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Journal articles on the topic 'Laguerre polynomial'

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Published: 4 June 2021

Last updated: 25 July 2025

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1

Jagtap, R. R., P. G. Andhare та S. B. Gaikwad. "A New General Integral Transform of Modified Laguerre’s Polynomial 𝐿𝐚,𝐛,𝐜,𝐧(𝑡)". Indian Journal Of Science And Technology 17, № 39 (2024): 4066–72. http://dx.doi.org/10.17485/ijst/v17i39.1251.

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Objectives: The core objective of this paper is to introduce the explicit Modified Laguerre polynomial 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡) and obtain a new general integral transform of Modified Laguerre’s Polynomial, by using this general transform, some transforms of Laguerre’s Polynomial are identified as special cases. Method: We derive transforms of the polynomials 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡), 𝐿𝑛 (𝑡) by using this new integral transform approach and found the transforms like Laplace, 𝛼-Laplace, Sawi, Elzaki, Sumudu, Natural, Aboodh, Pourreza, Mohand, G_transform, Kamal transforms and their relationship for the same. Findings:

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2

R, R. Jagtap, G. Andhare P та B. Gaikwad S. "A New General Integral Transform of Modified Laguerre's Polynomial 𝐿𝐚,𝐛,𝐜,𝐧(𝑡)". Indian Journal of Science and Technology 17, № 39 (2024): 4066–72. https://doi.org/10.17485/IJST/v17i39.1251.

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Abstract <strong>Objectives:</strong> The core objective of this paper is to introduce the explicit Modified Laguerre polynomial 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡) and obtain a new general integral transform of Modified Laguerre’s Polynomial, by using this general transform, some transforms of Laguerre’s Polynomial are identified as special cases. <strong>Method:</strong> We derive transforms of the polynomials 𝐿𝑎,𝑏,𝑐,𝑛 (𝑡), 𝐿𝑛 (𝑡) by using this new integral transform approach and found the transforms like Laplace, 𝛼-Laplace, Sawi, Elzaki, Sumudu, Natural, Aboodh, Pourreza, Mohand, G_tran

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3

Alam, Noor, Shahid Ahmad Wani, Waseem Ahmad Khan, Fakhredine Gassem та Anas Altaleb. "Exploring Properties and Applications of Laguerre Special Polynomials Involving the Δh Form". Symmetry 16, № 9 (2024): 1154. http://dx.doi.org/10.3390/sym16091154.

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The primary objective of this research is to introduce and investigate novel polynomial variants termed Δh 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 Δh 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

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4

DUNKL, CHARLES F. "A LAGUERRE POLYNOMIAL ORTHOGONALITY AND THE HYDROGEN ATOM." Analysis and Applications 01, no. 02 (2003): 177–88. http://dx.doi.org/10.1142/s0219530503000132.

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The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis, an analogous strange orthogonality is obtained for Meixner polynomials.

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5

Khan, Adnan, Muhammad Kalim, Ali Akgül та Fahd Jarad. "The Extended Laguerre Polynomials A q , n α x Involving F q q , q > 2". Journal of Function Spaces 2022 (15 квітня 2022): 1–14. http://dx.doi.org/10.1155/2022/7326760.

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In this paper, for the proposed extended Laguerre polynomials A α q , n x , the generalized hypergeometric function of the type F q q , q > 2 and extension of the Laguerre polynomial are introduced. Similar to those related to the Laguerre polynomials, the generating function, recurrence relations, and Rodrigue’s formula are determined. Some corollaries are also discussed at the end.

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6

Blel, Mongi. "On some m-symmetric generalized hypergeometric d-orthogonal polynomials." Filomat 38, no. 4 (2024): 1279–89. http://dx.doi.org/10.2298/fil2404279b.

