A set of orthogonal polynomials L_{n}(x) that are solutions to the ordinary differential equation:

2
d y dy
x --- + (1 - x) -- + ny = 0
2 dx
dx

They are generated by the Rodrigues formula:

n
exp(x) d n
L (x) = ------ --- (x exp(-x))
n n! n
dx

The Laguerre polynomials, like many orthogonal polynomials, also satisfy a three-term recurrence relation:

(2n + 1 - x)L (x) - nL (x)
n n-1
L (x) = -----------------------------
n+1 n+1

It is possible to generate all the polynomials given that the first three polynomials are:

L (x) = 1
0
L (x) = -x + 1
1
2
L (x) = (x - 4x + 2)/2
2

The Laguerre polynomials are related to the confluent hypergeometric function by: L_{n}(x) = (k+1)_{n}/n! _{1}F_{1}(-n; 1; x), where (a)_{n} is the Pochhammer symbol.