Page 319 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Appendix F

Laguerre and associated Laguerre polynomials

Laguerre polynomials
The Laguerre polynomials L k (r) are de®ned by means of the generating
function g(r, s)
1
e ÿrs=(1ÿs) X s k
g(r, s) L k (r) (F:1)
1 ÿ s k!
kˆ0
where 0 < r < 1 and where jsj , 1 in order to ensure convergence of the in®nite
series. Since the right-hand term is a Taylor series expansion of g(r, s), the Laguerre
polynomials are given by

k k
@ g(r, s) @ e ÿrs=(1ÿs)
L k (r) ˆ ˆ (F:2)
@s k @s k 1 ÿ s
sˆ0 sˆ0
r
To evaluate L k (r) from equation (F.2), we ®rst factor out e in the generating
function and expand the remaining exponential function in a Taylor series
á á
1
1
e r ÿr=(1ÿs) e r X (ÿ1) á r á r X (ÿ1) r ÿ(á‡1)
g(r, s) ˆ e ˆ ˆ e (1 ÿ s)
1 ÿ s 1 ÿ s á! 1 ÿ s á!
áˆ0 áˆ0
We then take k successive derivatives of g(r, s) with respect to s
á á
1
@ g(r, s) r X (ÿ1) r ÿ(á‡2)
ˆ e (á ‡ 1)(1 ÿ s)
@s á!
áˆ0
2
á á
1
@ g(r, s) r X (ÿ1) r ÿ(á‡3)
ˆ e (á ‡ 1)(á ‡ 2)(1 ÿ s)
@s 2 á!
áˆ0
. . .
á á
k
1
@ g(r, s) X (ÿ1) r (á ‡ k)!
ˆ e r (1 ÿ s) ÿ(á‡k‡1)
@s k á! á!
áˆ0
When the kth derivative is evaluated at s ˆ 0, we have
á
1 (ÿ1) (á ‡ k)!
X
L k (r) ˆ e r r á (F:3)
(á!) 2
áˆ0
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