Page 311 - Determinants and Their Applications in Mathematical Physics

296 6. Applications of Determinants in Mathematical Physics

where τ j is a function which appears in the Neugebauer solution and is
defined in (6.2.20).

(−1) j−1 M j (c)x r
2n
w r = j . (6.10.27)
ε ∗
j=1 j
Then,
x i − x j = c i − c j , independent of z,
ρ
2
ε j ε =1 + x . (6.10.28)
∗
j j
(m)
Now, let H (ε) denote the determinant of order 2n whose column vectors
2n
are defined as follows:
(m) 2 m−1 2
C (ε)= ε j c j ε j c ε j ··· c ε j 1 c j c ··· c 2n−m−1 T ,
j j
j j j
2n
1 ≤ j ≤ 2n. (6.10.29)
Hence,

m−1

2 T
(m) 1 1 c j c j c j 2 2n−m−1
C = ··· 1 c j c ··· c (6.10.30)
ε ε j ε j ε j ε j
j j j
2n
1 2 m−1 2 2n−m−1 T
= 1 c j c ··· c ε j c j ε j c ε j ··· c .
j j ε j
j j
2n
ε j
But,
(2n−m) 2 2n−m−1 2
C (ε)= ε j c j ε j c ε j ··· c ε j 1 c j c ··· c m−1 T . (6.10.31)
j j
j j j
2n
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),
  −1

2n
(m) 1 m(2n−1) (2n−m)
H =(−1)   H (ε),
ε
2n ε j 2n
j=1
(n+1) (n−1)
H 1/ε H (ε)
2n = − 2n . (6.10.32)
(n) (n)
H 1/ε H (ε)
2n 2n
Theorem.
2 −m(m−1)/2
a. |w i+j−2 + w i+j | m =(−ρ ) {V 2n (c)} m−1 H (m) (ε),
1 2n
2 −m(m−1)/2
b. |w i+j−2 | m =(−ρ ) {V 2n (c)} m−1 H (m) .
ε
2n ∗
The determinants on the left are Hankelians.