Page 313 - Determinants and Their Applications in Mathematical Physics

298 6. Applications of Determinants in Mathematical Physics

where
m

E m = ε k r ,
r=1
m

K = (k r − 1). (6.10.39)
r=1
Applying Theorem (c) in Section 4.1.8 on Vandermondian identities,
m−1
V (c k m+1 ,c k m+2 ,...,c k 2n ){V 2n (c)}
. (6.10.40)
K
)
Y m =(−1) E m
V (c k 1 ,c k 2 ,...,c k m
Hence,
m−1
(−1) {V 2n (c)}
K
W m =
ρ m(m−1)
1≤k 1 <k 2 <...<k m ≤2n
). (6.10.41)
·E m V (c k 1 ,c k 2 ,...,c k m )V (c k m+1 ,c k m+2 ,...,c k 2n
(m)
Using the Laplace formula (Section 3.3) to expand H (ε) by the first
2n
m rows and the remaining (2n − m) rows and referring to the exercise at
the end of Section 4.1.8,
(m)
H (ε)= N 12···m;k 1 ,k 2 ,...,k m A 12···m;k 1 ,k 2 ,...,k m ,
2n
1≤k 1 <k 2 <···<k m ≤2n
(6.10.42)
where
),
N 12···m;k 1 ,k 2 ,...,k m = E m V (c k 1 ,c k 2 ,...,c k m
R
A 12···m;k 1 ,k 2 ,...,k m =(−1) M 12···m;k 1 ,k 2 ,...,k m
R
=(−1) V (c k m+1 ,c k m+2 ,...,c k 2n ), (6.10.43)
where M is the unsigned minor associated with the cofactor A and R is
the sum of their parameters. Referring to (6.10.39),
m

1
R = m(m +1)+
2 k r
r=1
1
= K + m(m − 1). (6.10.44)
2
Hence,
(m)
H (ε)=(−1) R E m V (c k 1 ,c k 2 ,...,c k m )
2n
1≤k 1 <k 2 <···<k m ≤2n
)
·V (c k m+1 ,c k m+2 ,...,c k 2n
2 m(m−1)/2
(−ρ )
= W m , (6.10.45)
{V 2n (c)} m−1
which proves part (a) of the theorem. Part (b) can be proved in a similar
manner.