Page 169 - Determinants and Their Applications in Mathematical Physics

154 4. Particular Determinants

The theorem follows from (4.12.2) and (4.12.3) after replacing x by
x −1 .

In the next theorem, P m (x) is the Legendre polynomial.
Theorem 4.55.
2 2 n(n−1)/2
−(n−1)
|P m (x)| n =2 (x − 1) .
0≤m≤2n−2
First Proof. Let
2 −m/2
φ m (x)=(1 − x ) P m (x).
Then

φ (x)= mFφ m−1 (x)
m
where
2 −3/2
F =(1 − x )
φ 0 = P 0 (x)=1. (4.12.4)

Hence, if A = |φ m (x)| n , then A = 0 and A = |φ m (0)| n .
2 m/2
|P m (x)| n = (1 − x ) φ m (x) , 0 ≤ m ≤ 2n − 2

n
2 n(n−1)/2
=(1 − x ) |φ m (x)| n
2 n(n−1)/2
=(1 − x ) |φ m (0)| n
2 n(n−1)/2
=(1 − x ) |P m (0)| n .
The formula
2
n(n−1)/2 −(n−1)
|P m (0)| n =(−1) 2
is proved in Theorem 4.50 in Section 4.11.3 on determinants with binomial
and factorial elements. The theorem follows.
Other functions which contain orthogonal polynomials and which satisfy
the Appell equation are given by Carlson.
The second proof, which is a modified and detailed version of a proof
outlined by Burchnall with an acknowledgement to Chaundy, is preceded
by two lemmas.

Lemma 4.56. The Legendre polynomial P n (x) is equal to the coefficient
1
of t in the polynomial expansion of [(u + t)(v + t)] , where u = (x +1)
n
n
2
1
and v = (x − 1).
2
Proof. Applying the Rodrigues formula for P n (x) and the Cauchy
integral formula for the nth derivative of a function,
1 2
P n (x)= D (x − 1) n
n
2 n!
n