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The Laguerre polynomial method has been successfully used to investigate the dynamic responses of a half-space. However, it fails to obtain the correct stress at the interfaces in a layered half-space, especially when there are significant differences in material properties. Therefore, a coupled Legendre-Laguerre polynomial method with analytical integration is proposed. The Rayleigh waves in a one-dimensional (1D) hexagonal quasicrystal (QC) layered half-space with an imperfect interface are investigated. The correctness is validated by comparison with available results. Its computation efficiency is analyzed. The dispersion curves of the phase velocity, displacement distributions, and stress distributions are illustrated. The effects of the phonon-phason coupling and imperfect interface coefficients on the wave characteristics are investigated. Some novel findings reveal that the proposed method is highly efficient for addressing the Rayleigh waves in a QC layered half-space. It can save over 99% of the computation time. This method can be expanded to investigate waves in various layered half-spaces, including earth-layered media and surface acoustic wave (SAW) devices. [ABSTRACT FROM AUTHOR]

In this paper, we apply the incomplete Pochhammer matrix symbols to introduce incomplete Lauricella matrix functions (ILMFs) of n variables. Additionally, the integral formula, recursion formula, differentiation formula, and matrix differential equation of ILMFs are all derivations of specific attributes. In addition, we demonstrate a connection between these matrix functions and other matrix functions, such as the incomplete gamma matrix function, the Laguerre, the Bessel, and the modified Bessel matrix functions. [ABSTRACT FROM AUTHOR]

Based on the extensive application of the q$$ q $$‐series and q$$ q $$‐polynomials including (for example) the q$$ q $$‐Laguerre polynomials in several fields of the mathematical and physical sciences, we attach great importance to the equations and related application issues involving the q$$ q $$‐Laguerre polynomials. The mission of this paper is to find the general q$$ q $$‐operational equation together with the expansion issue of the bivariate q$$ q $$‐Laguerre polynomials from the perspective of q$$ q $$‐partial differential equations. We also give some applications including some q$$ q $$‐Hille‐Hardy type formulas. In addition, we present the Rogers‐type formulas and the U(n+1)$$ U\left(n+1\right) $$‐type generating functions for the bivariate q$$ q $$‐Laguerre polynomials by the technique based upon q$$ q $$‐operational equations. Moreover, we derive a new generalized Andrews‐Askey integral and a new transformation identity involving the bivariate q$$ q $$‐Laguerre polynomials by applying q$$ q $$‐operational equations. U(n+1)$$ U\left(n+1\right) $$ [ABSTRACT FROM AUTHOR]

In this paper, we revisit the Stark effect of the hydrogen atom induced by a uniform static electric field. In particular, a general formula for the integral of associated Laguerre polynomials was derived by applying the method for Hermite polynomials of degree n proposed in the work (Anh-Tai T.D. et al., 2021 AIP Advances 11 085310). The quadratic Stark effect is obtained by applying this formula and the time-independent non-degenerate perturbation theory to hydrogen. Using the Siegert state method, numerical calculations are performed and serve as data for benchmarking. The comparisons are then illustrated for the ground and some highly excited states to provide an insightful look at the applicable limit and precision of the quadratic Stark effect formula for other atoms with comparable properties. [ABSTRACT FROM AUTHOR]

We provide a refined combinatorial identity for the set of partitions of |$\{1,\dots , n\}$|⁠ , which plays an important role in investigating several limit theorems related to finite free convolutions. Firstly, we present the finite free analogue of Sakuma and Yoshida's limit theorem. That is, we provide the limit of |$\{D_{1/m}((p_{d}^{\boxtimes _{d}m})^{\boxplus _{d}m})\}_{m\in{\mathbb{N}}}$| as |$m\rightarrow \infty $| in two cases: (i) |$m/d\rightarrow t$| for some |$t>0$| or (ii) |$m/d\rightarrow 0$|⁠. The second application presents a central limit theorem for finite free multiplicative convolution. We establish a connection between this theorem and the multiplicative free semicircular distributions through combinatorial identities. Our last result gives alternative proofs for Kabluchko's limit theorems concerning the unitary Hermite and the Laguerre polynomials. [ABSTRACT FROM AUTHOR]

