88 4. Particular Determinants
which is centrosymmetric and can therefore be expressed as the prod-
uct of two determinants of lower order. Tn is also persymmetric about
its secondary diagonal.
LetAn,Bn, andEndenote Hankel matrices defined as follows:
An=
[
ti+j− 2
]
n
,
Bn=
[
ti+j− 1
]
n
,
En=
[
ti+j
]
n
. (4.5.8)
Then, the factors ofTncan be expressed as follows:
T 2 n− 1 =
1
2
|Tn− 1 −En− 1 ||Tn+An|,
T 2 n=|Tn+Bn||Tn−Bn|. (4.5.9)
Let
Pn=
1
2
|Tn−En|=
1
2
|t|i−j|−ti+j|n,
Qn=
1
2
|Tn+An|=
1
2
|t|i−j|+ti+j− 2 |n,
Rn=
1
2
|Tn+Bn|=
1
2
|t|i−j|+ti+j− 1 |n,
Sn=
1
2
|Tn−Bn|=
1
2
|t|i−j|−ti+j− 1 |n, (4.5.10)
Un=Rn+Sn,
Vn=Rn−Sn. (4.5.11)
Then,
T 2 n− 1 =2Pn− 1 Qn,
T 2 n=4RnSn
=U
2
n
−V
2
n
. (4.5.12)
Theorem.
a.T 2 n− 1 =Un− 1 Un−Vn− 1 Vn,
b.T 2 n=PnQn+Pn− 1 Qn+1.
Proof. Applying the Jacobi identity (Section 3.6),
∣
∣
∣
∣
T
(n)
11
T
(n)
1 n
T
(n)
n 1
T
(n)
nn
∣
∣
∣
∣
=TnT
(n)
1 n, 1 n
.
But
T
(n)
11
=T
(n)
nn
=Tn− 1 ,
T
(n)
n 1
=T
(n)
1 n
,
T
(n)
1 n, 1 n
=Tn− 2.
Hence,
T
2
n− 1
=TnTn− 2 +
(
T
(n)
1 n