Determinants and Their Applications in Mathematical Physics

5.8 Some Applications of Algebraic Computing 231

5.8.4 Hankel Determinants with Symmetric Toeplitz Elements

The symmetric Toeplitz determinantTn(Section 4.5.2) is defined as follows:

Tn=|t|i−j||n,

with

T 0 =1. (5.8.13)

For example,

T 1 =t 0 ,

T 2 =t

2

0

−t

2

1

,

T 3 =t

3

0 −^2 t 0 t

2

1 −t 0 t

2

2 +2t

2

1 t 2 , (5.8.14)

etc. In each of the following three identities, the determinant on the left

is a Hankelian with symmetric Toeplitz elements, but when the rows

or columns are interchanged they can also be regarded as second-order

subdeterminants of|T|i−j||n, which is a symmetric Toeplitz determinant

with symmetric Toeplitz elements. The determinants on the right are

subdeterminants ofTnwith a common principal diagonal.

∣

∣

∣

∣

T 0 T 1

T 1 T 2

∣

∣

∣

∣

=−|t 1 |

2

,

∣

∣

∣

∣

T 1 T 2

T 2 T 3

∣

∣

∣

∣

=−

∣

∣

∣

∣

t 1 t 0

t 2 t 1

∣

∣

∣

∣

2 , ∣ ∣ ∣ ∣

T 2 T 3

T 3 T 4

∣

∣

∣

∣

=−

∣ ∣ ∣ ∣ ∣ ∣

t 1 t 0 t 1

t 2 t 1 t 0

t 3 t 2 t 1

∣ ∣ ∣ ∣ ∣ ∣

2

. (5.8.15)

Conjecture.

∣

∣

∣

∣

Tn− 1 Tn

Tn Tn+1

∣

∣

∣

∣

=−

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

t 1 t 0 t 1 t 2 ··· tn− 2

t 2 t 1 t 0 t 1 ··· tn− 3

t 3 t 2 t 1 t 0 ··· tn− 4

t 4 t 3 t 2 t 1 ··· tn− 5

··· ··· ··· ··· ··· ···

tn tn− 1 tn− 2 tn− 3 ··· t 1

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

2

n

.

Other relations of a similar nature include the following:

∣

∣

∣

∣

T 0 T 1

T 2 T 3

∣

∣

∣

∣

=

∣ ∣ ∣ ∣ ∣ ∣

t 0 t 1 t 2

t 1 t 0 t 1

t 2 t 1

∣ ∣ ∣ ∣ ∣ ∣

, ∣ ∣ ∣ ∣ ∣ ∣

T 1 T 2 T 3

T 2 T 3 T 4

T 3 T 4 T 5

∣ ∣ ∣ ∣ ∣ ∣

has a factor

∣ ∣ ∣ ∣ ∣ ∣

t 0 t 1 t 2

t 1 t 2 t 3

t 2 t 3 t 4

∣ ∣ ∣ ∣ ∣ ∣

. (5.8.16)