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In [9] I. Lamiri and M. Ouni state some characterization theorems for d-orthogonal polynomials of Hermite, Gould-Hopper and Charlier type polynomials. In [3] Y. Ben Cheikh I. Lamiri and M.Ouni give a characterization theorem for some classes of generalized hypergeometric polynomials containing for example, Gegenbauer polynomials, Gould-Hopper polynomials, Humbert polynomials, a generalization of Laguerre polynomials and a generalization of Charlier polynomials. In this work, we introduce a new class D of generalized hypergeometric m-symmetric polynomial sequence containing the Hermite polynomi

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7

Hajmirzaahmad, Mojdeh. "Laguerre polynomial expansions." Journal of Computational and Applied Mathematics 59, no. 1 (1995): 25–37. http://dx.doi.org/10.1016/0377-0427(94)00020-2.

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8

Asad, A. "Two variables generalized Laguerre polynomials." Carpathian Mathematical Publications 13, no. 1 (2021): 134–41. http://dx.doi.org/10.15330/cmp.13.1.134-141.

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The objective of this paper is to introduce and study the generalized Laguerre polynomial for two variables. We prove that these polynomials are characterized by the generalized hypergeometric function. An explicit representation, generating functions and some recurrence relations are shown. Moreover, these polynomials appear as solutions of some differential equations.

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9

Khan, Adnan, M. Haris Mateen, Ali Akgül та Md Shajib Ali. "The K Extended Laguerre Polynomials Involving A α r , n , k x F r r , r > 2". Advances in Mathematical Physics 2022 (5 вересня 2022): 1–10. http://dx.doi.org/10.1155/2022/6815685.

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In this manuscript, we present the generalized hypergeometric function of the type F r r , r > 2 and extension of the K Laguerre polynomial for the K extended Laguerre polynomials A r , n , k α x . Additionally, we describe the K generating function, K recurrence relations, and K S Rodrigues formula.

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10

Baleanu, D., A. H. Bhrawy, and T. M. Taha. "A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/413529.

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This paper deals with modified generalized Laguerre spectral tau and collocation methods for solving linear and nonlinear multiterm fractional differential equations (FDEs) on the half line. A new formula expressing the Caputo fractional derivatives of modified generalized Laguerre polynomials of any degree and for any fractional order in terms of the modified generalized Laguerre polynomials themselves is derived. An efficient direct solver technique is proposed for solving the linear multiterm FDEs with constant coefficients on the half line using a modified generalized Laguerre tau method.

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11

Lévai, Géza. "Potentials from the Polynomial Solutions of the Confluent Heun Equation." Symmetry 15, no. 2 (2023): 461. http://dx.doi.org/10.3390/sym15020461.

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Polynomial solutions of the confluent Heun differential equation (CHE) are derived by identifying conditions under which the infinite power series expansions around the z=0 singular point can be terminated. Assuming a specific structure of the expansion coefficients, these conditions lead to four non-trivial polynomials that can be expressed as special cases of the confluent Heun function Hc(p,β,γ,δ,σ;z). One of these recovers the generalized Laguerre polynomials LN(α), and another one the rationally extended X1 type Laguerre polynomials L^N(α). The two remaining solutions represent previously

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12

Batra, Saniya, and Prakriti Rai. "A Class of Laguerre-Based Generalized Humbert Polynomials." International Journal of Differential Equations 2021 (July 29, 2021): 1–8. http://dx.doi.org/10.1155/2021/4324466.

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Several mathematicians have extensively investigated polynomials, their extensions, and their applications in various other research areas for a decade. Our paper aims to introduce another such polynomial, namely, Laguerre-based generalized Humbert polynomial, and investigate its properties. In particular, it derives elementary identities, recursive differential relations, additional symmetry identities, and implicit summation formulas.

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13

QUESNE, C. "RATIONALLY-EXTENDED RADIAL OSCILLATORS AND LAGUERRE EXCEPTIONAL ORTHOGONAL POLYNOMIALS IN kTH-ORDER SUSYQM." International Journal of Modern Physics A 26, no. 32 (2011): 5337–47. http://dx.doi.org/10.1142/s0217751x11054942.

