Determinants and Their Applications in Mathematical Physics

4.14 Casoratians — A Brief Note 169

Qn=

1

2

|qij|n, (4.13.19)

where

pij=t|i−j|−ti+j,

qij=t|i−j|+ti+j− 2 , (4.13.20)

appear in Section 4.5.2 as factors of a symmetric Toeplitz determinant.

Put

tr=ω

r

ur, (ω

2

=−1).

Then,

pij=ω

i+j− 2

α

′

ij,

qij=ω

i+j− 2

αij, (4.13.21)

whereα

′

ij

andαijare defined in (4.13.15) and (4.13.2), respectively. Hence,

referring to the corollaries in Theorems 4.60 and 4.61,

Pn=

1

2

∣

∣

ω

i+j− 2

α

′

ij

∣

∣

n

=

1

2

ω

n(n−1)

|α

′

ij

|n

=(−1)

n(n−1)/ 2

2

n

2

− 1

|ki+j+ki+j− 2 |n. (4.13.22)

Qn=

1

2

|ω

i+j− 2

αij|n

=(−1)

n(n−1)/ 2

2

(n−1)

2

|ki+j− 2 |n. (4.13.23)

SincePnandQneach have a factorω

n(n−1)

andn(n−1) is even for all

values ofn, these formulas remain valid whenωis replaced by (−ω) and

are applied in Section 6.10.5 on the Einstein and Ernst equations.

4.14 Casoratians — A Brief Note

The CasoratianKn(x), which arises in the theory of difference equations,

is defined as follows:

Kn(x)=|fi(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)

.....................................

fn(x) fn(x+1) ··· fn(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.