# Laguerre Polynomials

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## 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.

## 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

*L _{0}(x) = 1, L_{1}(x) = x – 1, l_{2}(x) = x^{2} – 4x + 2*

*L _{3}(x) = x^{3} – 9x^{2} + 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

*xy ^{n} + (1 – x)y + ny = 0*

The Laguerre polynomials are recursively defined by the formula

*L _{n+1}(x) = (x – 2n – 1)L_{n}(x) – n^{2}L_{n–1}(x)*

### REFERENCE

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