
30.11. Legendre Polynomials, Legendre Functions, and Associated Legendre Functions 1539
30.11.2 Associated Legendre Functions with Integer Indices
and Real Argument
◮ Formulas for associated Legendre functions. Differential equation.
The associated Legendre functions P
m
n
(x) of order m are defined by the formulas
P
m
n
(x) = (1 − x
2
)
m/2
d
m
dx
m
P
n
(x), n = 1, 2, 3, . . . , m = 0, 1, 2, . . .
It is assumed by definition that P
0
n
(x) = P
n
(x).
Properties:
P
m
n
(x) = 0 if m > n, P
m
n
(−x) = (−1)
n−m
P
m
n
(x).
The associated Legendre functions P
m
n
(x) have exactly n − m real zeros, which lie on
the interval − 1 < x < 1.
The associated Legendre functions P
m
n
(x) with low indices: ...