The Schrödinger equation for the hydrogen atom enclosed by an impenetrable spherical cavity is solved through a Finite‐Differences approach to gain an insight on the actual nature and structure of the ansatz wavefunction cutoff factor widely used in an ad hoc manner in corresponding variational calculations to comply with the Dirichlet boundary conditions. The results of this work provide a theoretical foundation for the choice of the appropriate analytical cutoff functions that fulfill the boundary conditions. We find three different regions for the behavior of the cutoff functions. Small cavity radius where the cutoff function has a parabolic behavior, an intermediate region where the cutoff function is quasi‐linear, and a large cavity region where the cutoff function is a step‐like function. We deduce the traditional linear and quadratic cutoff functions used in the literature as well as its validity region for the confining radius. Finally, we provide a mathematical deduction of the exact cutoff function in terms of the nodal structure of the free hydrogenic wavefunctions and a relation to the Laguerre polynomials for some cavity radii where the free atomic energy level coincides with a confined energy level. We find that the cutoff function transit over several unconfined solutions in terms of its nodal structure as the system is compressed. [ABSTRACT FROM AUTHOR]

Considering the approximation of , with , by the partial sums of the Fourier series in a system of polynomials orthogonal in the sense of Sobolev and generated by a system of classical Laguerre polynomials, we obtain some estimates for the convergence rate of to . [ABSTRACT FROM AUTHOR]

Consider a situation where one is interested in estimating the density of a survival time that is subject to random right censoring and measurement errors. This happens often in practice, like in public health (pregnancy length), medicine (duration of infection), ecology (duration of forest fire), among others. We assume a classical additive measurement error model with Gaussian noise and unknown error variance and a random right censoring scheme. Under this setup, we develop minimal conditions under which the assumed model is identifiable when no auxiliary variables or validation data are available, and we offer a flexible estimation strategy using Laguerre polynomials for the estimation of the error variance and the density of the survival time. The asymptotic normality of the proposed estimators is established, and the numerical performance of the methodology is investigated on both simulated and real data on gestational age. [ABSTRACT FROM AUTHOR]

For an odd prime p, let E p (X) = ∑ n = 0 ∞ a n X n ∈ F p [ [ X ] ] denote the reduction modulo p of the Artin-Hasse exponential series. It is known that there exists a series G (X p) ∈ F p [ [ X ] ] , such that L p - 1 (- T (X)) (X) = E p (X) · G (X p) , where T (X) = ∑ i = 1 ∞ X p i and L p - 1 (α) (X) denotes the (generalized) Laguerre polynomial of degree p - 1 . We prove that G (X p) = ∑ n = 0 ∞ (- 1) n a np X np , and show that it satisfies G (X p) G (- X p) T (X) = X p. [ABSTRACT FROM AUTHOR]

In this paper, we give a criterion to prove boundedness results for several operators from the Hardy‐type space H1((0,∞)d,γα)$H^1((0,\infty)^d,\gamma _\alpha)$ to L1((0,∞)d,γα)$L^1((0,\infty)^d,\gamma _\alpha)$ and also from L∞((0,∞)d,γα)$L^\infty ((0,\infty)^d,\gamma _\alpha)$ to the space of functions of bounded mean oscillation BMO((0,∞)d,γα)$\textup {BMO}((0,\infty)^d,\gamma _\alpha)$, with respect to the probability measure dγα(x)=∏j=1d2Γ(αj+1)xj2αj+1e−xj2dxj$d\gamma _\alpha (x)=\prod _{j=1}^d\frac{2}{\Gamma (\alpha _j+1)} x_j^{2\alpha _j+1} \text{e}^{-x_j^2} dx_j$ on (0,∞)d$(0,\infty)^d$ when α=(α1,⋯,αd)$\alpha =(\alpha _1, \dots,\alpha _d)$ is a multi‐index in −12,∞d$\left(-\frac{1}{2},\infty \right)^d$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood–Paley functions, multipliers of the Laplace transform type, fractional integrals, and variation operators in the Laguerre setting. [ABSTRACT FROM AUTHOR]

