Page 330 - Elements of Distribution Theory

P1: JZP
052184472Xc10 CUNY148/Severini May 24, 2005 2:50

316 Orthogonal Polynomials

Laguerre polynomials
The Laguerre polynomials, which will be denoted by L 0 , L 1 ,..., are orthogonal polyno-
mials with respect to the standard exponential distribution. Here we use the standardization
that

∞
2
L n (x) exp(−x) dx = 1, n = 0, 1,....
0
The Laguerre polynomials may be generated using the procedure described in Theo-
rem 10.3 using the function g(x) = x. That is, we may take
1 d n n
L n (x) = exp(x) x exp(−x), n = 0, 1,.... (10.11)
n! dx n
Theorem 10.9. The Laguerre polynomials are given by (10.11).

Proof. Using Leibnitz’s rule with (10.11) shows that
n
j

j n x
L n (x) = (−1)
j j!
j=0
and, hence, L n is a polynomial of degree n. Furthermore, it may be shown that
∞

2
L n (x) exp(−x) dx = 1;
0
this result is left as an exercise.
The result now follows from Theorem 10.3 provided that, for each n = 1, 2,...,
lim Q nmj (x) = lim Q nmj (x) = 0
x→0 x→∞
for j = 0, 1,..., n − 1 and m = 1, 2,..., n − 1, where

d m n
j
Q nmj (x) = x [x exp(−x)].
dx m
Note that, for m = 1, 2,..., n − 1, Q nmj (x)isalways of the form
x (n−m+ j) R(x)exp(−x)
where R is a polynomial of degree m. The result follows from the fact that n − m + j
≥ 1.

The following corollary simply restates the expression for L n derived in the proof of
Theorem 10.9.

Corollary 10.4. For each n = 0, 1,..., let L n denote the nth Laguerre polynomial. Then
n
j

j n x
L n (x) = (−1) .
j j!
j=0
Hence,
1 2 1 3 3 2
L 1 (x) = x − 1, L 2 (x) = x − 2x + 1, L 3 (x) =− x + x − 3x + 1.
2 6 2