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A previous study of exactly solvable rationally-extended radial oscillator potentials and corresponding Laguerre exceptional orthogonal polynomials carried out in second-order supersymmetric quantum mechanics is extended to kth-order one. The polynomial appearing in the potential denominator and its degree are determined. The first-order differential relations allowing one to obtain the associated exceptional orthogonal polynomials from those arising in a (k-1)th-order analysis are established. Some nontrivial identities connecting products of Laguerre polynomials are derived from shape invari

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14

Nahid, Tabinda, and Junesang Choi. "Certain Hybrid Matrix Polynomials Related to the Laguerre-Sheffer Family." Fractal and Fractional 6, no. 4 (2022): 211. http://dx.doi.org/10.3390/fractalfract6040211.

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The main goal of this article is to explore a new type of polynomials, specifically the Gould-Hopper-Laguerre-Sheffer matrix polynomials, through operational techniques. The generating function and operational representations for this new family of polynomials will be established. In addition, these specific matrix polynomials are interpreted in terms of quasi-monomiality. The extended versions of the Gould-Hopper-Laguerre-Sheffer matrix polynomials are introduced, and their characteristics are explored using the integral transform. Further, examples of how these results apply to specific memb

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15

Rajkovic, Predrag, Sladjana Marinkovic, and Miomir Stankovic. "Orthogonal polynomials with varying weight of Laguerre type." Filomat 29, no. 5 (2015): 1053–62. http://dx.doi.org/10.2298/fil1505053r.

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In this paper, we define and examine a new functional product in the space of real polynomials. This product includes the weight function which depends on degrees of the participants. In spite of it does not have all properties of an inner product, we construct the sequence of orthogonal polynomials. These polynomials can be eigenfunctions of a differential equation what was used in some considerations in the theoretical physics. In special, we consider Laguerre type weight function and prove that the corresponding orthogonal polynomial sequence is connected with Laguerre polynomials. We study

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16

KRASIKOV, ILIA. "DISCRETE ANALOGUES OF THE LAGUERRE INEQUALITY." Analysis and Applications 01, no. 02 (2003): 189–97. http://dx.doi.org/10.1142/s0219530503000120.

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It is shown that [Formula: see text], m = 0,1,…, where f(x) is either a real polynomial with only real zeros or an allied entire function of a special type, provided that the distance between two consecutive zeros of f(x) is at least [Formula: see text]. These inequalities are surprisingly similar discrete analogues of higher degree generalizations of the Laguerre and Turán inequalities. Applied to the classical discrete orthogonal polynomials, they yield sharp, explicit bounds, uniform in all parameters involved, on the polynomials and their extreme zeros. This is illustrated for the case of

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17

SRIVASTAVA, HARI MOHAN. "Some generating functions of the Bessel and related orthogonal polynomials." Applicable Nonlinear Analysis 1, no. 1 (2024): 1–19. http://dx.doi.org/10.69829/apna-024-0101-ta01.

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Our main object in this article is to present several families of generating functions for the simple Bessel polynomials and the generalized Bessel polynomials . We also investigate and present various potentially useful generating-function relationships involving such other orthogonal polynomial systems as (for example) the Jacobi, Laguerre and Hermite polynomials, together with their specialized and parametrically-varied forms.

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18

Rebocho, Maria Das Neves. "Laguerre–Hahn orthogonal polynomials on the real line." Random Matrices: Theory and Applications 09, no. 01 (2019): 2040001. http://dx.doi.org/10.1142/s2010326320400018.

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A survey is given on sequences of orthogonal polynomials related to Stieltjes functions satisfying a Riccati type differential equation with polynomial coefficients — the so-called Laguerre–Hahn class. The main goal is to describe analytical aspects, focusing on differential equations for those orthogonal polynomials, difference and differential equations for the recurrence coefficients, and distributional equations for the corresponding linear functionals.

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19

Yu, A. Long, and Jin Qiao Dai. "A Method to Compensate Thermocouple Sensor Non-Linearity Based on Lagurre Orthogonal Polynomial Basis Functions Neural Network." Advanced Materials Research 834-836 (October 2013): 896–99. http://dx.doi.org/10.4028/www.scientific.net/amr.834-836.896.