We obtain a formula describing the forward and backward wave profile for the solution of an initial–boundary value problem for the wave equation on an interval. The following combinations of boundary conditions are considered: (i) The first-kind condition at the left endpoint of the interval and the third-kind condition at the right endpoint. (ii) The second-kind condition at the left endpoint and the third-kind condition at the right endpoint. (iii) The first-kind condition at the left endpoint and the attached mass condition at the right endpoint. (iv) The second-kind condition at the left endpoint and the attached mass condition at the right endpoint. The formula contains finitely many arithmetic operations, elementary functions, quadratures, and transformations of the independent argument of the initial data such as the multiplication by a number and taking the integer part of a number. [ABSTRACT FROM AUTHOR]

The field of multiplicative analysis has recently garnered significant attention, particularly in the context of solving multiplicative differential equations (MDEs). The symmetry concept in MDEs facilitates the determination of invariant solutions and the reduction of these equations by leveraging their intrinsic symmetrical properties. This study is motivated by the need for efficient methods to address MDEs, which are critical in various applications. Our novel contribution involves leveraging the fundamental properties of orthogonal polynomials, specifically Laguerre polynomials, to derive new solutions for MDEs. We introduce the definitions of Laguerre multiplicative differential equations and multiplicative Laguerre polynomials. By applying the power series method, we construct these multiplicative Laguerre polynomials and rigorously prove their basic properties. The effectiveness of our proposed solution is validated through illustrative examples, demonstrating its practical applicability and potential for advancing the field of multiplicative analysis. [ABSTRACT FROM AUTHOR]

This article defines two common q-orthogonal polynomials: homogeneous q-Laguerre polynomials and homogeneous little q-Jacobi polynomials. They can be viewed separately as solutions to two q-partial differential equations. Furthermore, an analytic function satisfies a certain system of q partial differential equations if and only if it can be expanded in terms of homogeneous q-Laguerre polynomials or homogeneous little q-Jacobi polynomials. As applications, several generalized Ramanujan q-beta integrals and Andrews-Askey integrals are obtained. [ABSTRACT FROM AUTHOR]

AbstractThis article investigates the thermal, variable viscosity axial Couette flow between two con- centric circular cylinders, taking into account both the Navier and the dynamic slip boundary conditions. In order to put in evidence the effect of dynamic slip on the behavior of the solution, we have kept the flowing fluid to a simple Newtonian viscous fluid. The nonlinear governing equations for momentum and energy balance are solved under startup condition using the Laplace transform (LT) technique, in conjunction with the Homotopy Perturbation Method (HPM). The first two orders of approximation for temperature and velocity are obtained, as well as the entropy generation in the space occupied by the flow and the volumetric flux. Numerical results and graphical representations of the solution are provided and discussed to depict the effects of the slip parameters on the velocity and the temperature profiles, on the entropy generation rate and on the volumetric flux. [ABSTRACT FROM AUTHOR]

This study introduces a novel process identification method aimed at overcoming the challenge of accurately estimating process models when faced with deterministic disturbances, a common limitation in conventional identification methods. The proposed method tackles the difficult modeling problems due to deterministic disturbances by representing the disturbances as a linear combination of Laguerre polynomials and applies an integral transform with frequency weighting to estimate the process model in a numerically robust and stable manner. By utilizing a least squares approach for parameter estimation, it sidesteps the complexities inherent in iterative optimization processes, thereby ensuring heightened accuracy and robustness from a numerical analysis perspective. Comprehensive simulation results across various process types demonstrate the superior capability of the proposed method in accurately estimating the model parameters, even in the presence of significant deterministic disturbances. Moreover, it shows promising results in providing a reasonably accurate disturbance model despite structural disparities between the actual disturbance and the model. By improving the precision of process models under deterministic disturbances, the proposed method paves the way for developing refined and reliable control strategies, aligning with the evolving demands of modern industries and laying solid groundwork for future research aimed at broadening application across diverse industrial practices. [ABSTRACT FROM AUTHOR]

