212
17 Laguerre Polynomials
17.1 Laguerre Polynomials
17.1.1 General Aspects
Generalized Laguerre polynomials L α
n (x), n = 0, 1, · · · are polynomials of degree
n in x. Its Rodrigues equation [1, 2] is given by
L
α
n (x) =
exp(x)x −n
n!
d n
dx n (exp(−x)x
n+α ) .
(17.1)
Please note, there are different definitions and scaling factors in literature.
Generalized Laguerre polynomials play an important rôle as part of the solution
of the radial Schrödinger equation
−
1
2
d 2
dr 2 +
l(l + 1)
2r 2 −
Z
r
− E
R nl (r) = 0 ,
(17.2)
with
R nl (r) =
2Z
na 0
3 (n − l − 1)!
2n (n + l)!
ρ
l exp(−ρ/2)L
2l+1
n−l−1 (ρ), with ρ =
2Zr
na 0
,
(17.3)
and a 0 the Bohr radius, Z the atomic number, n the principal quantum number, and
l the angular momentum.
The orthogonality relation for the generalized Laguerre polynomials reads
∞
0
exp(−x)x
α L
α
n (x)L
α
n dx =
Γ (n + α + 1)
n!
δ nn .
(17.4)
The following recursion relation allows a numerically stable evaluation of the
Laguerre polynomial for a given parameter α:
L
α
n+1 (x) =
1
n + 1
(2n + α + 1 − x)L
α
n (x) − (n + α)L
α
n−1 (x)
,
(17.5)
with
L
α
0 (x) = 1, L
α
1 (x) = 1 + α − x .
(17.6)
As discussed for orthogonal polynomials in general, the equation above can be
mapped on a recurrence formula for normalized generalized Laguerre polynomials
to derive the corresponding Jacobi matrix. The roots of the generalized Laguerre
polynomials are given by the eigenvalues of this matrix. The Jacobi matrix for the