Determinants and Their Applications in Mathematical Physics

154 4. Particular Determinants

The theorem follows from (4.12.2) and (4.12.3) after replacing xby

x

− 1

.

In the next theorem,Pm(x) is the Legendre polynomial.

Theorem 4.55.

|Pm(x)|n=2

−(n−1)

2

(x

2

−1)

n(n−1)/ 2

.

0 ≤m≤ 2 n− 2

First Proof.Let

φm(x)=(1−x

2

)

−m/ 2

Pm(x).

Then

φ

′

m(x)=mF φm−^1 (x)

where

F=(1−x

2

)

− 3 / 2

φ 0 =P 0 (x)=1. (4.12.4)

Hence, ifA=|φm(x)|n, thenA

′

= 0 andA=|φm(0)|n.

|Pm(x)|n=

∣

∣(1−x^2 )m/^2 φ

m(x)

∣

∣

n

, 0 ≤m≤ 2 n− 2

=(1−x

2

)

n(n−1)/ 2

|φm(x)|n

=(1−x

2

)

n(n−1)/ 2

|φm(0)|n

=(1−x

2

)

n(n−1)/ 2

|Pm(0)|n.

The formula

|Pm(0)|n=(−1)

n(n−1)/ 2

2

−(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 polynomialPn(x)is equal to the coefficient

oft
n
in the polynomial expansion of[(u+t)(v+t)]
n
, whereu=

1

2

(x+1)

andv=

1

2

(x−1).

Proof. Applying the Rodrigues formula forPn(x) and the Cauchy

integral formula for thenth derivative of a function,

Pn(x)=

1

2

n

n!

D

n

(x

2

−1)

n