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This paper presents a new method to compensate thermocouple sensor non-linearity based on laguerre orthogonal polynomial basis functions neural network. The mathematical model is established based on laguerre artificial neural network for non-linearity compensation. The principle and training algorithms of laguerre orthogonal polynomial basis functions neural network are introduced. The results show that non-linearity compensation method has the advantages of high precision, good real time and strong robustness, etc. The maximum non-linearity error is 0.17%.

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20

Yüzbaşı, Şuayip. "A Laguerre Approach for the Solutions of Singular Perturbated Differential Equations." International Journal of Computational Methods 14, no. 04 (2017): 1750034. http://dx.doi.org/10.1142/s0219876217500347.

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In this paper, a Laguerre method is presented to solve singularly perturbated two-point boundary value problems. By means of the matrix relations of the Laguerre polynomials and their derivatives, original problem is transformed into a matrix equation. Later, we use collocation points in the matrix equation and thus the considered problem is reduced to a system of linear algebraic equations. The solution of this system gives the coefficients of the desired approximate solution. Also, an error estimation based on the residual function is introduced for the method. The Laguerre polynomial soluti

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21

Dogruer, CU. "Constrained model predictive control of a vehicle suspension using Laguerre polynomials." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 234, no. 6 (2019): 1253–68. http://dx.doi.org/10.1177/0954406219889078.

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In this paper, a fast constrained model predictive control algorithm was designed for the active suspension of a half-car model to increase the controller bandwidth so that high frequency displacement disturbance coming from the road can be rejected. To this end, a quasi-LTI model of a semi-active suspension model was controlled by a model predictive controller with orthogonal Laguerre polynomials. With the use of Laguerre polynomials, it has been shown that the optimization parameter set could be made minimal, and thereby it has been shown that on-line optimization takes less time. With numer

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22

Zhang, Bo, Honghang Tu, Liangjuan Li, Jiangong Yu, and Jun Dai. "Rayleigh Waves Propagating in the Functionally Graded One-Dimensional Hexagonal Quasicrystal Half-Space." Crystals 13, no. 8 (2023): 1205. http://dx.doi.org/10.3390/cryst13081205.

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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 t

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23

Lin, Chih-Hong. "Hybrid Recurrent Laguerre-Orthogonal-Polynomial NN Control System Applied in V-Belt Continuously Variable Transmission System Using Particle Swarm Optimization." Mathematical Problems in Engineering 2015 (2015): 1–17. http://dx.doi.org/10.1155/2015/106707.

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Because the V-belt continuously variable transmission (CVT) system driven by permanent magnet synchronous motor (PMSM) has much unknown nonlinear and time-varying characteristics, the better control performance design for the linear control design is a time consuming procedure. In order to overcome difficulties for design of the linear controllers, the hybrid recurrent Laguerre-orthogonal-polynomial neural network (NN) control system which has online learning ability to respond to the system’s nonlinear and time-varying behaviors is proposed to control PMSM servo-driven V-belt CVT system under

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Lin, Chih-Hong. "Application of a V-belt continuously variable transmission system by using a composite recurrent Laguerre orthogonal polynomial neural network control system and modified particle swarm optimization." Journal of Vibration and Control 23, no. 9 (2015): 1437–62. http://dx.doi.org/10.1177/1077546315593839.

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The nonlinear and time-varying characteristics of the V-belt continuously variable transmission system driven by a permanent magnet synchronous motor (PMSM) are unknown, therefore, improving the control performance of the linear control design is time-consuming. To overcome difficulties in the design of a linear controller for the PMSM servo-driven V-belt continuously variable transmission system with the lumped nonlinear load disturbances, a composite recurrent Laguerre orthogonal polynomial neural network (NN) control system which has online learning capability to respond the nonlinear time-

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Banerjee, Pradipto, and Ranjan Bera. "An irreducibility question concerning modifications of Laguerre polynomials." International Journal of Number Theory 16, no. 05 (2019): 1031–51. http://dx.doi.org/10.1142/s1793042120500530.