In this research article, we present an optimization algorithm aimed at finding the optimal solution for nonlinear 2-dimensional fractional optimal control problems that arise from nonlinear fractional dynamical systems governed by Caputo derivatives under Goursat–Darboux conditions. The system dynamics are described by equations such as the Klein–Gordon, convection–diffusion, and diffusion–wave equations. Our algorithm utilizes a novel class of basis functions called generalized Laguerre polynomials (GLPs), which are an extension of the traditional Laguerre polynomials. To begin, we introduce the GLPs and their properties, and we develop several new operational matrices specifically tailored for these basis functions. Next, we expand the state and control functions using the GLPs, with the coefficients and control parameters remaining unknown. This expansion allows us to transform the original problem into an algebraic system of equations. To facilitate this transformation, we employ operational matrices of Caputo derivatives, the rule of 2D Gauss–Legendre quadrature, and the method of Lagrange multipliers. [ABSTRACT FROM AUTHOR]

The known asymptotic relations interconnecting Jacobi, Laguerre, and Hermite classical orthogonal polynomials are generalized to the corresponding exceptional orthogonal polynomials of codimension m. It is proved that Xm-Laguerre exceptional orthogonal polynomials of type I, II, or III can be obtained as limits of Xm-Jacobi exceptional orthogonal polynomials of the same type. Similarly, Xm-Hermite exceptional orthogonal polynomials of type III can be derived from Xm-Jacobi or Xm-Laguerre ones. The quadratic transformations expressing Hermite classical orthogonal polynomials in terms of Laguerre ones is also extended to even X2m-Hermite exceptional orthogonal polynomials. [ABSTRACT FROM AUTHOR]

A Generalized Morse Potential (GMP) is an extension of the Morse Potential (MP) with an additional exponential term and an additional parameter that compensate for MP's erroneous behavior in the long range part of the interaction potential. Because of the additional term and parameter, the vibrational levels of the GMP cannot be solved analytically, unlike the case for the MP. We present several numerical approaches for solving the vibrational problem of the GMP based on Galerkin methods, namely, the Laguerre Polynomial Method (LPM), the Symmetrized LPM, and the Polynomial Expansion Method (PEM), and apply them to the vibrational levels of the homonuclear diatomic molecules B2, O2, and F2, for which high level theoretical near full configuration interaction (CI) electronic ground state potential energy surfaces and experimentally measured vibrational levels have been reported. Overall, the LPM produces vibrational states for the GMP that are converged to within spectroscopic accuracy of 0.01 cm−1 in between 1 and 2 orders of magnitude faster and with much fewer basis functions/grid points than the Colbert–Miller Discrete Variable Representation (CN-DVR) method for the three homonuclear diatomic molecules examined in this study. A Python library that fits and solves the GMP and similar potentials can be downloaded from https://gitlab.com/gds001uw/generalized-morse-solver. [ABSTRACT FROM AUTHOR]

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples.

Modeling several physical events leads to the Bagley–Torvik equation (BTE). In this study, we have taken into account the BTE, including the Caputo–Fabrizio and Atangana–Baleanu derivatives. It becomes challenging to find the analytical solution to these kinds of problems using standard methods in many circumstances. Therefore, to arrive at the required outcome, numerical techniques are used. The Laplace transform is a promising method that has been utilized in the literature to address a variety of issues that come up when modeling real-world data. For complicated functions, the Laplace transform approach can make the analytical inversion of the Laplace transform excessively laborious. As a result, numerical techniques are utilized to invert the Laplace transform. The numerical inverse Laplace transform is generally an ill-posed problem. Numerous numerical techniques for inverting the Laplace transform have been developed as a result of this challenge. In this article, we use the Weeks method, which is one of the most efficient numerical methods for inverting the Laplace transform. In our proposed methodology, first the BTE is transformed into an algebraic equation using Laplace transform. Then the reduced equation solved the Laplace domain. Finally, the Weeks method is used to convert the obtained solution from the Laplace domain into the real domain. Three test problems with Caputo–Fabrizio and Atangana–Baleanu derivatives are considered to demonstrate the accuracy, effectiveness, and feasibility of the proposed numerical method.

The main objective of this study is to present a computationally efficient numerical method for solving fractional-order differential equations with initial conditions. The proposed method is based on the newly developed generalized derivative operational matrix and generalized integral operational matrix derived from Laguerre polynomials, which belong to the class of orthogonal polynomials. Through the utilization of these operational matrices, the fractional-order problems can be transformed into a system of Sylvester-type matrix equations. This system is easily solvable using any computational software, thereby providing a practical framework for solving such equations. The results obtained are compared against various benchmarks, including an existing exact solution, Podlubny numerical techniques, analytical and numerical solvers, and reported solutions from stochastic techniques employing hybrid approaches. This comparative analysis serves to validate the accuracy of our proposed design scheme.

