Page 161 - Determinants and Their Applications in Mathematical Physics

146 4. Particular Determinants

Proof. Apply the Jacobi identity (Section 3.6.1) to A r , where r ≥ i +1:
(r)
A A (r) (r)

ij ir = A r A ,
(r) (r) ir,jr
A

rj A rr
(r−1)
= A r A ,
ij
(r) (r−1) (r) (r)
A r−1 A − A r A = A A . (4.11.47)
ij ij ir jr
Scale the cofactors and refer to Theorems 4.48 and 4.49a:

A − A ij = A r A ri A rj
ij
r−1
r r r
A r−1
=2 −(4r−5) A A rj
ri
r r
=2λ r−1,i−1 λ r−1,j−1 . (4.11.48)
Hence,
n n

2 λ r−1,i−1 λ r−1,j−1 = A − A ij
ij
r−1
r
r=i+1 r=i+1
= A − A ij
ij
n
i
= A − 2 2(i−1) λ i−1,j−1 , (4.11.49)
ij
n
which yields a formula for the scaled cofactor A . The stated formula for
ij
n
(n)
the simple cofactor A follows from Theorem 4.49a.
ij
Let
E n = |P m (0)| n , 0 ≤ m ≤ 2n − 2, (4.11.50)
where P m (x) is the Legendre polynomial [Appendix A.5]. Then,
P 2m+1 (0)=0,
P 2m (0) = ν m . (4.11.51)
Hence,

ν 0 • ν 1 • ν 2 ···

• ν 1 • ν 2 • ···

ν 1 • ν 2 • ν 3 ···
. (4.11.52)
E n =
• ν 2 • ν 3 • ···

ν 2 • ν 3 • ν 4 ···
........................

n
Theorem 4.51.
2
n(n−1)/2 −(n−1)
E n = |P m (0)| n =(−1) 2 .