1.1 Basic Concepts and Formulae
7
L n (x) = e
x d
n
dx n (x
n e
−x ) (Rodrigue’s formula)
(1.43)
The first few polynomials are:
L o (x) = 1, L 1 (x) = −x + 1, L 2 (x) = x
2
− 4x + 2
L 3 (x) = −x
3
+ 9x
2
− 18x + 6, L 4 (x) = x
4
− 16x
3
+ 72x
2
− 96x + 24 (1.44)
Generating function:
e
−xs/(1−s)
1 − s
=
∞
n=0
L n (x)s
n
n!
(1.45)
Recurrence formulas:
L n+1 (x) − (2n + 1 − x)L n (x) + n
2 L n−1 (x) = 0
x L
n (x) = nL n (x) − n
2 L n−1 (x)
(1.46)
Orthonormal properties:
∞
0
e
−x L m (x)L n (x) dx = 0 m = n
(1.47)
∞
0
e
−x
{L n (x)}
2 dx = (n!)
2
(1.48)
Bessel functions: (J n (x))
Differential equation of order n
x
2 y
+ x y
+ (x
2
− n
2 )y = 0 n ≥ 0
(1.49)
Expansion formula:
J n (x) =
∞
k=0
(−1)
k (x/2)
2k−n
k!Γ(k + 1 − n)
(1.50)
Properties:
J −n (x) = (−1)
n J n (x) n = 0, 1, 2, . . .
(1.51)
J
o (x) = −J 1 (x)
(1.52)
J n+1 (x) =
2n
x
J n (x) − J n−1 (x)
(1.53)
Generating function:
e
x(s−1/s)/2
=
∞
n=−∞
J n (x)t
n
(1.54)