Page 310 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Appendix E

Legendre and associated Legendre polynomials

Legendre polynomials
l
The Legendre polynomials P l (ì) may be de®ned as the coef®cients of s in an
in®nite series expansion of a generating function g(ì, s)
1
X l
2 ÿ1=2
g(ì, s) (1 ÿ 2ìs ‡ s ) P l (ì)s (E:1)
lˆ0
where ÿ1 < ì < 1 and jsj , 1 in order for the in®nite series to converge.
We may also expand g(ì, s) by applying the standard formula

1
n
X n d f X n : : : :
1
z 1 3 5 (2n ÿ 1)
z
f (z) (1 ÿ z) ÿ1=2 ˆ ˆ
n! dz n n! 2 n
nˆ0 zˆ0 nˆ0
1
X (2n)! n
ˆ z
2n
2 (n!) 2
nˆ0
If we set z ˆ s(2ì ÿ s), then g(ì, s) becomes
1
X (2n)!
n
g(ì, s) ˆ s (2ì ÿ s) n
2n
2 (n!) 2
nˆ0
n
With the use of the binomial expansion (A.2), the factor (2ì ÿ s) can be further
expanded as
n á
X (ÿ1) n!
n nÿá á
(2ì ÿ s) ˆ (2ì) s
á!(n ÿ á)!
áˆ0
so that
1 n á nÿá
X X (ÿ1) (2n)!ì
g(ì, s) ˆ s n‡á
2 n‡á n!á!(n ÿ á)!
nˆ0 áˆ0
l
We next collect all the coef®cients of s for some arbitrary l and replace the summation
over n with a summation over l. Since n ‡ á ˆ l, when n ˆ l,we have á ˆ 0; when
n ˆ l ÿ 1, we have á ˆ 1; and so on until n ˆ l ÿ M, á ˆ M, where M < l ÿ M or
M < l=2. The summation over á terminates at á ˆ M,with M ˆ l=2 for l even and
M ˆ (l ÿ 1)=2 for l odd, because á cannot be greater than n. The result is
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