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This paper addresses a question recently posed by Hajir concerning the irreducibility of certain modifications [Formula: see text] of generalized Laguerre polynomials [Formula: see text] where [Formula: see text] is an integer. For a fixed [Formula: see text], we obtain lower bounds [Formula: see text] on [Formula: see text] in terms of [Formula: see text] such that [Formula: see text] is irreducible over the rationals for all [Formula: see text]. Furthermore, for [Formula: see text], it is shown that [Formula: see text] is either irreducible or is a product of a linear polynomial and a polyno

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26

Yan, Xu. "Biorthogonal polynomial systems associate with Hermitepolynomials and generalized Laguerre polynomials." SCIENTIA SINICA Mathematica 50, no. 4 (2020): 513. http://dx.doi.org/10.1360/ssm-2020-0074.

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27

Molla, Hasib Uddin, and Goutam Saha. "Numerical Approximation of Fredholm Integral Equation (FIE) of 2nd Kind using Galerkin and Collocation Methods." GANIT: Journal of Bangladesh Mathematical Society 38 (January 14, 2019): 11–25. http://dx.doi.org/10.3329/ganit.v38i0.39782.

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In this research work, Galerkin and collocation methods have been introduced for approximating the solution of FIE of 2nd kind using LH (product of Laguerre and Hermite) polynomials which are considered as basis functions. Also, a comparison has been done between the solutions of Galerkin and collocation method with the exact solution. Both of these methods show the outcome in terms of the approximate polynomial which is a linear combination of basis functions. Results reveal that performance of collocation method is better than Galerkin method. Moreover, five different polynomials such as Leg

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28

Özat, Zeynep, Bayram Çekim, Mehmet Ali Özarslan, and Francesco Aldo Costabile. "Truncated-Exponential-Based General-Appell Polynomials." Mathematics 13, no. 8 (2025): 1266. https://doi.org/10.3390/math13081266.

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In this paper, a new and general form of truncated-exponential-based general-Appell polynomials is introduced using the two-variable general-Appell polynomials. For this new polynomial family, we present an explicit representation, recurrence relation, shift operators, differential equation, determinant representation, and some other properties. Finally, two special cases of this family, truncated-exponential-based Hermite-type and truncated-exponential-based Laguerre–Frobenius Euler polynomials, are introduced and their corresponding properties are obtained.

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Макаров, Володимир. "Операторна функція Рімана. Перетворення Келі". Ukrains’kyi Matematychnyi Zhurnal 77, № 2 (2025): 139–43. https://doi.org/10.3842/umzh.v77i2.8859.

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UDC 510 We study a new class of special functions called Laguerre–Bessel polynomials (PLBs), which play the same role as Laguerre polynomials in the case of application of the Cayley transformation to the evaluation of the operator exponent. The generating function for PLBs is a Bessel operator function of the first kind of zero order obtained by replacing its operator argument by its Cayley transformation. It is shown that the PLBs coincide with a new class of 2-orthogonal polynomials with an accuracy of up to a linear transformation. It is shown that the PLBs are classical in the Hahn–Maroni

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30

Exton, H. "Summation formulae involving the Laguerre polynomial." Journal of Computational and Applied Mathematics 100, no. 2 (1998): 225–27. http://dx.doi.org/10.1016/s0377-0427(98)00132-0.

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31

CHENG, YAN-FU, and TONG-QING DAI. "EXACT SOLUTIONS OF THE SCHRÖDINGER EQUATION FOR A NEW RING-SHAPED NONHARMONIC OSCILLATOR POTENTIAL." International Journal of Modern Physics A 23, no. 12 (2008): 1919–27. http://dx.doi.org/10.1142/s0217751x08039621.

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The bound state solutions of the Schrödinger equation with a new ring-shaped nonharmonic potential are presented using exactly the Nikiforov–Uvarov method. It is found that the solutions of the angular wave function can be expressed by Jacobi polynomial and radial wave functions are given by the generalized Laguerre polynomials. We also discuss the special case for α = 0 and β = 0 respectively.

