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 It might be outdated or ideologically biased.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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The m-th derivative of a generalized Laguerre polynomial satisfies the relation
For example, when p is odd and [absolute value of x] < 1, the roots of the generalized Laguerre polynomial [L.sup.(2q+1).sub.p] (x) (see Appendix A) could be used for [[tau].sup.c] and [[tau].sup.s].
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On specializing the coefficients A(N,k) suitably, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] yields a number of known polynomials such as Laguerre polynomial, Hermite polynomial, Jacobi polynomial, Brafman polynomial, Gould and Hopper polynomials and others.
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