Page 320 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Laguerre and associated Laguerre polynomials 311

Using equation (A.1) we note that
á
1
1
d k k ÿr d k X (ÿ1) á á‡k X (ÿ1) (á ‡ k)! á
(r e ) ˆ r ˆ r (F:4)
dr k dr k á! (á!) 2
áˆ0 áˆ0
Combining equations (F.3) and (F.4), we obtain the formula for the Laguerre
polynomials
k ÿr
L k (r) ˆ e r d k (r e ) (F:5)
dr k
Another relationship for the polynomials L k (r) can be obtained by expanding the
generating function g(r, s) in equation (F.1) using (A.1)
á á
e ÿrs=(1ÿs) X (ÿ1) r s á
1
g(r, s) ˆ ˆ
1 ÿ s á! (1 ÿ s) á‡1
áˆ0
The factor (1 ÿ s) ÿ(á‡1) may be expanded in an in®nite series using equation (A.3) to
obtain
1
X (á ‡ â)! â
ÿ(á‡1)
(1 ÿ s) ˆ s
á!â!
âˆ0
so that g(r, s) becomes
1 1 á á
X X (ÿ1) (á ‡ â)!r
g(r, s) ˆ s á‡â
(á!) â!
2
áˆ0 âˆ0
k
We next collect all the coef®cients of s for an arbitrary k, so that á ‡ ⠈ k, and
replace the summation over á by a summation over k. When á ˆ k, the index â equals
zero; when á ˆ k ÿ 1, the index â equals one; and so on until we have á ˆ 0 and
⠈ k. Thus, the result of the summation is
1 k kÿâ kÿâ
X X (ÿ1) r k
g(r, s) ˆ k! s
2
[(k ÿ â)!] â!
kˆ0 âˆ0
k
Since the Laguerre polynomial L k (r) divided by k! is the coef®cient of s in the
expansion (F.1) of the generating function, we have
k kÿâ
X (ÿ1)
L k (r) ˆ (k!) 2 r kÿâ
[(k ÿ â)!] â!
2
âˆ0
If we let k ÿ ⠈ ã and replace the summation over â by a summation over ã,we
obtain the desired result
k ã
X (ÿ1) ã
2
L k (r) ˆ (k!) r (F:6)
2
(ã!) (k ÿ ã)!
ãˆ0
A third relationship for the polynomials L k (r) can be obtained by expanding the
derivative in equation (F.5), using (A.4), to give
k á k kÿá ÿr k kÿá á k
X k! d r d e X (ÿ1) k! d r
r
L k (r) ˆ e ˆ
á!(k ÿ á)! dr á dr kÿá á!(k ÿ á)! dr á
áˆ0 áˆ0
k
We now observe that the operator [(d=dr) ÿ 1] may be expanded according to the
binomial theorem (A.2) as