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Rani, Kumari. "The Polynomial Set Rnk{(xn), y} in terms of the Sister Celin Polynomials Cj (x1r)." APPLIED SCIENCE PERIODICAL XXIV, no. 2, May 2022 (2022): 28–36. https://doi.org/10.5281/zenodo.15113873.

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              <em>The object of this paper is systematic study of multivariable generalized hypergeometric polynomial set Rnk{(xn), y}, obtained with the help of generating function involving Lauricella function in the notation of Burchanall and Chaundy [3], where n denotes the order of the polynomial set. The polynomial happens to be generalization of as many as thirty eight orthogonal and non-orthogonal polynomial set such as Legendre, Laguerre, Bateman, Jacobi, Tchebichef, Meixner, Lahiri, Charlier, Jackson, Hermite, Sylvester, etc.</em> <em>   

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QUESNE, C. "HIGHER-ORDER SUSY, EXACTLY SOLVABLE POTENTIALS, AND EXCEPTIONAL ORTHOGONAL POLYNOMIALS." Modern Physics Letters A 26, no. 25 (2011): 1843–52. http://dx.doi.org/10.1142/s0217732311036383.

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Exactly solvable rationally-extended radial oscillator potentials, whose wave functions can be expressed in terms of Laguerre-type exceptional orthogonal polynomials, are constructed in the framework of kth-order supersymmetric quantum mechanics, with special emphasis on k = 2. It is shown that for μ = 1, 2, and 3, there exist exactly μ distinct potentials of μth type and associated families of exceptional orthogonal polynomials, where μ denotes the degree of the polynomial gμ arising in the denominator of the potentials.

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Hetyei, Gábor. "Meixner polynomials of the second kind and quantum algebras representing su(1, 1)." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2117 (2009): 1409–28. http://dx.doi.org/10.1098/rspa.2009.0497.

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We show how Viennot’s combinatorial theory of orthogonal polynomials may be used to generalize some recent results of Sukumar and Hodges (Hodges & Sukumar 2007 Proc. R. Soc. A 463 , 2401–2414 ( doi:10.1098/rspa.2007.0001 ); Sukumar & Hodges 2007 Proc. R. Soc. A 463 , 2415–2427 ( doi:10.1098/rspa.2007.0003 )) on the matrix entries in powers of certain operators in a representation of su(1, 1). Our results link these calculations to finding the moments and inverse polynomial coefficients of certain Laguerre polynomials and Meixner polynomials of the second kind. As an immediate consequen

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35

Singh, B. "On the Non-Null Distribution of Anova F-Ratio in One Way Unbalanced Random Model." Calcutta Statistical Association Bulletin 36, no. 1-2 (1987): 57–62. http://dx.doi.org/10.1177/0008068319870106.

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Expressions for the non-null distribution of analysis of variance F-ratio and the power of F-test in the unbalanced one-way random model have been obtained. The exact values of power of the test have been worked out, numerically, and compared with those obtained by using ( i) the Laguerre polynomial and (ii) the two moments approximations. The computed results reveal that the two moments approximation, for the non-null distribution of F-ratio, is quite good and there is no need for the Laguerre polynomial approimation, which is computationally very tedious for practical applications.

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Fan, Hong-Yi, and Rui He. "Photon-added chaotic field and its damping in a thermo enviroment." Canadian Journal of Physics 93, no. 4 (2015): 456–59. http://dx.doi.org/10.1139/cjp-2014-0405.

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We propose a new photon field in quantum field theory, named Laguerre-polynomial-weighted chaotic field. This field will emerge when an initial photon-added chaotic field, which is represented by its density operator [Formula: see text], dissipates in an amplitude damping channel described by time evolution equation [Formula: see text], where κ is a damping coefficient, that is, the initial field will evolve into [Formula: see text] with [Formula: see text], where Ls is the s-order Laguerre-polynomial, and : : denotes normal ordering. We employ the method of summation within an ordered product

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Petković, Miodrag S. "Laguerre-like inclusion method for polynomial zeros." Journal of Computational and Applied Mathematics 152, no. 1-2 (2003): 451–65. http://dx.doi.org/10.1016/s0377-0427(02)00723-9.

