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5.8 Some Applications of Algebraic Computing 231
5.8.4 Hankel Determinants with Symmetric Toeplitz Elements
The symmetric Toeplitz determinant T n (Section 4.5.2) is defined as follows:
T n = |t |i−j| | n ,
with
T 0 =1. (5.8.13)
For example,
T 1 = t 0 ,
2
2
T 2 = t − t ,
0 1
3
2
2
2
T 3 = t − 2t 0 t − t 0 t +2t t 2 , (5.8.14)
0 1 2 1
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 of T n with a common principal diagonal.
T 0 2
T 1 = −|t 1 | ,
T 1 T 2
2
T 1 1
t
T 2 = − t 0 ,
T 2 T 3 t 2 t 1
2
1 t 1
t t 0
T 2
= − t 2 t 1
T 3 . (5.8.15)
T 3 T 4
t 0
t 3 t 2 t 1
Conjecture.
2
t 1 t 0 t 1 t 2 ··· t n−2
t 2 t 1 t 0 t 1 ··· t n−3
T n−1 t 3 t 2 t 1 t 0 ··· t n−4
.
T n
T n+1 t 4 t 3 t 2 t 1 ··· t n−5
= −
T n
··· ··· ··· ··· ··· ···
t n−1 t n−2 t n−3 ··· t 1
t n
n
Other relations of a similar nature include the following:
0 t 2
t t 1
T 0
= t 1 t 0
T 1 ,
T 2 T 3
t 1
t 2 t 1
t
T 1 T 2 0 t 1
T 3 t 2
has a factor t . (5.8.16)
T 2 T 3 1 t 2
T 4 t 3
T 3 T 4 T 5 t 2 t 3 t 4