Mathematical Methods for Physicists: A concise introduction - Site Map

LAGUERRE'S EQUATION
from which we ®nd that the indicial equation is 2 ˆ 0. And then (7.54) reduces to
X 1
kˆ0
‰k 2 a k x k1 ‡… k†a k x k Šˆ0:
Changing k 1tok 0 in the ®rst term, then renaming k 0 ˆ k, we obtain
X 1
kˆ0
f…k ‡ 1† 2 a k‡1 ‡… k†a k gx k ˆ 0;
whence the recurrence relations are
a k‡1 ˆ
k
…k ‡ 1† 2 a k:
…7:55†
When is a positive integer n, the recurrence relations give a k‡1 ˆ a k‡2 ˆˆ0,
and
a 1 ˆ n
1 2 a …n 1†
0; a 2 ˆ
2 2 a 1 ˆ …1†2 …n 1†n
…1 2† 2 a 0 ;
a 3 ˆ
…n 2†
3 2 a 2 ˆ …1†3 …n 2†…n 1†n
…1 2 3† 2 a 0 ; etc:
In general
a k ˆ…1† k …n k ‡ 1†…n k ‡ 2†…n 1†n
…k!† 2 a 0 : …7:56†
We usually choose a 0 ˆ…1†n!, then the polynomial solution of Eq. (7.52) is given
by
( )
L n …x† ˆ…1† n x n n2
1! xn1 ‡ n2 …n 1† 2
x n2 ‡‡…1† n n! : …7:57†
2!
This is called the Laguerre polynomial of degree n. We list the ®rst four Laguerre
polynomials below:
L 0 …x† ˆ1; L 1 …x† ˆ1 x; L 2 …x† ˆ2 4x ‡ x 2 ; L 3 …x† ˆ6 18x ‡ 9x 2 x 3 :
This is given by
The generating function for the Laguerre polynomials L n …x†
…x; z† ˆexz=…1z†
1 z
ˆ X1
nˆ0
L n …x†
z n :
n!
…7:58†
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