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He, Rui, Hong-Yi Fan, Jun Song, and Jun Zhou. "Laguerre-Polynomial-Weighted Two-Mode Squeezed State." International Journal of Theoretical Physics 55, no. 7 (2016): 3329–37. http://dx.doi.org/10.1007/s10773-016-2962-6.

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Dang, Hong Gang, Xiao Ya Yang, and Wan Sheng He. "Analysis of a Nonlinear System with a Random Parameter." Applied Mechanics and Materials 721 (December 2014): 366–69. http://dx.doi.org/10.4028/www.scientific.net/amm.721.366.

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In this paper, a nonlinear system with random parameter, which is called stochastic fractional-order complex Lorenz system, is investigated. The Laguerre polynomial approximation method is used to study the system. Then, the stochastic fractional-order system is reduced into the equivalent deterministic one with Laguerre approximation. The ensemble mean and sample responses of the stochastic system can be obtained.

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Campos, R. G., and L. A. Avila. "Some properties of orthogonal polynomials satisfying fourth order differential equations." Glasgow Mathematical Journal 37, no. 1 (1995): 105–13. http://dx.doi.org/10.1017/s0017089500030445.

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In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2–8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type,

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Dai, Liu, Zhi-Hua Xiao, Ren-Zheng Zhang, and Yao-Lin Jiang. "Laguerre-Gramian-based structure-preserving model order reduction for second-order form systems." Transactions of the Institute of Measurement and Control 44, no. 5 (2021): 1113–24. http://dx.doi.org/10.1177/01423312211050732.

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A new structure-preserving model order reduction technique based on Laguerre-Gramian for second-order form systems is presented in this article. The main task of the proposed approach is to use the Laguerre polynomial expansion of the matrix exponential function to obtain the approximate low-rank decomposition of the Gramians for the equivalent first-order representation of the original second-order form system. The approximate balanced system is generated by a balancing transformation which is directly computed from the expansion coefficients of impulse responses in the space spanned by Lague

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42

Alhaidari, A. D. "Evaluation of integrals involving orthogonal polynomials: Laguerre polynomial and Bessel function example." Applied Mathematics Letters 20, no. 1 (2007): 38–42. http://dx.doi.org/10.1016/j.aml.2006.02.020.

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43

Le Blanc, Richard. "Jaynes-Gibbs Entropic Convex Duals and Orthogonal Polynomials." Entropy 24, no. 5 (2022): 709. http://dx.doi.org/10.3390/e24050709.

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The univariate noncentral distributions can be derived by multiplying their central distributions with translation factors. When constructed in terms of translated uniform distributions on unit radius hyperspheres, these translation factors become generating functions for classical families of orthogonal polynomials. The ultraspherical noncentral t, normal N, F, and χ2 distributions are thus found to be associated with the Gegenbauer, Hermite, Jacobi, and Laguerre polynomial families, respectively, with the corresponding central distributions standing for the polynomial family-defining weights

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Rebocho, Maria das Neves, and Nicholas Witte. "Integrable differential systems for deformed Laguerre–Hahn orthogonal polynomials." Nonlinearity 36, no. 7 (2023): 3542–71. http://dx.doi.org/10.1088/1361-6544/acd4a3.

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Abstract Our work studies sequences of orthogonal polynomials { P n ( x ) } n ⩾ 0 of the Laguerre–Hahn class, whose Stieltjes functions satisfy a Riccati type differential equation with polynomial coefficients, which are subject to a deformation parameter t. We derive systems of differential equations and give Lax pairs, yielding nonlinear differential equations in t for the recurrence relation coefficients and Lax matrices of the orthogonal polynomials. A specialisation to a non semi-classical case obtained via a Möbius transformation of a Stieltjes function related to a deformed Jacobi weigh

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Da, Cheng, Qian-Fan Chen, Peng-Fei Zhang, and Hong-Yi Fan. "Time evolution law of the Laguerre-polynomial-weighted chaotic photon field in an amplitude-damping channel." Canadian Journal of Physics 93, no. 3 (2015): 283–89. http://dx.doi.org/10.1139/cjp-2014-0278.