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 L2$$ {L}^2 $$‐norm, which shows the spectral rate of convergence is attained. Moreover, several numerical experiments are presented to validate the theoretical results. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

The normalization of the radial functions R n ℓ (r) for the solution of Schrödinger's equation for the Coulomb potential usually proceeds by appealing to the properties of Associated Laguerre polynomials. In this paper we show how to effect the normalization directly from the overall form of the solution and the recursion relation for its series part. Our approach should be applicable to similar problems, such as the harmonic oscillator, and can serve to offer students an alternate method of establishing fully-normalized wavefunctions without invoking the properties of special functions. [ABSTRACT FROM AUTHOR]

In this paper, two model order reduction methods based on Laguerre orthogonal polynomials for parabolic equation constrained optimal control problems are studied. The spatial discrete scheme of the cost function subject to a parabolic equation is obtained by Galerkin approximation, and then the coupled ordinary differential equations of the optimal original state and adjoint state with initial value and final value conditions are obtained by Pontryagin's minimum principle. For this original system, we propose two kinds of model order reduction methods based on the differential recurrence formula and integral recurrence formula of Laguerre orthogonal polynomials, respectively, and they are totally different from these existing researches on model order reduction only for initial value problems. Furthermore, we prove the coefficient-matching properties of the outputs between the reduced system and the original system. Finally, two numerical examples are given to verify the feasibility of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a pioneering numerical technique that combines generalized Laguerre polynomials with an operational matrix of fractional integration to address fractional models in electrical circuits. Specifically focusing on Resistor-Inductor (RL), Resistor-Capacitor (RC), Resonant (Inductor-Capacitor) (LC), and Resistor-Inductor-Capacitor (RLC) circuits within the framework of the Caputo derivative, our approach aims to enhance the accuracy of numerical solutions. We meticulously construct an operational matrix of fractional integration tailored to the generalized Laguerre basis vector, facilitating a transformation of the original fractional differential equations into a system of linear algebraic equations. By solving this system, we obtain a highly accurate approximate solution for the electrical circuit model under consideration. To validate the precision of our proposed method, we conduct a thorough comparative analysis, benchmarking our results against alternative numerical techniques reported in the literature and exact solutions where available. The numerical examples presented in our study substantiate the superior accuracy and reliability of our generalized Laguerre-enhanced operational matrix collocation method in effectively solving fractional electrical circuit models. [ABSTRACT FROM AUTHOR]

This paper discusses the linear fractional Fredholm-Volterra integro-differential equations (IDEs) considered in the Caputo sense. For this purpose, Laguerre polynomials have been used to construct an approximation method to obtain the solutions of the linear fractional Fredholm-Volterra IDEs. By this approximation method, the IDE has been transformed into a linear algebraic equation system using appropriate collocation points. In addition, a novel and exact matrix expression for the Caputo fractional derivatives of Laguerre polynomials and an associated explicit matrix formulation has been established for the first time in the literature. Furthermore, a comparison between the results of the proposed method and those of methods in the literature has been provided by implementing the method in numerous examples. [ABSTRACT FROM AUTHOR]

In this paper we investigate a family of bivariate orthogonal functions arising as a generalization of Koornwinder polynomials in two variables. General properties like recurrence relations and partial differential equations are introduced. Some special cases are considered and a limit relation of these functions is studied. As a consequence, a new class of bivariate orthogonal polynomials is presented. [ABSTRACT FROM AUTHOR]

We give an analog of exceptional polynomials in the matrix‐valued setting by considering suitable factorizations of a given second‐order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence are discussed in detail. We give an example of matrix‐valued exceptional Laguerre polynomials of arbitrary size. [ABSTRACT FROM AUTHOR]

