Laguerre polynomials

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! ₁F₁(-n; 1; x), where (a)_n is the Pochhammer symbol.