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We examine how a Laguerre-polynomial-weighted chaotic photon field (LPWCPF), whose density operator is [Formula: see text], evolves in an amplitude-damping channel. By using a newly derived generating function of two-variable Hermite polynomials we obtain the evolution law of LPWCPF, which turns out to be a new LPWCPF with a new parameter, depending on 1 − e−2κt, where κ represents decay rate. The technique of integration (summation) within an ordered product of operators is used in our discussions.

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Sghaier, Mabrouk, and Lamaa Khaled. "Orthogonal polynomials associated with an inverse spectral transform. The cubic case." Filomat 31, no. 8 (2017): 2477–97. http://dx.doi.org/10.2298/fil1708477s.

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The purpose of this work is to give some new algebraic properties of the orthogonality of a monic polynomial sequence {Qn}n ? o defined by Qn(X) = Pn(X) + SnPn-1(X) + tnPn-2(X) + rnPn-3(X), n ? 1, where rn ? 0, n ? 3, and {Pn}n?0 is a given sequence of monic orthogonal polynomials. Essentially, we consider some cases in which the parameters rn, sn, and tn can be computed more easily. Also, as a consequence, a matrix interpretation using LU and UL factorization is done. Some applications for Laguerre, Bessel and Tchebychev orthogonal polynomials of second kind are obtained.

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Natanson, Gregory. "Quantization of rationally deformed Morse potentials by Wronskian transforms of Romanovski-Bessel polynomials." Acta Polytechnica 62, no. 1 (2022): 100–117. http://dx.doi.org/10.14311/ap.2022.62.0100.

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The paper advances Odake and Sasaki’s idea to re-write eigenfunctions of rationally deformed Morse potentials in terms of Wronskians of Laguerre polynomials in the reciprocal argument. It is shown that the constructed quasi-rational seed solutions of the Schrödinger equation with the Morse potential are formed by generalized Bessel polynomials with degree-independent indexes. As a new achievement we can point to the construction of the Darboux-Crum net of isospectral rational potentials using Wronskians of generalized Bessel polynomials with no positive zeros. One can extend this isospectral f

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Hajir, Farshid. "Algebraic Properties of a Family of Generalized Laguerre Polynomials." Canadian Journal of Mathematics 61, no. 3 (2009): 583–603. http://dx.doi.org/10.4153/cjm-2009-031-6.

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Abstract.We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers r, n ≥ 0, we conjecture that is a ℚ-irreducible polynomial whose Galois group contains the alternating group on n letters. That this is so for r = n was conjectured in the 1950's by Grosswald and proven recently by Filaseta and Trifonov. It follows fromrecent work of Hajir andWong that the conjecture is true when r is large with respect to n ≥ 5. Here we verify it in three situations: (i) when n is large with respect to r, (ii) when r ≤ 8, and (iii) when n

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Shah, Syed Ali Haider, and Shahid Mubeen. "Expressions of the Laguerre polynomial and some other special functions in terms of the generalized Meijer $ G $-functions." AIMS Mathematics 6, no. 11 (2021): 11631–41. http://dx.doi.org/10.3934/math.2021676.

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<abstract><p>In this paper, we investigate the relation of generalized Meijer $ G $-functions with some other special functions. We prove the generalized form of Laguerre polynomials, product of Laguerre polynomials with exponential functions, logarithmic functions in terms of generalized Meijer $ G $-functions. The generalized confluent hypergeometric functions and generalized tricomi confluent hypergeometric functions are also expressed in terms of the generalized Meijer $ G $-functions.</p></abstract>

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Nishino, Akinori, Hideaki Ujino, and Miki Wadati. "Rodrigues Formula for the Nonsymmetric Multivariable Laguerre Polynomial." Journal of the Physical Society of Japan 68, no. 3 (1999): 797–802. http://dx.doi.org/10.1143/jpsj.68.797.

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