Laguerre polynomials

About: Laguerre polynomials is a research topic. Over the lifetime, 5868 publications have been published within this topic receiving 88892 citations. The topic is also known as: Laguerre polynomials.

Distributions of Matrix Variates and Latent Roots Derived from Normal Samples

Abstract: The paper is largely expository, but some new results are included to round out the paper and bring it up to date. The following distributions are quoted in Section 7. 1. Type $_0F_0$, exponential: (i) $\chi^2$, (ii) Wishart, (iii) latent roots of the covariance matrix. 2. Type $_1F_0$, binomial series: (i) variance ratio, $F$, (ii) latent roots with unequal population covariance matrices. 3. Type $_0F_1$, Bessel: (i) noncentral $\chi^2$, (ii) noncentral Wishart, (iii) noncentral means with known covariance. 4. Type $_1F_1$, confluent hypergeometric: (i) noncentral $F$, (ii) noncentral multivariate $F$, (iii) noncentral latent roots. 5. Type $_2F_1$, Gaussian hypergeometric: (i) multiple correlation coefficient, (ii) canonical correlation coefficients. The modifications required for the corresponding distributions derived from the complex normal distribution are outlined in Section 8, and the distributions are listed. The hypergeometric functions $_pF_q$ of matrix argument which occur in the multivariate distributions are defined in Section 4 by their expansions in zonal polynomials as defined in Section 5. Important properties of zonal polynomials and hypergeometric functions are quoted in Section 6. Formulae and methods for the calculation of zonal polynomials are given in Section 9 and the zonal polynomials up to degree 6 are given in the appendix. The distribution of quadratic forms is discussed in Section 10, orthogonal expansions of $_0F_0$ and $_1F_1$ in Laguerre polynomials in Section 11 and the asymptotic expansion of $_0F_0$ in Section 12. Section 13 has some formulae for moments.

Log-Gases and Random Matrices

Abstract: Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years. Log-Gases and Random Matrices gives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painleve transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, making Log-Gases and Random Matrices an indispensable reference work, as well as a learning resource for all students and researchers in the field.

A MATLAB differentiation matrix suite

Abstract: A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.

System identification using Laguerre models

Abstract: The traditional approach of expanding transfer functions and noise models in the delay operator to obtain linear-in-the-parameters predictor models leads to approximations of very high order in cases of rapid sampling and/or dispersion in time constants. By using prior information about the time constants of the system more appropriate expansions, related to Laguerre networks, are introduced and analyzed. It is shown that the model order can be reduced, compared to ARX (FIR, AR) modeling, by using Laguerre models. Furthermore, the numerical accuracy of the corresponding linear regression estimation problem is improved by a suitable choice of the Laguerre parameter. Consistency (error bounds), persistence of excitation conditions. and asymptotic statistical properties are investigated. This analysis is based on the result that the covariance matrix of the regression vector of a Laguerre model has a Toeplitz structure. >

The Eigenfunctions of Laplace's Tidal Equations over a Sphere

Abstract: Numerical calculations are presented for the eigenvalues of Laplace’s tidal equations governing a thin layer of fluid on a rotating sphere, for a complete range of the parameter e ( omega = rate of rotation, R = radius, g = gravity, h — depth of fluid layer). The corresponding eigenfunctions or ‘Hough functions’ are shown graphically for the lower modes of oscillation. Negative values of e, which have application in problems involving forced motions, are also considered The calculations reveal many asymptotic forms of the solution for various limiting values of e. The corresponding analytical expressions are derived in the present paper. Thus, as e0 through positive values we have the well-known waves of the first and second class respectively, which were found by Margules and Hough. These can be represented in terms of spherical harmonics. As e-> + oo there are three distinct asymptotic forms. In each of these the energy is concentrated near the equator. In the first type, the kinetic energy is three times the potential energy. In the other two types the kinetic and potential energies are equal. The waves of the second type are all propagated towards the west. The waves of the third type are Kelvin waves propagated eastwards along the equator. All three types are described in terms of Hermite polynomials. As e 0 through negative values there is only one asymptotic form of solution, representing motions which are analytically continuous with Hough’s ‘waves of the second class’. As e —> — oc there are three different asymptotic forms, in each of which the energy tends to be concentrated near the poles of rotation. In the first two types the energy is mainly kinetic and the motion is in inertial circles. In the third type the energy is mainly potential. The modes tend to occur in pairs of almost the same frequency, one being symmetric and the other antisymmetric about the equator. The analytical forms of the solutions involve generalized Laguerre polynomials. In the special case of zonal oscillations, the first two limiting forms as E -> — oo go over into a different form in which the frequency tends to zero as e tends to a finite negative value. In this case the third type does not occur. The way in which the various asymptotic solutions are connected can be traced in figures 1 to 6 (e > 0) and figures 16 to 21 (e 4 4 are tabulated in tables 1 to 10. The eigenfunctions for the lower modes are presented graphically.