1.1 Basic Concepts and Formulae 5
First few Hermite’s polynomials are:
Ho(x)= 1 ,H 1 (x)= 2 x,H 2 (x)= 4 x^2 − 2
H 3 (x)= 8 x^3 − 12 x,H 4 (x)= 16 x^4 − 48 x^2 + 12 (1.23)
Generating function:
e^2 tx−t
2
=
∑∞
n= 0
Hn(x)tn
n!
(1.24)
Recurrence formulas:
Hn′(x)= 2 nHn− 1 (x)
Hn+ 1 (x)= 2 xHn(x)− 2 nHn− 1 (x) (1.25)
Orthonormal properties:
∫∞
−∞
e−x
2
Hm(x)Hn(x)dx= 0 m
=n (1.26)
∫∞
−∞
e−x
2
{Hn(x)}^2 dx= 2 nn!
√
π (1.27)
Legendre functions:
Differential equation of ordern:
(1−x^2 )y′′− 2 xy′+n(n+1)y= 0 (1.28)
whenn= 0 , 1 , 2 ,...we get Legendre polynomialsPn(x).
Pn(x)=
1
2 nn!
dn
dxn
(x^2 −1)n (1.29)
First few polynomials are:
Po(x)= 1 ,P 1 (x)=x,P 2 (x)=
1
2
(3x^2 −1)
P 3 (x)=
1
2
(5x^3 − 3 x),P 4 (x)=
1
8
(35x^4 − 30 x^2 +3) (1.30)
Generating function:
1
√
1 − 2 tx+t^2
=
∑∞
n= 0
Pn(x)tn (1.31)