6
1 Mathematical Physics
Recurrence formulas:
x P
n (x) − P
n−1 (x) = n P n (x)
P
n+1 (x) − P
n−1 (x) = (2n + 1)P n (x)
(1.32)
Orthonormal properties:
1
−1
P m (x)P n (x) dx = 0 m = n
(1.33)
1
−1
{P n (x)}
2 dx =
2
2n + 1
(1.34)
Other properties:
P n (1) = 1, P n (−1) = (−1)
n
, P n (−x) = (−1)
n P n (x)
(1.35)
Associated Legendre functions:
Differential equation:
(1 − x
2 )y
− 2x y
+
l(l + 1) −
m
2
1 − x 2
y = 0
(1.36)
P
m
l (x) = (1 − x
2 )
m/2 d
m
dx m P l (x)
(1.37)
where P l (x) are the Legendre polynomials stated previously, l being the positive
integer.
P
o
l (x) = P l (x)
(1.38)
and P
m
l (x) = 0 if m > n
(1.39)
Orthonormal properties:
1
−1
P
m
n (x)P
m
l (x) dx = 0 n = l
(1.40)
1
−1
{P
m
l (x)}
2 dx =
2
2l + 1
(l + m)!
(l − m)!
(1.41)
Laguerre polynomials:
Differential equation:
x y
+ (1 − x)y
+ ny = 0
(1.42)
if n = 0, 1, 2, . . . we get Laguerre polynomials given by