Hermite polynomials (thing)

A class of orthogonal polynomials H_n(x) that are solutions to the following ordinary differential equation:

 2
d y      dy
--- - 2x -- + ny = 0
  2      dx
dx

The polynomials are given by the Rodrigues formula:

                       n
            n      2  d        2
H (x) = (-1)  exp(x ) ---exp(-x )
 n                      n
                      dx

and satisfy the three-term recurrence relation:

H   (x) = 2xH (x) - 2nH   (x)
 n+1         n         n-1

They are also orthogonal over the range (-∞, ∞) with weight exp(-x²):

 ∞                 2           n
∫  H (x)H (x)exp(-x ) dx = δ  2 n!sqrt(π)
 -∞ m    n                  mn

The first few polynomials are:

H (x) = 1
 0
H (x) = 2x
 1
          2
H (x) = 4x  - 2
 2
          3
H (x) = 8x  - 12x
 3

These polynomials are also related to the confluent hypergeometric function by the relation:

          n
         2 sqrt(π)                2
H (x) = ----------- M(-n/2; 1/2; x ) -
 n        γ(1-n/2)
          n+1
         2   sqrt(π)                  2
        -------------xM((1-n)/2; 3/2; x )
           γ(-n/2)