Page 103 - Determinants and Their Applications in Mathematical Physics

88 4. Particular Determinants

which is centrosymmetric and can therefore be expressed as the prod-
uct of two determinants of lower order. T n is also persymmetric about
its secondary diagonal.
Let A n , B n , and E n denote Hankel matrices defined as follows:

,
A n = t i+j−2
n

,
B n = t i+j−1
n

. (4.5.8)
E n = t i+j
n
Then, the factors of T n can be expressed as follows:
1
T 2n−1 = |T n−1 − E n−1 ||T n + A n |,
2
T 2n = |T n + B n ||T n − B n |. (4.5.9)
Let
1
1
P n = |T n − E n | = |t |i−j| − t i+j | n ,
2 2
1
1
Q n = |T n + A n | = |t |i−j| + t i+j−2 | n ,
2 2
1
1
R n = |T n + B n | = |t |i−j| + t i+j−1 | n ,
2 2
1
1
S n = |T n − B n | = |t |i−j| − t i+j−1 | n , (4.5.10)
2 2
U n = R n + S n ,
V n = R n − S n . (4.5.11)
Then,
T 2n−1 =2P n−1 Q n ,
T 2n =4R n S n
2
2
= U − V . (4.5.12)
n n
Theorem.
a. T 2n−1 = U n−1 U n − V n−1 V n ,
b. T 2n = P n Q n + P n−1 Q n+1 .
Proof. Applying the Jacobi identity (Section 3.6),
(n) (n)
T T
11
(n)
1n = T n T .
(n)
T n1 T nn
(n) 1n,1n
But
(n) (n)
T = T = T n−1 ,
11 nn
(n) (n)
T = T ,
n1 1n
(n)
T = T n−2 .
1n,1n
Hence,

T 2 = T n T n−2 + T (n) 2 . (4.5.13)
n−1 1n