`laguerreL`

Laguerre polynomial, plain or generalized.

Syntax

L = laguerreL(n, x)

Laguerre polynomial L_n(x).

L = laguerreL(n, a, x)

Generalized Laguerre polynomial L_n^a(x).

Description

The Laguerre polynomials are orthogonal on [0, Inf) with weight exp(-x). The plain family starts with L_0 = 1, L_1 = 1 - x, and obeys (n + 1)*L_(n+1) = (2*n + 1 - x)*L_n - n*L_(n-1). The generalized family with parameter a has weight x^a*exp(-x), and laguerreL(n, 0, x) is the plain one.

They describe the radial part of the hydrogen atom wave functions and are used in Gauss-Laguerre quadrature.

Input arguments

n: Degree, a nonnegative integer.
a: Parameter of the generalized polynomial.
x: Point(s) of evaluation or a symbolic variable.

Output arguments

L: Numbers with the size of x, or a symbolic polynomial.

Example

Plain and generalized

syms x
laguerreL(2, x)
laguerreL(2, 1, x)
laguerreL(3, 0.5)

Output

ans = x^2/2 - 2*x + 1
ans = x^2/2 - 3*x + 3
ans = -0.1458

laguerreL