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. 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

) 2

. (4.5.13)