Associated Laguerre functions

18.8 ASSOCIATED LAGUERRE FUNCTIONS
which trivially rearranges to give the recurrence relation (18.115).
To obtain the recurrence relation (18.116), we begin by differentiating the generating
function (18.114) with respect to x, which yields
∂G
he−xh/(1−h)
=
Ln hn ,
=−
∂x
(1 − h)2
and thus we have
−h
Ln hn = (1 − h)
Ln hn .
Equating coefficients of hn on each side then gives
−Ln−1 = Ln − Ln−1 ,
which immediately simplifies to give (18.116). 18.8 Associated Laguerre functions
The associated Laguerre equation has the form
xy + (m + 1 − x)y + ny = 0;
(18.118)
it has a regular singularity at x = 0 and an essential singularity at x = ∞. We
restrict our attention to the situation in which the parameters n and m are both
non-negative integers, as is the case in nearly all physical problems. The associated
Laguerre equation occurs most frequently in quantum-mechanical applications.
Any solution of (18.118) is called an associated Laguerre function.
Solutions of (18.118) for non-negative integers n and m are given by the
associated Laguerre polynomials
m
Lm
n (x) = (−1)
dm
Ln+m (x),
dxm
(18.119)
where Ln (x) are the ordinary Laguerre polynomials.§
Show that the functions Lmn (x) defined in (18.119) are solutions of (18.118).
Since the Laguerre polynomials Ln (x) are solutions of Laguerre’s equation (18.107), we
have
xLn+m + (1 − x)Ln+m + (n + m)Ln+m = 0.
Differentiating this equation m times using Leibnitz’ theorem and rearranging, we find
(m+1)
(m)
xL(m+2)
n+m + (m + 1 − x)Ln+m + nLn+m = 0.
On multiplying through by (−1)m and setting Lmn = (−1)m L(m)
n+m , in accord with (18.119),
we obtain
x(Lmn ) + (m + 1 − x)(Lmn ) + nLmn = 0,
which shows that the functions Lmn are indeed solutions of (18.118). §
m
m
Note that some authors define the associated Laguerre polynomials as Lm
n (x) = (d /dx )Ln (x),
m m
which is thus related to our expression (18.119) by Lm
n (x) = (−1) Ln+m (x).
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SPECIAL FUNCTIONS
In particular, we note that L0n (x) = Ln (x). As discussed in the previous section,
Ln (x) is a polynomial of order n and so it follows that Lm
n (x) is also. The first few
associated Laguerre polynomials are easily found using (18.119):
Lm
0 (x) = 1,
Lm
1 (x) = −x + m + 1,
2
2!Lm
2 (x) = x − 2(m + 2)x + (m + 1)(m + 2),
3
2
3!Lm
3 (x) = −x + 3(m + 3)x − 3(m + 2)(m + 3)x + (m + 1)(m + 2)(m + 3).
Indeed, in the general case, one may show straightforwardly, from the definition
(18.119) and the expression (18.111) for the ordinary Laguerre polynomials, that
Lm
n (x) =
n
(−1)k
k=0
(n + m)!
xk .
k!(n − k)!(k + m)!
(18.120)
18.8.1 Properties of associated Laguerre polynomials
The properties of the associated Laguerre polynomials follow directly from those
of the ordinary Laguerre polynomials through the definition (18.119). We shall
therefore only briefly outline the most useful results here.
Rodrigues’ formula
A Rodrigues’ formula for the associated Laguerre polynomials is given by
Lm
n (x) =
ex x−m dn n+m −x
(x e ).
n! dxn
(18.121)
It can be proved by evaluating the nth derivative using Leibnitz’ theorem (see
exercise 18.7).
Mutual orthogonality
In section 17.4, we noted that the associated Laguerre equation could be transformed into a Sturm–Liouville one with p = xm+1 e−x , q = 0, λ = n and ρ = xm e−x ,
and its natural interval is thus [0, ∞]. Since the associated Laguerre polynomials
Lm
n (x) are solutions of the equation and are regular at the end-points, those
with the same m but differing values of the eigenvalue λ = n must be mutually
orthogonal over this interval with respect to the weight function ρ = xm e−x , i.e.
∞
m
m −x
Lm
dx = 0
if n = k.
n (x)Lk (x)x e
0
This result may also be proved directly using the Rodrigues’ formula (18.121), as
may the normalisation condition when k = n.
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18.8 ASSOCIATED LAGUERRE FUNCTIONS
Show that
∞
I≡
Lmn (x)Lmn (x)xm e−x dx =
0
(n + m)!
.
n!
(18.122)
Using the Rodrigues’ formula (18.121), we may write
∞
dn
(−1)n ∞ dn Lmn n+m −x
1
Lmn (x) n (xn+m e−x ) dx =
x e dx,
I=
n! 0
dx
n!
dxn
0
where, in the second equality, we have integrated by parts n times and used the fact that
the boundary terms all vanish. From (18.120) we see that dn Lmn /dxn = (−1)n . Thus we have
∞
(n + m)!
1
,
xn+m e−x dx =
I=
n! 0
n!
where, in the second equality, we use the expression (18.153) defining the gamma function
(see section 18.12). The above orthogonality and normalisation conditions allow us to expand any
(reasonable) function in the interval 0 ≤ x < ∞ in a series of the form
f(x) =
∞
an Lm
n (x),
n=0
in which the coefficients an are given by
∞
n!
m −x
an =
f(x)Lm
dx.
n (x)x e
(n + m)! 0
We note that it is sometimes convenient to define the orthogonal associated
m/2 −x/2 m
e
Ln (x), which may also be used to produce a
Laguerre functions φm
n (x) = x
series expansion of a function in the interval 0 ≤ x < ∞.
Generating function
The generating function for the associated Laguerre polynomials is given by
∞
G(x, h) =
e−xh/(1−h)
n
=
Lm
n (x)h .
m+1
(1 − h)
(18.123)
n=0
This can be obtained by differentiating the generating function (18.114) for the
ordinary Laguerre polynomials m times with respect to x, and using (18.119).
Use the generating function (18.123) to obtain an expression for Lmn (0).
From (18.123), we have
∞
n=0
Lmn (0)hn =
1
(1 − h)m+1
= 1 + (m + 1)h +
(m + 1)(m + 2) 2
(m + 1)(m + 2) · · · (m + n) n
h + ··· +
h + ··· ,
2!
n!
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