Laguerre Polynomials

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Laguerre polynomials

[lə′ger ‚päl·ə′nō·mē·əlz]
A sequence of orthogonal polynomials which solve Laguerre's differential equation for positive integral values of the parameter.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Laguerre Polynomials


(named after the French mathematician E. Laguerre, 1834–86), a special system of polynomials of successively increasing degree. For n = 0, 1, 2, … the Laguerre polynomials Ln(x) can be defined by the formula

The first few Laguerre polynomials are

L0(x) = 1, L1(x) = x – 1, l2(x) = x2 – 4x + 2

L3(x) = x3 – 9x2 + 18x – 6

The Laguerre polynomials are orthogonal on the half-line x ≥ 0 with respect to the weight function e–x and are solutions of the differential equation

xyn + (1 – x)y + ny = 0

The Laguerre polynomials are recursively defined by the formula

Ln+1(x) = (x – 2n – 1)Ln(x) – n2Ln–1(x)


Lebedev, N. N. Spetsial’nye funktsii i ikh prilozheniia, 2nd ed. Moscow-Leningrad, 1963.
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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Wang [26] studied the Fisher equation on a semi-infinite domain using the generalized Laguerre functions. Wang and Jiao [27] considered the Fisher equation on unbounded domain using the generalized Hermite functions.
In particular, the polyBergman spaces are spaces of Wavelet transform which is related to Laguerre functions, and the poly-Fock spaces are spaces of short-time Fourier transform which is related to Hermite functions.
The method of approximation analyzed in this paper is the Laguerre Impulse Response Approximation (LIRA) which is based on approximation of noninteger order system impulse response with Laguerre functions. Early works in this area were unsuccessful in approximation of [alpha] order integrators [18, 19].
There are two parameters in this model: the number of Laguerre orthogonal functions K and the pole of Laguerre functions [lambda]([absolute value of [lambda]] < 1).
Very good results for such problems have been achieved by using nonclassical orthogonal basis sets for systems [9], mapped orthogonal systems [16,17], Laguerre functions [10], mapped Legendre functions [11], and mapped Fourier sine series [12].
Different approaches have been considered for reducing the online computation time by discusses the Optimal Model Predictive Control (OMPC) which gives stability and recursive feasibility guarantee and is often assumed in modern MPC approaches.MPC implementation based on Laguerre functions also called Laguerre optimal model predictive control (LOMPC) has been discussed in [11,12] enables the capture of near optimal control trajectory with less number of parameters as compared with conventional MPC algorithms.
The Rao-Wilton-Glisson (RWG) [26] functions are used as the spatial expansion and testing functions, and the weighted Laguerre functions are used as the temporal expansion and testing functions.
In this paper, we propose an efficient and accurate method to apply FDTD for transient wave propagation in a general dispersive media using the MOD scheme with the associate Laguerre functions. This MOD methodology has been successfully implemented in a FDTD formulation [10-14] and in integral equations dealing with skin effects in conductors, and propagation in non-dispersive dielectric and in a dispersive media [15-17].
The most important special functions are known as: Bessel functions, Hermite functions, Legendre functions, Laguerre functions, Chebyshev functions etc., [10].
Lindsay, and V Pavlov, (2005), 'Smooth estimation of yield curves by Laguerre functions', In: MODSIM 05--International Congress on Modelling and Simulation Advances and Applications for Management and Decision Making, 12 December-15 December, 2005, Australia, Victoria, Melbourne.