Page 322 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Laguerre and associated Laguerre polynomials 313
ÿrs=(1ÿs) j j k
j
1
d e (ÿ1) s ÿrs=(1ÿs) X j s
g(r, s; j) ˆ ˆ e ˆ L (r) (F:10)
k
dr j 1 ÿ s (1 ÿ s) j‡1 k!
kˆ j
The summation in the right-hand term begins with k ˆ j, since j cannot exceed k.
j
We can write an explicit series for L (r) by substituting equation (F.6) into (F.9)
k
k ã j k ã
X (ÿ1) d X (ÿ1)
j 2 ã 2 ãÿ j
L (r) ˆ (k!) r ˆ (k!) r
k (ã!) (k ÿ ã)! dr j ã!(k ÿ ã)!(ã ÿ j)!
2
ãˆ0 ㈠j
The summation over ã now begins with the term 㠈 j because the earlier terms
vanish in the differentiation. If we let ã ÿ j ˆ á and replace the summation over ã by
a summation over á,we have
kÿ j j‡á á
X (ÿ1) r
j 2
L (r) ˆ (k!) (F:11)
k á!(k ÿ j ÿ á)!(j ‡ á)!
áˆ0
For the purpose of deriving some useful relationships involving the polynomials
j
j
L (r), we de®ne the polynomial Ë (r)as
k i
i!
j j
Ë (r) L i‡ j (r) (F:12)
i
(i ‡ j)!
If we replace the dummy index of summation k in equation (F.10) by i, where
i ˆ k ÿ j, then (F.10) takes the form
j j
1
1
X L (r) X
i‡ j i‡ j j Ë (r) i
i
g(r, s; j) ˆ s ˆ s s
(i ‡ j)! i!
iˆ0 iˆ0
j
Thus, Ë (r) are just the coef®cients in a Taylor series expansion of the function
i
s ÿ j g(r, s; j) and are, therefore, given by
@ i
j ÿ j
Ë (r) ˆ s g(r, s; j)
i
@s i
sˆ0
Substituting for g(r, s; j) using equation (F.10), we obtain
" #
@ i e ÿrs=(1ÿs)
j j
Ë (r) ˆ (ÿ1)
i i
@s (1 ÿ s) j‡1
sˆ0
" #
@ i e e
r ÿr=(1ÿs)
ˆ (ÿ1) j
@s i (1 ÿ s) j‡1
sˆ0
" #
1
@ i X (ÿr) á
ˆ (ÿ1) j e r
@s i á!(1 ÿ s) ᇠj‡1
áˆ0 sˆ0
á
1 (ÿr) (á ‡ j ‡ i)!
X
j r
ˆ (ÿ1) e (1 ÿ s) ÿ(ᇠj‡i‡1)
á! (á ‡ j)!
áˆ0 sˆ0
1
X (á ‡ j ‡ i)!
j r
ˆ (ÿ1) e (ÿr) á (F:13)
á!(á ‡ j)!
áˆ0
We next note that