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4.14 Casoratians — A Brief Note  169

                                         1
                                    Q n = |q ij | n ,              (4.13.19)
                                         2
          where
                                 p ij = t |i−j| − t i+j ,
                                 q ij = t |i−j| + t i+j−2 ,        (4.13.20)
          appear in Section 4.5.2 as factors of a symmetric Toeplitz determinant.
            Put
                                             2
                               t r = ω u r ,  (ω = −1).
                                     r
          Then,
                                              α ,
                                   p ij = ω i+j−2
                                               ij
                                   q ij = ω i+j−2 α ij ,           (4.13.21)
          where α and α ij are defined in (4.13.15) and (4.13.2), respectively. Hence,

                 ij
          referring to the corollaries in Theorems 4.60 and 4.61,

                                     α
                               ω
                        P n =  1   i+j−2
                              2        ij n
                              1
                           = ω  n(n−1)
                              2      |α | n
                                       ij
                                          2
                                 n(n−1)/2 n −1
                           =(−1)        2    |k i+j + k i+j−2 | n .  (4.13.22)
                              1
                        Q n = |ω i+j−2
                              2      α ij | n
                                              2
                                 n(n−1)/2 (n−1)
                           =(−1)        2      |k i+j−2 | n .      (4.13.23)
          Since P n and Q n each have a factor ω n(n−1)  and n(n − 1) is even for all
          values of n, these formulas remain valid when ω is replaced by (−ω) and
          are applied in Section 6.10.5 on the Einstein and Ernst equations.
          4.14 Casoratians — A Brief Note
          The Casoratian K n (x), which arises in the theory of difference equations,
          is defined as follows:

                   K n (x)= |f i (x + j − 1)| n
                             f 1 (x)  f 1 (x +1)  ···  f 1 (x + n − 1)

                             f 2 (x)  f 2 (x +1)  ···  f 2 (x + n − 1)

                                                                .
                         =
                             .....................................
                            f n (x) f n (x +1) ··· f n (x + n − 1)

                                                               n
          The role played by Casoratians in the theory of difference equations is
          similar to the role played by Wronskians in the theory of differential equa-
          tions. Examples of their applications are given by Milne-Thomson, Brand,
          and Browne and Nillsen. Some applications of Casoratians in mathematical
          physics are given by Hirota, Kajiwara et al., Liu, Ohta et al., and Yuasa.
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