The associated Legendre function P_{n}^{m} has the following recurrence
formulae:
(C.1)
(C.2)
Multiplying (C.1) by
one obtains
(n+m+1)(n+1-m)R_{n+1,m} - (2n+1) x_{3}R_{n,m} + r^{2}R_{n-1,m}=0.
Multiplying (C.2) by
,
setting
m=n+1 and using (A.2) one obtains
In the same manner, one has
r^{2}S_{n+1,m}-(2n+1) x_{3}S_{n,m} + (n+m)(n-m)S_{n-1,m}=0 .
Using above formulae one can compute R_{n,m} and S_{n,m} as follows:
Computation of R_{n,m}
1.
R_{0,0}=1.
2.
Compute R_{n,n} recursively for
n=1,2,....,P
using