Determinants and Their Applications in Mathematical Physics

296 6. Applications of Determinants in Mathematical Physics

whereτj is a function which appears in the Neugebauer solution and is

defined in (6.2.20).

wr=

2 n

∑

j=1

(−1)

j− 1

Mj(c)x

r

j

ε

∗

j

. (6.10.27)

Then,

xi−xj=

ci−cj

ρ

, independent ofz,

εjε

∗

j=1+x

2

j. (6.10.28)

Now, letH

(m)

2 n

(ε) denote the determinant of order 2nwhose column vectors

are defined as follows:

C

(m)

j

(ε)=

[

εj cjεj c

2

jεj···c

m− 1

j εj^1 cj c

2

j···c

2 n−m− 1

j

]T

2 n

,

1 ≤j≤ 2 n. (6.10.29)

Hence,

C

(m)

j

(

1

ε

)

=

[

1

εj

cj

εj

c

2

j

εj

···

c

m− 1

j

εj

1 cj c

2

j

···c

2 n−m− 1

j

]T

2 n

(6.10.30)

=

1

εj

[

1 cj c

2

j···c

m− 1

j εj cjεj c

2

jεj···c

2 n−m− 1

j εj

]T

2 n

.

But,

C

(2n−m)

j (ε)=

[

εj cjεj c

2

jεj···c

2 n−m− 1

j εj^1 cjc

2

j···c

m− 1

j

]T

2 n

. (6.10.31)

The elements in the last column vector are a cyclic permutation of the

elements in the previous column vector. Hence, applying Property (c(i))

in Section 2.3.1 on the cyclic permutation of columns (or rows, as in this

case),

H

(m)

2 n

(

1

ε

)

=(−1)

m(2n−1)





2 n

∏

j=1

εj





− 1

H

(2n−m)

2 n

(ε),

H

(n+1)

2 n

(

1 /ε

)

H

(n)

2 n

(

1 /ε

) =−

H

(n−1)

2 n

(ε)

H

(n)

2 n

(ε)

. (6.10.32)

Theorem.

a.|wi+j− 2 +wi+j|m=(−ρ

2

)

−m(m−1)/ 2

{V 2 n(c)}

m− 1

H

(m)

2 n

(ε),

b.|wi+j− 2 |m=(−ρ

2

)

−m(m−1)/ 2

{V 2 n(c)}

m− 1

H

(m)

2 n

(

1

ε

∗

)

.

The determinants on the left are Hankelians.