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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
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