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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]

A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space C ([ 0 , T ] × [ 0 , T ]). The existence of at least one solution to the QIE is discussed in this article. Our evidence depends on a suitable combination of the measures of the noncompactness approach and the fixed-point principle of Darbo. The quadratic integral equation can be used to derive a system of integral equations of the second kind using the quadrature method. With the aid of two different polynomials, Laguerre and Hermite, the system of integral equations is solved using the collocation method. In each numerical approach, the estimation of the error is discussed. Finally, using some examples, the accuracy and scalability of the proposed method are demonstrated along with comparisons. Mathematica 11 was used to obtain all of the results from the techniques that were shown. [ABSTRACT FROM AUTHOR]

The primary objective of this research is to introduce and investigate novel polynomial variants termed Δ h Laguerre polynomials. This unique polynomial type integrates the monomiality principle alongside operational rules. Through this innovative approach, the study delves into uncharted territory, unveiling fresh insights that build upon prior research endeavours. Notably, the Δ h Laguerre polynomials exhibit significant utility in the realm of quantum mechanics, particularly in the modelling of entropy within quantum systems. The research meticulously unveils explicit formulas and elucidates the fundamental properties of these polynomials, thereby forging connections with established polynomial categories. By shedding light on the distinct characteristics and functionalities of the Δ h Laguerre polynomials, this study contributes significantly to their comprehension and application across diverse mathematical and scientific domains. [ABSTRACT FROM AUTHOR]

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]

On the half line, we introduce a new sequence of near-best uniform approximation polynomials, easily computable by the values of the approximated function at a truncated number of Laguerre zeros. Such approximation polynomials come from a discretization of filtered Fourier–Laguerre partial sums, which are filtered using a de la Vallée Poussin (VP) filter. They have the peculiarity of depending on two parameters: a truncation parameter that determines how many of the n Laguerre zeros are considered, and a localization parameter, which determines the range of action of the VP filter we will apply. As n → ∞ , under simple assumptions on such parameters and the Laguerre exponents of the involved weights, we prove that the new VP filtered approximation polynomials have uniformly bounded Lebesgue constants and uniformly convergence at a near–best approximation rate, for any locally continuous function on the semiaxis. The numerical experiments have validated the theoretical results. In particular, they show a better performance of the proposed VP filtered approximation versus the truncated Lagrange interpolation at the same nodes, especially for functions a.e. very smooth with isolated singularities. In such cases, we see a more localized approximation and a satisfactory reduction of the Gibbs phenomenon. [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]

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]

I. I. Sharapudinov, when studying the approximation properties of partial sums of a special series in Laguerre polynomials, introduced a weighted best approximation characteristic that depends on a parameter . In the present paper, we prove a Jackson-type theorem for this characteristic for . [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]

This study focuses on calculating solutions for fractional kinetic equations involving Laguerre polynomials and their fractional derivatives. By leveraging the Sumudu transform technique, we derive these solutions in the form of the Mittag‐Leffler function. Our investigation includes graphical representations generated using MATLAB to illustrate the behavior of these solutions under varying parametric conditions. It is essential to note that the results obtained in this study are exceptionally versatile and have the potential to yield both established and potentially novel findings in this field of research. [ABSTRACT FROM AUTHOR]

n recent years, some applications of Lah numbers were discovered in the real world problem of telecommunications and optics. The aim of this paper is to study the λ-analogues of Lah numbers and Lah-Bell polynomials which are λ-analogues of the Lah numbers and and LahBell polynomials. Here we note that λ-analogues appear when we replace the falling factorials by the generalized falling factorials in the defining equations. By using generating function method, we study some properties, explicit expressions, generating functions and Dobinski-like formulas for those numbers and polynomials. We also treat the more general λ-analogues of r-Lah numbers and r-extended λ-Lah-Bell polynomials. In addition, we show that the expectations of two random variables, both associated with the Poisson random variable with parameter α/λ, are equal to the λ-analogue of the Lah-Bell polynomial evaluated at α for one and the r-extended λ-Lah-Bell polynomial evaluated at α for the other. [ABSTRACT FROM AUTHOR]

We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al. [Appl. Math. Comput., 333 (2018), pp. 490--505] by introducing a unifying abstract framework, and we derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, we consider a reformulation in the space of absolutely continuous functions via integration. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations when the collocation nodes are chosen as the family of scaled zeros or extrema of Laguerre polynomials. This ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. The result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations. The new approach used to prove convergence also provides the exact location of the spectrum of the differentiation matrices for the Laguerre zeros and extrema, adding new insights into properties that are important in the numerical solution of differential equations by pseudospectral methods. [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]

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]

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.

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]

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.

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]

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 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]

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 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]

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.

A quadratic integral Equation (QIE) of the second kind with continuous kernels is solved in the space C([0,T]×[0,T]). The existence of at least one solution to the QIE is discussed in this article. Our evidence depends on a suitable combination of the measures of the noncompactness approach and the fixed-point principle of Darbo. The quadratic integral equation can be used to derive a system of integral equations of the second kind using the quadrature method. With the aid of two different polynomials, Laguerre and Hermite, the system of integral equations is solved using the collocation method. In each numerical approach, the estimation of the error is discussed. Finally, using some examples, the accuracy and scalability of the proposed method are demonstrated along with comparisons. Mathematica 11 was used to obtain all of the results from the techniques that were shown.

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]