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(-x^{2}):

∞ 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)