximation schemes to obtain the mass spectra of heavy quarkonium system. The relativistic properties using Klein-Gordon or Dirac equations can be explored to obtain the mass spectra of light quarkonia. Finally, the information entailed in the normalized wave-functions can also be studied.

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.

On computing modified moments for half-range Hermite weights

AbstractIn this paper, we consider the computation of the modified moments for the system of Laguerre polynomials on the real semiaxis with the Hermite weight. These moments can be used for the computation of integrals with the Hermite weight on the real semiaxis via product rules. We propose a new computational method based on the construction of the null-space of a rectangular matrix derived from the three-term recurrence relation of the system of orthonormal Laguerre polynomials. It is shown that the proposed algorithm computes the modified moments with high relative accuracy and linear complexity. Numerical examples illustrate the effectiveness of the proposed method.

Rayleigh Waves Propagating in the Functionally Graded One-Dimensional Hexagonal Quasicrystal Half-Space

For the manufacturing and optimization of quasicrystal structures, Rayleigh waves propagating in the functionally graded one-dimensional hexagonal quasicrystal half-space are investigated. The analytical Laguerre orthogonal polynomial method is employed to solve dynamic equations of wave motion, which greatly improves the computational efficiency. Dispersion curves and displacement distributions are illustrated. The influences of the phonon–phason coupling effect, inhomogeneity, and quasiperiodic direction on wave characteristics are analyzed. Some new results are obtained: (1) Compared with the classical Laguerre polynomial method, the improvement in computational efficiency of the analytical Laguerre polynomial method is more than 99%. (2) The energy penetration depth of phason modes is greater than that of phonon modes. The results lay a theoretical foundation for designing and optimizing SAW devices.

The Black–Scholes equation in finance: Quantum mechanical approaches

In this paper, the Black–Scholes equation of the option pricing theory in order to minimize the risk through the stocks is studied. The solutions are obtained in terms of exceptional Laguerre polynomials. Moreover, higher-order supesymmetric representations are studied with a special case of third order. The Darboux transformation of the heat equation linked to the Black–Scholes system is given and a new potential is shown.

Moment-based approach to the flux-tube linear gyrokinetic model

This work reports on the development and numerical implementation of the linear electromagnetic gyrokinetic (GK) model in a tokamak flux-tube geometry using a moment approach based on the expansion of the perturbed distribution function on a velocity-space Hermite–Laguerre polynomials basis. A hierarchy of equations of the expansion coefficients, referred to as the gyro-moments (GMs), is derived. We verify the numerical implementation of the GM hierarchy in the collisionless limit by performing a comparison with the continuum GK code GENE, recovering the linear properties of the ion temperature gradient, trapped electron, kinetic ballooning and microtearing modes, as well as the collisionless damping of zonal flows. An analysis of the distribution functions and ballooning eigenmode structures is performed. The present investigation reveals the ability of the GM approach to describe fine velocity-space-scale structures appearing near the trapped and passing boundary and kinetic effects associated with parallel and perpendicular particle drifts. In addition, the effects of collisions are studied using advanced collision operators, including the GK Coulomb collision operator. The main findings are that the number of GMs necessary for convergence decreases with plasma collisionality and is lower for pressure gradient-driven modes, such as in H-mode pedestal regions, compared with instabilities driven by trapped particles and magnetic gradient drifts often found in the core. The accuracy of approximations often used to model collisions (relative to the GK Coulomb operator) is studied in the case of trapped electron modes, showing differences between collision operator models that increase with collisionality and electron temperature gradient, consistent with the results of Pan et al. (Phys. Rev. E, vol. 103, 2021, L051202). Such differences are not observed in other edge microinstabilities, such as microtearing modes. The importance of a proper collision operator model is also confirmed by analysing the collisional damping of geodesic acoustic modes and zonal flows.

Spectral-Domain Method of Moments for the Modal Analysis of Line Waveguides

A rigorous full-wave modal analysis based on the method of moments in the spectral domain is presented for line waveguides constituted by two-part impedance planes with arbitrary anisotropic surface impedances. An integral equation is formulated by introducing an auxiliary current sheet on one of the two half planes and extending the impedance boundary condition of the complementary half plane to hold on the entire plane. The equation is then discretized with the method of moments in the spectral domain, by employing exponentially weighted Laguerre polynomials as entire-domain basis functions and performing a Galerkin testing. Numerical results for both <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bound</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">leaky</i> line waves are presented and validated against independent results, obtained for isotropic surface impedances with the analytical Sommerfeld–Maliuzhinets method and for the general anisotropic case with a commercial electromagnetic simulator. The proposed approach is computationally efficient, can accommodate the presence of spatial dispersion, and offers physical insight into the modal propagation regimes.