In this article, we get certain integral representations of the multi-index Wright generalized Bessel function by making use of the extended beta function. This function is presented as a part of the generalized Bessel–Maitland function obtained by taking the extended fractional derivative of the generalized Bessel–Maitland function developed by Özarsalan and Özergin [M. Ali Özarslan and E. Özergin, Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model. 52 2010, 9–10, 1825–1833]. In addition, we demonstrate the exciting connections of the multi-index Wright generalized Bessel function with Laguerre polynomials and Whittaker function. Further, we use the generalized Wright hypergeometric function to calculate the Mellin transform and the inverse of the Mellin transform. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the Tricomi continuants, a family of tridiagonal determinants forming a Sheffer sequence closely related to the Tricomi polynomials and the Laguerre polynomials. Specifically, we obtain the main umbral operators associated with these continuants and establish some of their basic relations. Then, we obtain a Turan-like inequality, some congruences, some binomial identities (including a Carlitz-like identity), and some relations with the Cayley continuants. Furthermore, we show that the infinite Hankel matrix generated by the Tricomi continuants has an LDU-Sheffer factorization, while the infinite Hankel matrix generated by the shifted Tricomi continuants has an LTU-Sheffer factorization. Finally, by the first factorization, we obtain the linearization formula for the Tricomi continuants and its inverse. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this research is to present a new generalization of d-orthogonal $ (d\geq 2) $ (d ≥ 2) polynomials of Laguerre type by utilizing a suitable generating function from Sheffer class and employing operational rules associated with the lowering and raising operators that satisfy d-orthogonality. We derive several properties of these polynomials and establish the recurrence relation. Moreover, we provide explicit, connection and inversion formulas, the $ (d+1) $ (d + 1) -order differential equation, and the canonical d-dimensional functional vector. [ABSTRACT FROM AUTHOR]

In 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. [ABSTRACT FROM AUTHOR]

This study investigates an extension of the extended Hurwitz-Lerch zeta function. along with related integral images and derivatives. by extending the extended beta function. Also established is a link between the extended Hurwitz-Lerch zeta function and the Laguerre polynomials. It 5.P f is also demonstrated how to use the enlarged Hurwitz-Lerch zeta function Cv.Xcx; c. a.p.q) to probability distributions. Some (old and new) observations are offered here as specific illustrations of our theories. [ABSTRACT FROM AUTHOR]

Considering the approximation of a function from a Sobolev space by the partial sums of Fourier series in a system of Sobolev orthogonal polynomials generated by classical Laguerre polynomials, we obtain an estimate for the convergence rate of the partial sums to . [ABSTRACT FROM AUTHOR]

The aim of this work is to establish a connection between Bohr's radius and the analytic and normalized solutions of two differential second-order differential equations, namely y ″ (z) + a (z) y ′ (z) + b (z) y (z) = 0 and z 2 y ″ (z) + a (z) y ′ (z) + b (z) y (z) = d (z) . Using differential subordination, we find the upper bound of the Bohr and Rogosinski radii of the normalized solution F (z) of the above differential equations. We construct several examples by judicious choice of a (z) , b (z) and d (z) . The examples include several special functions like Airy functions, classical and generalized Bessel functions, error functions, confluent hypergeometric functions and associate Laguerre polynomials. [ABSTRACT FROM AUTHOR]

We consider the asymptotic limits where certain parameters in the definitions of the Laguerre and Jacobi ensembles diverge. In these limits, Dette, Imhof, and Nagel proved that, up to a linear transformation, the joint probability distributions of the ensembles become more and more concentrated around the zeros of the Laguerre and Jacobi polynomials, respectively. In this paper, we improve the concentration bounds. Our proofs are similar to those in the original references, but the error analysis is improved and arguably simpler. For the first and second moments of the Jacobi ensemble, we further improve the concentration bounds implied by our aforementioned results. [ABSTRACT FROM AUTHOR]

In this paper, we establish L p -boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions. [ABSTRACT FROM AUTHOR]

In this paper, new fast computing methods for partial differential equations with variable coefficients are studied and analyzed. They are two kinds of two‐sided Krylov enhanced proper orthogonal decomposition (KPOD) methods. First, the spatial discrete scheme of an advection‐diffusion equation is obtained by Galerkin approximation. Then, an algorithm based on a two‐sided KPOD approach involving the block Arnoldi and block Lanczos processes for the obtained time‐varying equations is put forward. Moreover, another type of two‐sided KPOD algorithm based on Laguerre orthogonal polynomials in frequency domain is provided. For the two kinds of two‐sided KPOD methods, we present a theoretical analysis for the moment matching of the discrete time‐invariant transfer function in time domain and give the error bound caused by the reduced‐order projection between the Galerkin finite element solution and the approximate solution of the two‐sided KPOD method. Finally, the feasibility of four two‐sided KPOD algorithms is verified by several numerical results with different inputs and setting parameters. [ABSTRACT FROM AUTHOR]

