1.1 Basic Concepts and Formulae
5
First few Hermite’s polynomials are:
H o (x) = 1, H 1 (x) = 2x, H 2 (x) = 4x
2
− 2
H 3 (x) = 8x
3
− 12x, H 4 (x) = 16x
4
− 48x
2
+ 12
(1.23)
Generating function:
e
2t x−t
2 =
∞
n=0
H n (x)t
n
n!
(1.24)
Recurrence formulas:
H
n (x) = 2n H n−1 (x)
H n+1 (x) = 2x H n (x) − 2n H n−1 (x)
(1.25)
Orthonormal properties:
∞
−∞
e
−x
2 H m (x)H n (x) dx = 0 m = n
(1.26)
∞
−∞
e
−x
2 {H n (x)}
2 dx = 2
n n!
√ π
(1.27)
Legendre functions:
Differential equation of order n:
(1 − x
2 )y
− 2x y
+ n(n + 1)y = 0
(1.28)
when n = 0, 1, 2, . . . we get Legendre polynomials P n (x).
P n (x) =
1
2 n n!
d
n
dx n (x
2
− 1)
n
(1.29)
First few polynomials are:
P o (x) = 1, P 1 (x) = x, P 2 (x) =
1
2
(3x
2
− 1)
P 3 (x) =
1
2
(5x
3
− 3x), P 4 (x) =
1
8
(35x
4
− 30x
2
+ 3)
(1.30)
Generating function:
1
√
1 − 2t x + t 2
=
∞
n=0
P n (x)t
n
(1.31)