Laguerre Polynomials (redirected from Laguerre functions)

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.

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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 L_n(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

L_n+1(x) = (x – 2n – 1)L_n(x) – n²L_n–1(x)

REFERENCE

Lebedev, N. N. Spetsial’nye funktsii i ikh prilozheniia, 2nd ed. Moscow-Leningrad, 1963.