Laguerre Polynomials
(redirected from Laguerre function)Laguerre polynomials
[lə′ger ‚päl·ə′nō·mē·əlz] (mathematics)
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 It might be outdated or ideologically biased.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)
REFERENCE
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.