# Laguerre Polynomials

(redirected from Laguerre polynomial)

## 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 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.
Mentioned in ?
References in periodicals archive ?
Now consider a result [Prudnikov, et al (1986), p.579 (8)] connecting [.sub.1][F.SUB.1] and Laguerre polynomial
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].
where [L.sup.[alpha].sub.n](x) is generalized Laguerre polynomial with [alpha] > -1.
If [rho] is even, the roots [{[x.sup.([rho]).sub.j]]}.sup.[rho].sub.j=1] of the Sonin-Markov polynomial [p.sub.[rho]](x) [equivalent to] [p.sup.[beta].sub.[rho]](x) are all distinct from zero because the roots of every generalized Laguerre polynomial are all distinct from zero; see definition (21).
Liang, "A novel unconditionally stable PSTD method based on weighted laguerre polynomial expansion," Journal of Electromagnetic Waves and Applications vol.
So from (2) and the generating function of the Laguerre polynomial [L.sub.n](x) we may get
Let [l.sub.n](z) be the nth Laguerre polynomial. Then
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.
[x.sub.1] = 0 , and [x.sub.2], ..., [x.sub.N] are the roots of [L.sub.N-1](x), the Laguerre polynomial of degree N - 1, indexed in increasing order of magnitude.
The scaled Laguerre polynomial of degree l is defined by
(iii) Laguerre polynomial with the Gamma distribution is defined on [0, + [infinity]).

Site: Follow: Share:
Open / Close