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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.