Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
By writing the series for the exponential and collecting powers of z, you can verify
the ®rst few terms of the series. And it is also straightforward to show that
x @2 @
‡…1 x†
@x2 @x ‡ z @
@z ˆ 0:
Substituting the right hand side of Eq. (7.58), that is, …x; z† ˆP1
nˆ0 ‰L n…x†=n!Šz n ,
into the last equation we see that the functions L n …x† satisfy Laguerre's equation.
Thus we identify …x; z† as the generating function for the Laguerre polynomials.
Now multiplying Eq. (7.58) by z n1 and integrating around the origin, we
obtain
L n …x† ˆ n! I
2i
e
xz=…1z†
dz; …7:59†
n‡1
…1 z†z
which is an integral representation of L n …x†.
By di€erentiating the generating function in Eq. (7.58) with respect to x and z,
we obtain the recurrence relations
)
L n‡1 …x† ˆ…2n ‡ 1 x†L n …x†n 2 L n1 …x†;
…7:60†
nL n1 …x† ˆnLn1…x†L 0 n…x†:
0
Rodrigues' formula for the Laguerre polynomials L n …x†
The Laguerre polynomials are also given by Rodrigues' formula
L n …x† ˆe x d n
dx n …xn e x †:
…7:61†
To prove this formula, let us go back to the integral representation of L n …x†, Eq.
(7.59). With the transformation
xz
1 z ˆ s x or z ˆ s x ;
s
Eq. (7.59) becomes
I
L n …x† ˆn!ex
2i
s n e n
…s x†
n‡1
ds;
the new contour enclosing the point s ˆ x in the s plane. By Cauchy's integral
formula (for derivatives) this reduces to
L n …x† ˆe x d n
dx n …xn e x †;
which is Rodrigues' formula.
Alternatively, we can di€erentiate Eq. (7.58) n times with respect to z and
afterwards put z=0, and thus obtain
e x @ n h
lim
z!0 @z n …1 z†1 exp x i
ˆ L
1 z n …x†:
318