Appendix A: List of associated Legendre functions

Let x = cos($x=\cos(\theta)$), and . The associated Legendre functions P_n^m = P_n^m(x) are listed below, for n = 0,8 and m = 0,n, in Table 3 (n = 0,5) and Table 4 (n = 6,8). The associated Legendre functions with m = -M < 0 are simply given in terms of the equivalent functions with M > 0 according to

$$P_n^m=P_n^{-M} = (-1)^M \frac{(n-M)!}{(n+M)!} P_n^M.$$ (30)

Table 3: List of associated Legendre polynomials from n = 0 to n = 5.
n	m	Function
0	0	1
1	0	x
1	1	s
2	0
2	1	3 x s
2	2	3 (1-x2)
3	0	$\frac{1}{2} x (5x^2-3)$
3	1
3	2	15 x (1-x2)
3	3	15 s3
4	0
4	1
4	2
4	3	105xs3
4	4	105 s4
5	0
5	1
5	2
5	3
5	4	945xs4
5	5	945s5

Table 4: List of associated Legendre polynomials from n = 6 to n = 8.
n	m	Function
6	0
6	1
6	2	$\frac{105}{8}s^2(33x^4-18x^2+1)$
6	3	$\frac{315}{2}(11x^2-3)xs^3$
6	4	$\frac{945}{2}s^4(11x^2-1)$
6	5	10395xs5
6	6	10395s6
7	0	$\frac{1}{16}x(429x^6-693x^4+315x^2-35)$
7	1	$\frac{7}{16}s(429x^6-495x^4+135x^2-5)$
7	2	$\frac{63}{8} x s^2(143x^4-110x^2+15)$
7	3	$\frac{315}{8}s^3(143x^4-66x^2+3)$
7	4	$\frac{3465}{2}xs^4(13x^2-3)$
7	5	$\frac{10395}{2}s^5(13x^2-1)$
7	6	135,135xs6
7	7	135,135s7
8	0	$\frac{1}{128}(6435x^8-12,012x^6+6930x^4-1260x^2+35)$
8	1	$\frac{9}{16}xs(715x^6-1001x^4+385x^2-35)$
8	2	$\frac{315}{16}s^2(143x^6-143x^4+33x^2-1)$
8	3	$\frac{3465}{8}xs^3(39x^4-26x^2+3)$
8	4	$\frac{10395}{8}s^4(65x^4-26x^2+1)$
8	5	$\frac{135,135}{2}xs^5(5x^2-1)$
8	6	$\frac{135,135}{2}s^6(15x^2-1)$
8	7	2,027,025xs7
8	8	2,027,025s8