Modern technology in the field of images led to the issue of information transfer speed. As a result, there was need to find new and rapid technology in application to reach good results. In this paper, we propose new transformations derived from Laguerre polynomials. The image is analyzed using new wavelets to remove the noise from the color image and to avoid the problem of large data and the difficulty of transferring this data to keep it from being lost. The image is compressed to reduce the space occupied by this huge data with the help of the MATLAB program. The obtained results show the efficiency of our proposed method. [ABSTRACT FROM AUTHOR]

The main objective of this study is to present a computationally efficient numerical method for solving fractional-order differential equations with initial conditions. The proposed method is based on the newly developed generalized derivative operational matrix and generalized integral operational matrix derived from Laguerre polynomials, which belong to the class of orthogonal polynomials. Through the utilization of these operational matrices, the fractionalorder problems can be transformed into a system of Sylvester-type matrix equations. This system is easily solvable using any computational software, thereby providing a practical framework for solving such equations. The results obtained are compared against various benchmarks, including an existing exact solution, Podlubny numerical techniques, analytical and numerical solvers, and reported solutions from stochastic techniques employing hybrid approaches. This comparative analysis serves to validate the accuracy of our proposed design scheme. [ABSTRACT FROM AUTHOR]

The unsteady combustion of solid propellants under oscillating environments is the key to understanding the combustion instability inside solid rocket motors. The discontinuous Galerkin–finite element method (DG-FEM) is introduced to provide an efficient yet flexible numerical platform to investigate the combustion dynamics of solid propellants. The algorithm is developed for the classical unsteady model, the Zel'dovich–Novozhilov model. It is then validated based on a special analytical solution. The DG-FEM algorithm is then compared with the classical spectral method based on Laguerre polynomials. It is shown that the DG-FEM works more efficiently than the traditional spectral method, providing a more accurate solution with a lower computational cost. [ABSTRACT FROM AUTHOR]

Based on the extended fractional dimensional nonlinear Schrödinger equation and the variable separation method, a fractional accessible soliton solution with initial phase curvature is proposed for the first time. The soliton solution of the model is composed of hypergeometric functions and generalized Laguerre polynomials in fractional dimensional space, namely, Hypergeometric–Laguerre–Gaussian soliton. The theoretical results indicate that a series of different types of solitons are generated with the change of the beam parameters, forming a fractious family of solitons. At the same time, solitons produce a splitting phenomenon similar to that of the Hermitian beams. Additionally, the initial phase curvature also affects the stability of beam propagation, suppressing the formation of soliton. [ABSTRACT FROM AUTHOR]

The translation model is a useful tool for simulating non-Gaussian wind speed time series, it employs marginal transformation to fit the distribution function of wind speed, and deploys a Gaussian process with a specified autocorrelation function (ACF) ρ Z (τ) to match the ACF of wind speed. However, determining ρ Z (τ) requires solving an integral equation, which is not computationally convenient. In this paper, an easy-to-use method is presented for determining ρ Z (τ) . Based on Spearman's rank correlation coefficient, a rank ACF is proposed to characterize the time-dependent behavior of wind speed time series, whereby ρ Z (τ) can be specified by an analytical formula. In the case that only discrete values of ρ Z (τ) are available, a Laguerre polynomial based model is developed to fit ρ Z (τ) . Furthermore, if the distribution function of wind data is unknown, the kappa distribution is introduced to fit the quantile function of wind speed. Finally, a case study is conducted to check the proposed method, the results indicate that the rank correlation based translation model can yield a satisfactory representation for statistical features of wind speed time series. [ABSTRACT FROM AUTHOR]

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 -function. In this paper, we prove that these coefficients are non-vanishing for | x | > κ (n - 1) and κ ≈ 9.7225 . Numerical computations imply that 9.72245 is a lower bound for κ . 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. [ABSTRACT FROM AUTHOR]