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

10.2 General Systems of Orthogonal Polynomials 303

function. Let {p 0 , p 1 ,...} denote a system of orthogonal polynomials with respect to F.
Suppose there exists a polynomial g satisfying the following conditions:
(i) For each n = 0, 1,...,
1 d n n
f n (x) ≡ [g(x) p(x)]
p(x) dx n
is a polynomial of degree n.
(ii) For each n = 1, 2,... and each j = 0, 1,..., n − 1,
d m n j d m n
j
lim x [g(x) p(x)] = lim x [g(x) p(x)] = 0, m = 1, 2,..., n − 1.
x→b dx m x→a dx m
Then there exist constants c 0 , c 1 ,... such that f n = c n p n ,n = 0, 1,....

Proof. Fix n.By Theorem 10.1, it suffices to show that, for each j = 0, 1,..., n − 1,
b
j
x f n (x) dF(x) = 0.
a
Using integration-by-parts,
b b n
j j d n
x f n (x) dF(x) ≡ x n [g(x) p(x)] dx
a a dx
d n−1 n b b j−1 d n−1 n
j
= x [g(x) p(x)] − j x [g(x) p(x)] dx

dx n−1 a a dx n−1
b d n−1
n
=− j x j−1 n−1 [g(x) p(x)] dx.
a dx
Continuing in this way,
b d n d n− j b
j
n
n
x j n [g(x) p(x)] dx = (−1) j! n− j [g(x) p(x)] = 0.

a dx dx a
Since 0 ≤ n − j ≤ n − 1, the result follows.
Hence, when the conditions of Theorem 10.3 hold, we can take the orthogonal polyno-
mials to be
1 d n n
p n (x) = [g(x) p(x)] (10.4)
c n p(x) dx n
for some constants c 0 , c 1 ,.... This is known as Rodrigue’s formula.

Example 10.3 (Legendre polynomials). In order to determine Rodrigue’s formula for the
Legendre polynomials, it suffices to find a polynomial g such that
d n n
g(x)
dx n
is a polynomial of degree n and
d m n j d m n
j
lim x g(x) = lim x g(x) = 0
x→1 dx m x→−1 dx m
for j = 0, 1,..., n − 1 and m = 1, 2,..., n − 1.