Mathematical Methods for Physicists: A concise introduction - Site Map

SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
the last step resulting from integrating by parts m times. The integral on the right
hand side is zero when n > m and, since L n …x† is a polynomial of degree m in x, it
follows that
Z 1
0
e x L m …x†L n …x†dx ˆ 0
…m 6ˆ n†;
which is Eq. (7.62). The reader can also apply Eq. (7.63) to show that
Z 1
e x
0
fL n …x† g 2 dx ˆ…n!† 2 : …7:64†
Hence the functions fe x=2 L n …x†=n!g form an orthonormal system.
The associated Laguerre polynomials L m n …x†
Di€erentiating Laguerre's equation (7.52) m times by the Leibnitz theorem we
obtain
xD m‡2 y ‡…m ‡ 1 x†D m‡1 y ‡…n m†D m y ˆ 0 … ˆ n†
and writing z ˆ D m y we obtain
xD 2 z ‡…m ‡ 1 x†Dz ‡…n m†z ˆ 0:
…7:65†
This is Laguerre's associated equation and it clearly possesses a polynomial solution
z ˆ D m L n …x† L m n …x† …m n†; …7:66†
called the associated Laguerre polynomial of degree (n m). Using Rodrigues'
formula for Laguerre polynomial L n …x†, Eq. (7.61), we obtain
L m n …x† ˆ dm
dx m L n…x† ˆ dm d n
dx m ex dx n …xn e x † : …7:67†
This result is very useful in establishing further properties of the associated
Laguerre polynomials. The ®rst few polynomials are listed below:
L 0 0…x† ˆ1; L 0 1…x† ˆ1 x; L 1 1…x† ˆ1;
L 0 2…x† ˆ2 4x ‡ x 2 ; L 1 2…x† ˆ4 ‡ 2x; L 2 2…x† ˆ2:
Generating function for the associated Laguerre polynomials
The Laguerre polynomial L n …x† can be generated by the function
1
1 t
xt
exp ˆ X1
L
1 t
n …x† tn
n! :
nˆ0
320