A class of orthogonal polynomials Hn(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)